Geometry, Analysis & Gravitation

Schur multipliers were introduced by Schur in early 20th century and have since
then found a considerable number of applications in Analysis and enjoyed an intensive
development. Apart from the beauty of the subject itself, sources of interest in them
were connections with Perturbation Theory, Harmonic Analysis, the Theory of Opera-
tor Integrals and other. Schur multipliers have a simple definition: a bounded function
φ : N x N -> C (where N and C are the set of positive integers and complex numbers
respectively) is called a Schur multiplier if whenever a matrix (aij) gives rise to
a (bounded) transformation Sφ of the space $l_2$, the matrix (φ(i, j)aij) does so as
well. A characterisation of Schur multipliers was given by Grothendieck in his Resume.
If instead of $l_2$ we consider a pair of Hilbert spaces H1 = L2(X, μ), H2 = L2(Y, ν)
then there is also a method (due mainly to Birman and Solomyak) to relate to some
bounded functions φ on X x Y linear transformations Sφ on the space B(H1,H2) (these
transformations are called masurable Schur multipliers or, in a more general setting of
spectral measures μ, ν, double operator integrals). Namely one defines firstly a map Sφ
on Hilbert Schmidt operators multiplying their integral kernels by φ; if this map turns
out to be bounded in operator norm, extend it to the space K(H1,H2) of all compact
operators by continuity. Then Sφ is defined on B(H1,H2) as the second adjoint of
the constructed map of K(H1,H2). A characterisation of all such multipliers was first
established by Peller: Schur multipliers are precisely the functions of the form

φ(x, y) =Σ a_k(x)b_k(y)

such that (esssup Σ|a_k(x)|^2)(esssup Σ|b_k(x)|^2) < \infty.

We shall discuss results on Schur multipliers and the question for which φ the map
Sφ is closable in the operator norm or in the weak* topology of B(H1,H2). If φ is of
Toeplitz type, i.e. φ(x, y) = f(x - y) ( x, y in G), where G is a locally compact abelian
group then the question is related to certain questions about the Fourier algebra A(G);
if φ(x, y) is of the form (f(x)-f(y))/(x-y) then the property is related to "operator
smoothness" of f. This is a joint work with V.Shulman and I.Todorov.
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Abstract: In this talk I will start by revisiting the calculation of entanglement entropy in free Maxwell theory in 3+1 dimensional Minkowski spacetime. I will characterize the soft sector associated with a subregion and demonstrate that conformally soft mode configurations at the entangling surface, or equivalently correlated fluctuations in the large gauge charges of the subregion and its complement, give a non-trivial contribution to the entanglement entropy across a cut of future null infinity. I will conclude with some comments on the holographic description of bulk subregions in asymptotically flat spacetimes.

Abstract: We combine supersymmetric localization with the numerical conformal bootstrap to bound the scaling dimension and OPE coefficient of the lowest-dimension operator in N = 4 SU( N) super-Yang-Mills theory for a wide range of N and Yang-Mills couplings g. We find that our bounds are approximately saturated by weak coupling results at small g. Furthermore, at large N our bounds interpolate between integrability results for the Konishi operator at small g and strong-coupling results, including the first few stringy corrections, for the lowest-dimension double-trace operator at large g. In particular, our scaling dimension bounds describe the level splitting between the single- and double-trace operators at intermediate coupling.

Abstract: Dark matter is a century-long unsolved mystery. Plenty of observational evidence points to its existence through gravitational interactions with ordinary matter, and numerous theories have been proposed to explore its compositions, but its nature remains unknown. Gravitational-wave detectors such as LIGO offer a direct probe for dark matter, leveraging their exceptional sensitivities to spacetime distortions. This work tested a specific dark matter candidate, macroscopic dark matter domain walls arisen from a scalar field theory. Mining 2 years of observed data, we find no evidence for dark matter domain walls. Nevertheless, we advanced our understanding of the candidate dark matter theory by several orders of magnitude. 

Abstract:
During the first half of the fourth LIGO-Virgo-KAGRA observing run, we observed gravitational waves from merging binaries roughly every three days. While this routine detection promises exciting results, it is becoming a significant challenge to analyze all events using our most sophisticated theoretical models. In this talk, I will describe how to overcome these challenges using deep learning techniques for rapid, amortized Bayesian inference. This approach uses simulated data to train neural networks (such as normalizing flows) to represent the Bayesian posterior. Once trained, sampling becomes extremely fast. I will also describe how to establish full confidence in results using importance sampling, as well as initial results on population inference and future prospects to treat realistic noise.

Abstract: In this seminar, we shall discuss some recent results on the gap theorem of nonnegatively curved manifolds with small curvature in an average integral sense, which can be viewed as a Riemannian analog of the optimal gap result by Ni on Kahler manifolds. In dimension 3, we also establish a gap theorem for Ricci nonnegative manifolds with pointwise quadratic curvature decay and fast average integral curvature decay. This talk is based on some joint works with Man-Chun Lee. 

Abstract: The evolution of a two dimensional incompressible ideal fluid with smooth initial vorticity concentrated in small regions is well understood on finite intervals of time: it converges to a super position of Dirac deltas centered at collision-less solutions to the point vortex system, in the limit of vanishing regions. Even though  for generic initial conditions the vortex point system has a global smooth solution, much less is known on the long time behavior of the fluid vorticity.
We consider the case of two vortex pairs traveling in opposite directions. Using gluing methods we describe the global dynamics of this configuration. This work is in collaboration with J Davila (U. of Bath), M. del Pino (U of Bath) and S. Parmeshwar (Warwick University)

Abstract: Rigorously unravelling the intricate mathematical structure of space and time near the big bang singularity — the origin of the universe — has been an elusive task until recently. Using new Fuchsian partial differential equation techniques, we have proven a nonlinear stability result for the celebrated standard cosmological model — specifically for solutions of the fully coupled Einstein-matter equations. This not only rigorously validates some of the simplified assumptions of the standard model in certain regimes but also highlights new critical phenomena that are currently not well-understood. This is joint work with Todd Oliynyk from Monash.

Abstract:
The evolution of a network of interfaces by mean curvature flow features the
occurrence of topology changes and geometric singularities. As a
consequence, classical solution concepts for mean curvature flow are in
general limited to a finite time horizon. At the same time, the evolution
beyond topology changes can be described only in the framework of weak
solution concepts (e.g., Brakke solutions), whose uniqueness may fail.
Following the relative energy approach, we prove a quantitative stability
estimate holding up to the singular time at which a circular closed curve
shrinks to a point. This implies a weak-strong uniqueness principle for weak
BV solutions to planar multiphase mean curvature flow beyond circular
topology changes. We expect our method to have further applications to other
types of shrinkers.

This talk is based on a joint work with Julian Fischer, Sebastian Hensel and
Maximilian Moser

I will discuss the physics of high energy ("many-particle”) states from two complementary perspectives. First, I will present a new method for using data from conformal field theories to compute observables in more general QFTs, which can be used to numerically study many properties of many-particle states. Then I will consider an analytic approach to a particular set of many-particle states, those near threshold, where many features become largely theory-independent.

Abstract: In this talk I will discuss some recent results on the null and approximate boundary controllability properties of heat-like equations featuring a singular potential that diverges as the inverse square of the distance to the boundary of a domain in $\mathbb{R}^n$. For this purpose, I will establish two families of $H^1$-Carleman estimates for the associated operators involving weights with non-smooth powers of the boundary distance: a global estimate on convex domains, suitable for controls prescribed along the entire boundary, which yields a boundary null control result, and a local estimate that allows for a control localized to arbitrarily small portions on the boundary, leading to (local) unique continuation and approximate controllability. Additionally, I will describe the role of the boundary conditions, the potential strength and the geometry of the domain in our results. This is based on joint work with A. Enciso (ICMAT) and A. Shao (QMUL).

In this talk, I will show how one can construct counterexamples to unique continuation for critically singular wave operators fails from a timelike or null hypersurface, based on a paper written in collaboration with Arick Shao. In the first part of the talk, I will first emphasize the consequences of such a result on asymptotically Anti-de Sitter spacetimes and how these may yield potential mechanisms for counterexamples to the AdS/CFT correspondence. In the second part, I will briefly show the main steps of the construction, based on the classical result of Alinhac and Baouendi.

In the pre-talk, I will give a brief introduction to unique continuation problems and how these show up in Anti-de Sitter spacetimes.

Abstract: The study of optimal upper bounds for Laplace eigenvalues on closed surfaces under area constraint is a classical problem of spectral geometry. It is particularly interesting due to the fact that optimal metrics (if exist) correspond to branched minimal surface in n-dimensional sphere. In general, determining the geometric properties of these surfaces, such as embeddedness or the value of n are very challenging problems, where very few results are known. In the present talk we will discuss how one can sometimes resolve these issues, leading to new constructions of embedded minimal surfaces in the 3-sphere. Similar results will be outlined for Steklov eigenvalues that correspond to free boundary minimal immersions in the unit 3-ball.
Based on a joint work with R. Kusner, P. McGrath and D. Stern.

Abstract: Even for a mean curvature flow with uniformly bounded mean curvature, singularities may occur. For flows with these additional bounds, we can improve our understanding of singularities by incorporating the theory of varifolds with bounded mean curvature. In particular, tangent flows are necessarily static flows of minimal cones, and the tangent flow is unique if the cone has smooth link. In certain cases, we characterize the smooth minimal surfaces that pinch off at smaller scales around a singularity. We'll also discuss generalizations of these results to Brakke flows with high co-dimension and integral mean curvature bounds.

Abstract:

Decoherence and thermalisation of isolated many-particle quantum states are studied in many different subfields of physics, including high-energy physics. One of the most interesting case are Heavy Ion Collisions which can be holographically connected to string theory in Anti-de Sitter space and for which very detailed data exists. After a general introduction I will focus on the question whether SU(N) gauge theories behave as predicted by the Eigenstate Thermalization Hypothesis (ETH). To answer this question we have performed simulations for low-dimensional SU(2) gauge theories on digital computers (arXiv: 2308.16202) which gave encouraging results. As ETH makes predictions for energy eigenstates the most natural theoretical approach to study e.g. thermalization of QCD is the numnerical simulation of Hamiltonian lattice QCD on quantum computers which, however, is not yet possible. Investigating the validity of ETH on digital computers is an early step in this direction.

Abstract: We will discuss how amplitudes can be used to efficiently derive classical
gravitational-wave observables characterizing black hole binary encounters.
This technique is very flexible and can be applied to General Relativity, but
also to its extensions and, in the spirit of Effective Field Theory, can be used
to describe compact objects beyond Schwarzschild black holes. We will briefly
discuss some recent applications to spinning black holes and to the subleading
Post-Minkowsian waveforms.

Abstract: The linearisation of a second-order formulation of the conformal Einstein field equations (CEFEs) in Generalised Harmonic Gauge (GHG), with trace-free matter is derived. The linearised equations are obtained for a general background and then particularised for the study linear perturbations around a flat background —the inversion (conformal) representation of the Minkowski spacetime— and the solutions discussed. We show that the generalised Lorenz gauge (defined as the linear analogue of the GHG-gauge) propagates. Moreover, the equation for the conformal factor can be trivialised with an appropriate choice for the gauge source functions; this permits a scri-fixing strategy using gauge source functions for the linearised wave-like CEFE-GHG, which can in principle be generalised to the nonlinear case. As a particular application of the linearised equations, the far-field and compact source approximation is employed to derive quadrupole-like formulae for various conformal fields such as the perturbation of the rescaled Weyl tensor.

Abstract. Optimal transport tools have been extremely powerful to study Ricci curvature, in particular Ricci lower bounds in the non-smooth setting of metric measure spaces (which can be been thought as "non-smooth Riemannian manifolds”).
Since the geometric framework of general relativity is the one of Lorentzian manifolds (or space-times), and the Ricci curvature plays a prominent role in Einstein’s theory of gravity, it is natural too expect that optimal transport tools can be useful also in this setting. 
The goal of the talk is to introduce the topic and to report on recent progress.
More precisely: After recalling the general setting of Lorentzian pre-length spaces (introduced by Kunzinger-Sämann, after Kronheimer-Penrose), I will discuss some basics of optimal transport theory thereof in order to define "timelike Ricci curvature and dimension bounds” for a possibly non-smooth Lorentzian space. Some cases of such bounds have remarkable physical interpretations (like the attractive nature of gravity) and can be used to give a characterisation of the Einstein’s equations for a non-smooth space and to establish new isoperimetric-type inequalities in Lorentzian signature. Based partly on joint work with S. Suhr and partly on joint work with F. Cavalletti.

We will give an intuitive explanation of why and how matrices (or, more precisely, large-N gauge theories) can describe a black hole, without assuming knowledge of quantum mechanics and holographic duality.

Firstly, we explain an intuitive picture inroduced by Witten: diagonal entries of matrices describe particles and off-diagonal entries describe strings connecting particles. When many strings are excited, a lot of energy and entropy are packed in a small region and form black hole. 

Next, we consider classical dynamics of matrix model. Specifically, we colide two black holes. Using the energy conservation, equipartition law of energy and elementary school math, we show that black hole becomes colder after the merger. Matrices know black hole's negative heat capacity!

To gain a little bit more intuition, we will look at ants. Collective behavior of ants has a striking similarity to black hole. The mapping rule is ant -> particle, pheromone -> string, and ant trail -> black hole. Tuning parameters such as temperature or each ant's laziness, we can obtain three kinds of phase diagrams. Each of them has a counterpart in large-N gauge theories. 

If time permits, I will explain the mechanism applicable to strongly-coupled and highly quantum regime needed for quantitative agreement with Einstein gravity. (This part requires a good understanding of undergraduate-level quantum mechanics.)

We consider the renormalization group flow of a quantum field theory (QFT) in Anti-de Sitter (AdS) space. We derive sum rules that express UV data and the energy of a chosen eigenstate in terms of the spectral densities and certain correlation functions of the theory. In two dimensions, this leads to a bootstrap setup that involves the UV central charge and may allow us to follow a Renormalization Group (RG) flow non-perturbatively by continuously varying the AdS radius. Along the way, we establish the convergence properties of the newly discovered local block decomposition, which applies to three-point functions involving one bulk and two boundary operators.

Expanding Ricci solitons are central to our understanding of both the long-time behaviour of Ricci flow, and the short-time asymptotics of Ricci flow as it desingularises rough initial data. By utilising a one-to-one correspondence between complete Ricci flows and their weak initial data, we complete the classification of expanding Ricci solitons in two dimensions. This is joint work with Peter Topping.

Another recent application was the calculation of the Newman-Penrose Constants (NPC). These are five complex quantities defined on null infinity that are absolutely conserved if it is smooth. In stationary space-times, they can be written terms of mass and angular momentum moments, but their physical interpretation in non-stationary space-times is still lacking. We compute, for the first time, the NPC in a general setting and show that they remain constant, implying smoothness of null infinity to at least the level of our numerical precision.

Abstract: Consider a single photon living in curved space. It is described by Maxwell’s equations. We seek solutions harmonic in time. This reduces to the spectral problem for the operator curl, whose spectrum is, in general, asymmetric about zero (think particle/antiparticle).
Spectral asymmetry is a classical subject in analysis and geometry, whose ori- gins lie in the papers of Atiyah, Patodi and Singer. In this talk I will discuss a new approach to the study of spectral asymmetry based on the use of pseu- dodifferential techniques developed in a series of recent joint papers by Dmitri Vassiliev and myself.

We will discuss recent efforts to understand gravitational wave generation in Dark energy models. I will consider a class of alternative theories of gravity known as k-essence. This theory is a cosmologically relevant scalar-tensor theory that involves first-order derivative self-interactions, which pass all existing gravitational wave bounds and provides a screening mechanism. In this talk, I will present our numerical simulations of this theory considering three scenarios: non-linear stellar oscillations, gravitational collapse and binary neutron stars.

Abstract:
Extreme mass ratio inspirals (EMRIs) are expected to be a key source of gravitational waves for the LISA mission. In order to extract the maximum amount of information from EMRI observations by LISA, it is important to have an accurate prediction of the expected waveforms. In particular, it will be necessary to have waveforms that incorporate effects that appear at second order in the mass ratio. In this talk I will present the latest progress towards this goal, including recent results for the second-order gravitational-wave energy flux and for the gravitational waveform.

Vlasov-Poisson type systems are well known as kinetic models for plasma. The precise structure of the model differs according to which species of particle it describes, with the `classical’ version of the system describing the electrons in a plasma. The model for ions, however, includes an additional exponential nonlinearity in the equation for the electrostatic potential, which creates several mathematical difficulties. For this reason, the theory of the ionic system has so far not been as fully explored as the theory for the electron equation.
 
A plasma has a characteristic scale, the Debye length, which describes the scale of electrostatic interaction within the plasma. In real plasmas this length is typically very small, and in physics applications frequently assumed to be very close to zero. This motivates the study of the limiting behaviour of Vlasov-Poisson type systems as the Debye length tends to zero relative to the observation scale—known as the ‘quasi-neutral’ limit. In the case of the ionic model, the formal limit is the kinetic isothermal Euler system; however, this limit is highly non-trivial to justify rigorously and known to be false in general without very strong regularity conditions and/or structural conditions.
 
I will present a recent work, carried out in collaboration with Mikaela Iacobelli, in which we prove the quasi-neutral limit for the ionic Vlasov-Poisson system for a class of rough (L^\infty) data: that is, data that may be expressed as perturbations of an analytic function, small in the sense of Monge-Kantorovich distances. The smallness of the perturbation that we require is much less restrictive than in the previously known results.

Abstract: The mean curvature flow is an example of a geometric flow, where in this case one deforms a submanifold according to its mean curvature vector. Like many such flows though the mean curvature flow will develop singularities, where the flow “pinches.” The entropy, in the sense of Colding and Minicozzi, is an interesting area-like monotone quantity under the flow, for one because it can constrain what sorts of singularity models may arise, and has played an important role in many recent developments. In this talk, after introducing the relevant notions we’ll discuss some of these results, including some joint work with S. Wang.

Abstract:
The dynamics of a binary system moving in the background of a black hole is affected by tidal forces. In this talk, we derive the electric and magnetic tidal moments at quadrupole order induced by a Kerr black hole. We make use of these moments in the scenario of a hierarchical triple system made of a Kerr black hole and an extreme-mass ratio binary system consisting of a Schwarzschild black hole and a test particle. We study how the secular dynamics of the test particle in the binary system is distorted by the presence of tidal forces from a much larger Kerr black hole. We compute the shifts in the physical quantities for the secular dynamics of the test particle and show that they are gauge-invariant. In particular, we apply our formalism to the innermost stable circular orbit for the test particle and to the case of the photon sphere. 

Abstract: In this talk, I will give an overview of the stability problems for black hole solutions, starting with the mode stability results in black hole perturbation theory in the 80’s to more recent mathematical proofs, as the linear and the fully non-linear stability of black hole solutions require new mathematical techniques. Finally, I will present some aspects of our recent proof with Klainerman and Szeftel of the non-linear stability of the slowly rotating Kerr black hole. 

Abstract: A system of equations that serves as a model for the Einstein field equation in generalised harmonic gauge called the good-bad-ugly system is studied in the region close to null and spatial infinity in Minkowski spacetime. This analysis is performed using H. Friedrich's cylinder construction at spatial infinity and defining suitable conformally rescaled fields. The results are translated to the physical set up to investigate the relation between the polyhomogeneous expansions arising from the analysis of linear fields using the i0-cylinder framework and those obtained through a heuristic method based on Hörmander's asymptotic system.

Abstract: In this seminar I will review how the quantum path integral of 2 dimensional Jackiw-Teitelboim (JT) gravity (with arbitrary numbers of boundaries) is computed by a particular Hermitian matrix integral. This duality is perturbative, meaning the equivalence is demonstrated by matching the perturbation series order by order in both descriptions. Building on this, I will describe an altogether different class of matrix integrals, namely one involving unitary matrices, which is also dual to JT gravity when parameters are tuned in a particular way. This allows a non-perturbative view of the JT theory as a particular phase of a more general theory, tying into the structure of phase transitions more familiar in statistical physics.

Abstract:

We present recent results on time-scales separation in fluid
mechanics. The fundamental mechanism to detect in a precise
quantitative manner is commonly referred to as fluid mixing. Its
interaction with advection, diffusion and nonlocal effects produces a
variety of time-scales which explain many experimental and numerical
results related to hydrodynamic stability and turbulence theory.

In this talk, I will explain how uniformisation and conformal maps have been used and keep being used nowadays to obtain results in the spectral geometry of surfaces. Whether it is for shape optimisation or eigenvalue asymptotics, much can be said from the careful analysis of conformal factors. Along the way, this will also be a good opportunity to survey the field of two-dimensional spectral geometry.

Abstract: In this talk, I will present the family of potential wells to the initial boundary problem of semilinear wave equations. This gives the vacuum isolating solutions and threshold result of global existence and nonexistence of solutions.

Spontaneous scalarization is among the most interesting mechanisms to
endow compact objects with scalar hair while leaving the weak field
regime of gravity unaltered with respect to GR. In the present talk we
will discuss the dynamics of spontaneous scalarization of black holes
and neutron stars in extended scalar-tensor theories of gravity,
focusing primarily on some of the most interesting cases from an
astrophysical point of view. These include binary merger, stellar core
collapse, and gravitational phase transitions. The electromagnetic and
gravitational wave signatures, as well as the prospect for their
detectability, will be also discussed.

Abstract: In this talk, we model the gravitational collapse of stars in 
massive scalar-tensor gravity. In this theory, the two tensorial 
gravitational-wave polarization modes are complemented by a massive 
breathing mode. This latter mode is triggered by the spontaneous 
scalarization mechanism discovered by Damour and Esposito-Farese; its 
radiation exhibits a drastically different behaviour dominated by the 
dispersive character of the mass term which leads to quasi-monochromatic 
signals that can last years or even centuries. This smoking-gun effect 
offers unique opportunities to test this class of theories. We also 
briefly discuss the overlap of numerous such signals arising from multiple 
supernova events in the local universe and compare the resulting
gravitational-wave energy density with present constraints from LIGO-Virgo 
observations.

Abstract: In this talk I am going to analyse the compressible dissipative hydrodynamic model of crowd motion or of granular flow. The model resembles the famous Aw-Rascle model of traffic, except that the difference between the actual and the desired velocities (the offset function) is a gradient of the density function, and not a scalar. This modification gives rise to a dissipation term in the momentum equation that vanishes when the density is equal to zero.
I will compare the dissipative Aw-Rascle system with the compressible Euler and compressible Navier-Stokes equations, and back it up with two existence and ill-posedness results. In the last part of my talk I will explain the proof of conjecture made by Lefebvre-Lepot and Maury, that the hard congestion limit of this system (with singular offset function) leads to congested compressible/incompressible Euler equations.

Deviations from beta equilibrium impact on the fine-print details of the gravitational-wave signal from neutron-star binary inspiral and merger. During the inspiral phase, the individual neutron stars are cold enough that the Urca reactions cannot establish equilibrium on the inspiral timescale. This leads to the presence of composition g-modes which may become resonant as the system evolves through the sensitivity band of ground-based detectors. In contrast, during the hot post-merger phase the reactions are expected to be fast enough that they need to be accounted for (at least in parts of parameter space). In this talk I will tie together the two phases of the binary problem, outlining the role of composition effects during the inspiral as well as the emergence of an effective bulk viscosity after merger. Framing the discussion in the context of state-of-the-art merger simulations, I will suggest that the non-equilibrium physics impact on the gravitational-wave signal at the “few percent level”, likely undetectable with the LIGO instruments but plausibly within reach of third generation detectors like the Einstein Telescope and the Cosmic Explorer.  

ABSTRACT: The formation of Primordial black holes is naturally enhanced during the quark-hadron phase transition, because of the softening of the equation of state occurring during this epoch: at a scale between 1 and 3 solar masses, the threshold is reduced of about 10% with a corresponding abundance of primordial black holes significantly increased by more than 100 times. We show that a sub-population of primordial black black holes formed during the QCD epoch, in the solar mass range, is compatible with the current observational constraints, and could explain some of the interesting sources emitting gravitational waves detected by LIGO/VIRGO in the black hole mass gap, such as GW190814, and other light events

It has been shown that, for certain modified theories in which a scalar field is coupled with the Gauss-Bonnet curvature invariant, black hole spacetimes can undergo a tachyonic instability which culminates in a non-trivial stationary scalar configuration. This mechanism is known as spontaneous scalarization, and its onset is controlled by parameters such as the coupling strength, mass and spin of the black hole. Thus, dynamical binary systems allow for a variety of configurations that depend on these parameters. For example, two scalarized black holes can merge to form a larger remnant that has no scalar profile due to the weaker spacetime curvature near the horizon, a mechanism known as dynamical descalarization. In this talk, I will be discussing my work on modelling binary black hole systems in scalar-Gauss-Bonnet gravity to identify the possible configurations. Knowledge of these configurations can assist in the identification of the gravitational waveform modifications due to the scalar field and can provide estimates of the constraints on the theory.

Abstract: We will talk about an analogue of the Brakke's local regularity theorem for the $\epsilon$ parabolic Allen-Cahn equation. In particular, we show uniform $C_{2,\alpha}$ regularity for the transition layers converging to smooth mean curvature flows as $\epsilon$ tend to 0 under the almost unit-density assumption. This can be viewed as a diffused version of the Brakke regularity for the limit mean curvature flow. This talk is based on joint work with Huy Nguyen.

Abstract: In this talk, I will give an overview on "gravitational memory effects”. I will address the following questions: what are they? How can they be observed? and what are their implications for fundamental physics?  I will emphasise my contribution in this field.

abstract: I will present the derivation of the antipodal matching relations used to demonstrate the equivalence between soft graviton theorems and BMS charge conservation across spatial infinity. To this end I will provide a precise map between Bondi data at null infinity and Beig-Schmidt data at spatial infinity in a context appropriate to the gravitational scattering problem and celestial holography. I will also demonstrate that, among various proposals of BMS charges at null infinity found in the literature, only a subset match the conserved charges at spatial infinity and are therefore preferred from that perspective.

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Abstract: I will present the derivation of the antipodal matching relations used to demonstrate the equivalence between soft graviton theorems and BMS charge conservation across spatial infinity. To this end I will provide a precise map between Bondi data at null infinity and Beig-Schmidt data at spatial infinity in a context appropriate to the gravitational scattering problem and celestial holography. I will also demonstrate that, among various proposals of BMS charges at null infinity found in the literature, only a subset match the conserved charges at spatial infinity and are therefore preferred from that perspective.

Abstract: The exterior dynamics of black holes has played a major role
in holographic duality, describing the approach to thermal equilibrium
of strongly coupled media. The interior dynamics of black holes in a
holographic setting has, in contrast, been largely unexplored. I will
describe recent work investigating the classical interior dynamics of
various holographic black holes. I will discuss the nature of the
singularity, the absence of Cauchy horizons and a new kind of chaotic
behavior that emerges in the presence of charged scalar fields.

Abstract: The study of optimal upper bounds for Laplace eigenvalues on closed surfaces is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. Its most fascinating feature is the connection to the theory of minimal surfaces in spheres. Optimization of Steklov eigenvalues is an analogous problem on surfaces with boundary. It was popularised by A. Fraser and R. Schoen, who discovered its connection to the theory of free boundary surfaces in Euclidean balls. Despite many widely-known empiric parallels, an explicit link between the two problems was discovered only in the last two years. In the present talk, we will show how Laplace eigenvalues can be recovered as certain limits of Steklov eigenvalues and discuss the applications of this construction to the geometry of minimal surfaces. The talk is based on joint works with D. Stern.

I will review approaches to solving the perturbation equations in electromagnetism and linearized gravity on black hole spacetimes in the presence of sources, focussing particularly on recent work in the Lorenz (de Donder) gauge. I will present some partial results and review the challenges remaining. The overall aim is to extend the gravitational self-force programme to second order in the mass ratio, to accurately model Extreme Mass-Ratio Inspirals for LISA. 

Abstract: Non-shearing congruences of null geodesics on four-dimensional Lorentzian manifolds are fundamental objects of mathematical relativity. Their prominence in exact solutions to the Einstein field equations is supported by major results such as the Robinson, Goldberg-Sachs and Kerr theorems. Conceptually, they lie at the crossroad between Lorentzian conformal geometry and Cauchy-Riemann geometry, and are one of the original ingredients of twistor theory.
 
Identified as involutive totally null complex distributions of maximal rank, such congruences generalise to any even dimensions, under the name of Robinson structures. Nurowski and Trautman aptly described them as Lorentzian analogues of Hermitian structures. In this talk, I will give a survey of old and new results in the field.

Abstract: "We discuss recent developments in the analysis of linear and non-linear wave equations on the Schwarzschild de Sitter spacetime, which is a black hole spacetime with a positive cosmological constant. Specifically, we use only time domain methods, as opposed to the previous spectral approach, and give a new proof of exponential decay for the solutions of the linear wave equation and a new proof for the stability of the solutions of the quasilinear wave equation. Trapped null geodesics will make a frequent appearance in this talk."

Abstract: In this talk, I will propose a definition of asymptotic flatness at timelike infinity in four spacetime dimensions. I will also present a detailed study of the asymptotic equations of motion and the action of supertranslations on asymptotic fields. I will then show that the Lee-Wald symplectic form does not get contributions from future timelike infinity with appropriate boundary conditions. As a result, the “future charges” can be computed on any two-dimensional surface surrounding the sources at timelike infinity. Finally, I will present expressions for supertranslation and Lorentz charges.

Zoom: https://qmul-ac-uk.zoom.us/j/82729534665?pwd=UUE3b3pHbDZsSGdhVzF

When studying the Cauchy problem of general relativity we typically obtain L² bounds on the (Ricci) curvature tensor of spacelike hypersurfaces and its derivatives. In many situations it is useful to deduce from these H^{k} bounds that there exists coordinates on the spacelike hypersurface with optimal H^{k+2} bounds on the components of the induced Riemannian metric. The general idea is that this can be achieved using harmonic coordinates -- in which the principal terms of the Ricci curvature tensor are the Laplace-Beltrami operators of the metric components -- and standard elliptic regularity results. In this talk, I will make this idea concrete in the case of Riemannian 3-manifolds with a 2-sphere boundary, with Ricci curvature in L² and second fundamental form of the boundary in H^{1/2} both close to their respective Euclidean unit 3-disk values. The crux of the proof is a refined Bochner identity with boundary for harmonic functions. The cherry on the cake is that this result does not require any topology assumption on the Riemannian 3-manifold (apart from its boundary), and that we obtain -- as a conclusion -- that it must be diffeomorphic to the 3-disk. This talk is based on a result that I obtained in [Global nonlinear stability of Minkowski space for spacelike-characteristic initial data, Appendix A]

On a background Minkowski spacetime, the relativistic Euler equations are known, for a relatively general equation of state, to admit unstable homogeneous solutions with finite-time shock formation. By contrast, such shock formation can be suppressed on background cosmological spacetimes whose spatial slices expand at an accelerated rate. The critical case of linear, ie zero-accelerated, spatial expansion, is not as well understood. In this talk, I will present two recent works concerning the relativistic Euler and the Einstein-Dust equations for geometries expanding at a linear rate. This is based on joint works with David Fajman, Todd Oliynyk and Max Ofner.

With A. Deruelle, we define a Perelman like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.

Zoom details:

https://ucl.zoom.us/j/95150620185?pwd=THJ5R3B3cjBSclJ5azNXampGOGI0UT09
 
Meeting ID: 951 5062 0185
Passcode: 956606

About 121 years ago, Otto Zoll described a large family of rotationally symmetric Riemannian two-dimensional spheres whose geodesics are all closed and have the same period. Since then, a very rich (but yet incomplete) theory developed in order to construct and understand geometries (in a broad sense) with these special geodesic flows, also in higher dimensions. After working on certain systolic questions about minimal two-dimensional spheres in three-dimensional Riemannian spheres with R. Montezuma (UFC), and motivated by other interesting geometric reasons, I became convinced that another sort of higher dimensional generalisation of Zoll surfaces, within the theory of minimal submanifolds, deserved to be investigated on its own. In this talk, we will report on some of the results we proved about these new objects, including existence results, together with F. Codá Marques (Princeton) and A. Neves (UChicago).

Abstract: 100 years ago, Kasner discovered the first exact cosmological solutions to Einstein's field equations, revealing the presence of a striking new phenomenon, namely, the Big Bang singularity. Since then, it has been the object of study in a great deal of research on general relativity. However, the nature of the 'generic' Big Bang singularity remains a mystery. Rivaling scenarios are abound (monotonicity, chaos, spikes) that make the classification of all solutions a very intricate problem. I will give a historic overview of the subject and describe recent progress that confirms a small part of the conjectural picture.

Abstract: In the mean curvature flow, translating solutions are an important model for singularity formation. In this talk, we will consider the class of 2-dimensional mean curvature flow translators embedded in R^3 which have finite total curvature and describe their asymptotic structure, which turns out to be highly rigid. I will outline the proof of this asymptotic description, in particular focusing on some novel and unexpected features of the proof.

Abstract: Recently in their celestial holography programme, Strominger and coworkers attempt to provide a holographic description of conventional 4d gravity.  In their investigations, they uncovered a hidden w-infinity symmetry in their `celestial soft OPEs' for graviton scattering.  This talk will explain the origin of this symmetry in terms of old ideas of Newman and Penrose based on light-cone cuts of null infinity and their description in terms of asymptotic twistors and certain sigma models in asymptotic twistor space.  W_n symmetries were introduced by Zamolodchikov as higher spin symmetries in 2d conformal field theories.  These were given a geometric interpretation for n=infinity as area-preserving diffeomorphisms of the plane.  I will explain how the corresponding loop algebra becomes a hidden symmetry of self-dual gravity via Penrose's nonlinear graviton construction.  The action of this symmetry on the tree-level S-matrix of full gravity beyond the self-dual sector will then be obtained from its action on a sigma model in the asymtotic twistor space of a general space-time.  This talk is based on https://arxiv.org/abs/2110.06066 and https://arxiv.org/abs/2103.16984.

Abstract: After a brief discussion of the problem of relativistic fluids, I will talk about the 1D compressible, isentropic Euler equations in special relativity. In recent work, G.-Q. Chen and I have shown the existence of global entropy solutions to this system through a compensated compactness framework, which I will outline in this talk. We also demonstrate the convergence of relativistic solutions to their classical (Newtonian) counterparts, and so, in the final portion of my talk, I will briefly outline how this is achieved.

Abstract: In the context of holography, one would like to establish correspondence statements between asymptotically anti-de Sitter (aAdS) solutions of the Einstein-vacuum equations and suitable ‘data’ on the conformal boundary. To this end, progress has been made in the Riemannian (Biquard ‘08) and Stationary Lorentzian (Chruściel-Delay ‘11) cases. I will present recent unique continuation results for wave equations on aAdS backgrounds which lay the groundwork for progress in the non-stationary setting. In particular, I will describe how these preliminary results lead to a rigidity statement for locally AdS spacetimes based around conditions on the boundary data. This talk involves joint work with Arick Shao.

Abstract: We will present a collection of conjectures formulated with Gromov and other members of our IAS Emerging Topics Working Group on the limits of sequences of Riemannian manifolds with uniform lower bounds on their scalar curvature. We will survey results in special cases and present key theorems concerning volume preserving intrinsic flat convergence that have been applied to prove these special cases. For a complete list of papers about intrinsic flat convergence see here. 

https://ucl.zoom.us/j/92705362785?pwd=VGtKdVBuQzRFRVlXYTdXL1VDOURPUT09

Passcode: 873569 

Abstract:
Both in the context of QCD and CMT the AdS-CFT correspondence has been used to describe the behaviour of strongly coupled field theories via a classical gravity dual. While this gives a beautiful geometrization of certain problems in strongly coupled theories, an important long standing question is how similar the quantitative behaviour of holographic theories is to the usual QFTs we are typically interested in. We address this by studying a universal property of QFTs, the dependence of vaccum energy on the curvature of space, focussing on (2+1)-dimensions. I will detail how to numerically compute this Casimir energy in the free field setting, and how to compute it in the holographic context. We will then see a very surprising similarity in the detailed behaviour between holographic theories and free CFTs, particularly the free Dirac theory. We will also discuss some interesting geometric constraints on this energy in the holographic case both in vacuum and also at finite temperature, that arise from bulk considerations. Together with our numerical free field calculations this motivates an interesting conjecture for free theories.

Abstract:
I will discuss diffeomorphism invariant theories of gravity coupled to matter, with second order equations of motion. This includes Lovelock and Horndeski theories, as well as some effective field theories which include 4-derivative corrections to conventional 2-derivative theories. In such theories, generically causality is not determined by the null cone of the metric. I will describe a method for studying causality in these theories in a gauge-invariant way. For the class of theories with a single scalar field, I will present an explicit characterization of the causal cone in an arbitrary background. There is an interesting analogy with the cone associated with wave propagation in certain anisotropic elastic solids.

Stable constant mean curvature spheres encode important information on the asymptotic geometry of initial data sets for isolated gravitational systems. In this talk, I will present a short new proof (joint with M. Eichmair) based on Lyapunov-Schmidt reduction of the existence of an asymptotic foliation of such an initial data set by stable constant mean curvature spheres. In the case where the scalar curvature is non-negative, our method also shows that the leaves of this foliation are the only large stable constant mean curvature spheres that enclose the center of the initial data set. 

Abstract: For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G with compact, smooth orbit space, we show the following rigidity result:

Applications will be discussed. This is joint work with R. Lafuente.

https://ucl.zoom.us/j/92705362785?pwd=VGtKdVBuQzRFRVlXYTdXL1VDOURPUT09

Passcode: 873569

I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.  The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family.  This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

https://ucl.zoom.us/j/92705362785?pwd=VGtKdVBuQzRFRVlXYTdXL1VDOURPUT09

Passcode: 873569

Abstract: Four-dimensional complete hyperkaehler manifolds can be classified into ALE, ALF, ALG, ALG*, ALH, ALH* families. It has been conjectured that every ALG or ALG* hyperkaehler metric can be realized as a 4d Hitchin moduli space. I will describe ongoing work with Rafe Mazzeo, Jan Swoboda, and Hartmut Weiss to prove a special case of the conjecture, and some consequences. The hyperkaehler metrics on Hitchin moduli spaces are of independent interest, as the physicists Gaiotto—Moore—Neitzke give an intricate conjectural description of their asymptotic geometry.

Abstract: The study of p-harmonic functions on Riemannian manifolds has invoked the interest of mathematicians and physicists for nearly two centuries. Applications within physics can for example be found in continuum mechanics, elasticity theory, as well as two-dimensional hydrodynamics problems involving Stokes flows of incompressible Newtonian fluids.

In my talk I will focus on the construction of explicit p-harmonic functions on rank-one Lie groups of Iwasawa type. This joint work with Sigmundur Gudmundsson and Marko Sobak.

Abstract: I will present recent joint work with Valentino Tosatti in which we obtain a complete asymptotic expansion (locally uniformly away from the singular fibers) of Calabi-Yau metrics collapsing along a holomorphic fibration of a fixed compact Calabi-Yau manifold. The result is weaker than a standard asymptotic expansion in that the coefficient functions might still depend on the small parameter in some unknown way in the base variables. However, it is far stronger in that all terms including the remainder at each order are proved to be uniformly bounded in C^k for all k. We also calculate the first nontrivial coefficient in terms of the Kodaira-Spencer forms of the fibration.

Abstract: Let N be a compact Riemannian manifold of dimension 3 or higher, and g a Lipschitz non-negative (or non-positive) function on N. We prove that there exists a closed hypersurface M whose mean curvature attains the values prescribed by g (joint work with Neshan Wickramasekera, Cambridge). Except possibly for a small singular set (of codimension 7 or higher), the hypersurface M is C^2 immersed and two-sided (it admits a global unit normal); the scalar mean curvature at x is g(x) with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when g is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature. The construction of M is carried out largely by means of PDE principles: (i) a minmax for an Allen–Cahn (or Modica-Mortola) energy, involving a parameter that, when sent to 0, leads to an interface from which the desired PMC hypersurface is extracted; (ii) quasi-linear elliptic PDE and geometric-measure-theory arguments, to obtain regularity conclusions for said interface; (iii) parabolic semi-linear PDE (together with specific features of the Allen-Cahn framework), to tackle cancellation phenomena that can happen when sending to 0 the Allen-Cahn parameter.

Abstract : Let M be a compact 3-manifold with scalar curvature at least 1. We show that there exists a Morse function f on M, such that every connected component of every fiber of f has genus, area and diameter bounded by a universal constant. The proof uses Min-Max theory and Mean Curvature Flow. This is a joint work with Davi Maximo. Time permitting, I will discuss a related problem for macroscopic scalar curvature in metric spaces (joint with Boris Lishak, Alexander Nabutovsky and Regina Rotman).

COMPACTNESS AND PARTIAL REGULARITY THEORY OF RICCI FLOWS IN HIGHER DIMENSIONS

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected L^p-curvature bounds.

As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.

UNIQUENESS OF CERTAIN CYLINDRICAL TANGENT CONES

Leon Simon showed that if an area minimizing hypersurface
admits a cylindrical tangent cone of the form C x R, then this tangent
cone is unique for a large class of minimal cones C. One of the
hypotheses in this result is that C x R is integrable and this
excludes the case when C is the Simons cone over S^3 x S^3. The main
result in this talk is that the uniqueness of the tangent cone holds
in this case too. The new difficulty in this non-integrable situation
is to develop a version of the Lojasiewicz-Simon inequality that can
be used in the setting of tangent cones with non-isolated
singularities.

LIMITS OF MANIFOLDS WITH A KATO BOUND ON THE RICCI CURVATURE

Starting from Gromov pre-compactness theorem, a vast theory about the structure of limits of manifolds with a lower bound on the Ricci curvature has been developed thanks to the work of J. Cheeger, T.H. Colding, M. Anderson, G. Tian, A. Naber, W. Jiang. Nevertheless, in some situations, for instance in the study of geometric flows, there is no lower bound on the Ricci curvature. It is then important to understand what happens when having a weaker condition.

In this talk, we present new results about limits of manifolds with a Kato bound on the negative part of the Ricci tensor. Such bound is weaker than the previous L^p bounds considered in the literature (P. Petesern, G. Wei, G. Tian, Z. Zhang, C. Rose, L. Chen, C. Ketterer…). In the non-collapsing case, we recover part of the regularity theory that was known in the setting of Ricci lower bounds: in particular, we obtain that all tangent cones are metric cones, a stratification result and volume convergence to the Hausdorff measure. After presenting the setting and main theorem, we will focus on proving that tangent cones are metric cones, and in particular on the study of the appropriate monotone quantities that leads to this result.

In this talk, I will explain our recent work showing that mean curvature flow through neck-singularities is unique. The key is a classification result for ancient asymptotically cylindrical flows that describes all possible blowup limits near a neck-singularity. In particular, this confirms Ilmanen’s mean-convex neighborhood conjecture, and more precisely gives a canonical neighborhood theorem for neck-singularities. Furthermore, assuming the multiplicity-one conjecture, we conclude that for embedded two-spheres mean curvature flow through singularities is well-posed. The two-dimensional case is joint work with Choi and Hershkovits, and the higher-dimensional case is joint with Choi, Hershkovits and White.

RIGIDITY OF THE EUCLIDEAN HEAT KERNEL

It is a joint work with David Tewodrose (Bruxelles) https://arxiv.org/abs/1912.10759

I will explain that a metric measure space with Euclidean heat kernel is Euclidean. An almost rigidity result comes then for free, and this can be used to give another proof of Colding’s almost rigidity for complete manifold with non negative Ricci curvature and almost Euclidean growth.

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.

Abstract: It is known that the almost-Kähler anti-self-dual metrics on a given 4-manifold sweep out an open subset in the moduli space of anti-self-dual metrics. However, we show by example that this subset is not generally closed, and does not always sweep out entire connected components in the moduli space. The construction hinges on an unexpected link between harmonic functions on certain hyperbolic 3-manifolds and self-dual harmonic 2-forms on associated 4-manifolds.

Abstract: It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. I will first review some history of this question, and present some recent progress, ruling out positive scalar curvature on closed aspherical manifolds of dimensions 4 and 5 (as conjectured by Schoen-Yau and by Gromov). I will also discuss some related questions including the Urysohn width inequalities on manifolds with scalar curvature lower bounds. This talk is based on joint work with Otis Chodosh.

Zoom details:
https://ucl.zoom.us/j/97740010241?pwd=RElWTDQvSUFiTWNRVFpIOEMyNCtKUT09

Meeting ID: 977 4001 0241
Passcode: 261438

In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an epsilon-regularity theorem showing that such a space must be close to Euclidean space in a suitable sense. Interestingly, such a result is false with respect to the Gromov-Hausdorff and Intrinsic Flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. Instead, we introduce the notion of the d_p distance between (in particular) Riemannian manifolds, which measures the distance between W^{1,p} Sobolev spaces, and it is with respect to this distance that the epsilon regularity theorem holds. We will discuss various applications to manifolds with scalar curvature and entropy lower bounds, including a compactness and limit structure theorem for sequences, a uniform L^infinity Sobolev embedding, and a priori L^p scalar curvature bounds for p<1 This is joint work with Man-Chun Lee and Aaron Naber.

Although the words "Topology" and "Optimization" make perfect sense to a mathematician, the compound term "Topology Optimization" might raise eyebrows. Nevertheless, Topology Optimization is a well-defined concept in the field of computational design in engineering. In
computational design, the engineer relies on algorithms to automatically generate solutions to engineering design problems. In the case of Topology Optimization, this means to generate a shape (a.k.a. domain with piecewise-smooth boundary) that fulfills an engineering task (e.g. transmits torque) with optimal performance (e.g. is stiff and lightweight). I will explain this concept in more detail, from the mathematical formulation and the numerical
implementation to the application in real-world engineering design scenarios.

Bio

Abstract: 

In a lot of geometric situation we need to work with families of varieties. In this talk we focus on families of singular Kähler-Einstein metric. In particular we study the case of a family of Kähler varieties and we develop the first steps of pluripotential theory in family, which will allow us to have a control on the C^0 estimate when the complex structure varies. This type of result will be applied in different geometric contexts. This is a joint work with V. Guedj and H. Guenancia.

Mean curvature flow (MCF) is the gradient flow of the area functional; it moves the surface in the direction of steepest decrease of area. An important motivation for the study of MCF comes from its potential geometric applications, such as classification theorems and geometric inequalities. MCF develops “singularities” (curvature blow-up), which obstruct the flow from existing for all times and therefore understanding these high curvature regions is of great interest. This is done by studying ancient solutions, solutions that have existed for all times in the past, and which model singularities. In this talk we will discuss their importance and ways of constructing and classifying such solutions. In particular, we will focus on “collapsed” solutions and construct, in all dimensions n>=2, a large family of new examples, including both symmetric and asymmetric examples, as well as many eternal examples that do not evolve by translation. Moreover, we will show that collapsed solutions decompose “backwards in time” into a canonical configuration of Grim hyperplanes which satisfies certain necessary conditions. This is joint work with Mat Langford and Giuseppe Tinaglia.

The Bernstein theorem is a classical result for minimal graphs. It states that a globally defined solution of the minimal surface equation on $R^n$ must be linear, provided the dimension is small enough. We present an analogous theorem for two-valued minimal graphs, valid in dimension four. By definition two-valued functions take values in the unordered pairs of real numbers; they arise as the local model of branch point singularities. The plan is to juxtapose this with the classical single-valued theory, and explain where some of the difficulties emerge in the two-valued setting.

https://ucl.zoom.us/j/91921950801?pwd=bnEwUWsrODMybEUwR01GNURZcVAvQT09

Abstract:  In 1994, Guan published a series of papers constructing non-Kähler holomorphic symplectic manifolds, challenging a conjecture by Todorov. These examples, called now BG manifolds were given a more transparent presentation by Bogomolov in '96, which emphasizes the analogy with the Kodaira-Thurston example of non-Kähler symplectic surfaces. We will discuss some important properties of BG manifolds: deformation theory, which is quite similar to that of the hyperKaehler case, algebraic reduction and submanifolds.

Let Bun_G be the moduli stack of G-bundles on a compact Riemann surface. After reviewing (and motivating) the notion of “temperedness” appearing in the geometric Langlands program, I will discuss the proof of a conjecture of Gaitsgory stating that the constant D-module on Bun_G is anti-tempered. No prior familiarity with geometric Langlands will be assumed; rather, I’ll emphasize some key ingredients that might be of broader interest: a Serre duality in an unusual context and various cohomology vanishing computations.

Abstract: Given a vector bundle of arbitrary rank with ample determinant line
bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths – and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. Another outcome of the approach is a new concept of volume for vector bundles.

In 1981 Sacks and Uhlenbeck introduced their famous alpha-approximation of the Dirichlet energy for maps from surfaces and showed that critical points converge to a harmonic map (away from finitely many points). Now one can ask whether every harmonic map is captured by this limiting process. Lamm, Malchiodi and Micallef answered this for maps from the two sphere into the two sphere and showed that the Sacks-Uhlenbeck method produces only constant maps and rotations if the energy lies below a certain threshold. We investigate the same question for the epsilon-approximation of the Dirichlet energy.

Joint work with Tobias Lamm and Mario Micallef.

Abstract: We exhibit new obstructions to the desingularization of Einstein metrics in dimension 4. These obstructions are specific to the compact situation and raise the question of whether or not a sequence of Einstein metrics degenerating while bubbling out gravitational instantons has to be Kähler-Einstein. We then test these obstructions to discuss the possibility of producing a Ricci-flat but not Kähler metric by the promising desingularization configuration proposed by Page in 1981.

Abstract:  Allen-Cahn (AC) minimal hypersurfaces are limits of nodal sets of solutions to the AC equation. An important problem is to understand the local picture of this convergence. For instance, can we avoid the situation in which the nodal set looks like a multigraph over the limit hypersurface? General examples of this phenomenon, known as “multiplicity” or "interface foliation”,  exist when the limit hypersurface is unstable. Together with A. Neves and F. Marques we proved that, generically and in all dimensions, these are the only possible examples of interface foliation, i.e. generic stable AC minimal hypersurfaces can only occur with multiplicity one. We will discuss this and other topics.

It’s a long standing problem in Hodge theory to complete the image of a period map.  The latter arise in the study of algebraic moduli, and are proper holomorphic maps into locally homogeneous spaces that are subject to a differential constraint.  I’ll give a survey of the problem and then describe recent progress, with an emphasis on the role of complex geometry and Lie theory.  Joint with Mark Green and Phillip Griffiths.

Abstract: I will discuss recent work with Yevgeny Liokumovich and Luca Spolaor concerning generic regularity of min-max minimal hypersurfaces in the first dimension that they might be singular.

Abstract: The self-dual Yang-Mills-Higgs (or Ginzburg-Landau) functionals are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points (particularly those satisfying a first-order system known as the “vortex equations” in the Kahler setting) have long been studied as a basic model problem in gauge theory. In this talk, we will discuss joint work with Alessandro Pigati characterizing the behavior of critical points associated to line bundles over manifolds of arbitrary dimension. We show in particular that critical points give rise to minimal submanifolds of codimension two in certain natural scaling limits, and use this information to provide new constructions of codimension-two minimal varieties in arbitrary Riemannian manifolds.

A classic result in the study of Kähler metrics with special curvature properties is that the cscK equation can be realized as the moment map equation for an infinite-dimensional Kähler reduction. We present a natural hyperkähler extension of this moment map picture, obtaining a new system of equations reminiscent of Hitchin’s equations for Higgs bundles. We will discuss some recent existence results, particularly obstructions to solutions to the problem.

To keep up-to-date with the schedule, and to receive the Zoom link for each talk, you can subscribe to the B.O.W.L. mailing list on the page http://geometry.ulb.ac.be/bowl/

G2-MONOPOLES

This talk is aimed at reviewing what is known about G2-monopoles and motivate their study. After this, I will mention some recent results obtained in collaboration with Ákos Nagy and Daniel Fadel which investigate the asymptotic behavior of G2-monopoles. Time permitting, I will mention a few possible future directions regarding the use of monopoles in G2-geometry.

(This seminar is held jointly with the Relativity and Cosmology Seminar)

(This Geometry and Analysis Webinar is held online on MS Teams.)

(This Geometry and Analysis Webinar is held online on MS Teams.)

The index theorem  for compact manifolds with boundary, established by Atiyah-Patodi-Singer in 1975, is considered one of the most significant mathematical achievements of the 20th century.
An important and curious fact is that local boundary conditions are topologically obstructed for index formulae and non-local boundary conditions lie at the heart of this theorem.
Consequently, this has inspired the study of boundary value problems for first-order elliptic differential operators by many different schools, with a class of induced boundary operators taking centre stage in establishing non-local boundary conditions.

The work of Bär and Ballmann from 2012 is a modern and comprehensive framework that is useful to study  elliptic boundary value problems for first-order elliptic operators on manifolds with compact and smooth boundary.
As in the work of Atiyah-Patodi-Singer, a fundamental assumption in Bär-Ballmann is that the induced operator on the boundary can be chosen self-adjoint.
All  Dirac-type operators, which in particular includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator, are captured via this framework.

In contrast to the APS index theorem, which is essentially restricted to Dirac-type operators, the earlier index theorem of Atiyah-Singer  from 1968 on closed manifolds is valid for general first-order elliptic differential operators.
There are important operators from both geometry and physics which are more general than those captured by the state-of-the-art for BVPs and index theory.
A quintessential example is the Rarita-Schwinger operator on 3/2-spinors, which arises in physics for the study of the so-called delta baryons.
A fundamental and seemingly fatal obstacle to study BVPs for such operators is that the induced operator on the boundary may no longer be chosen self-adjoint, even if the operator on the interior is symmetric.

In recent work with Bär, we extend the Bär-Ballmann framework to consider general first-order elliptic differential operators by dispensing with the self-adjointness requirement for induced boundary operators.
Modulo a zeroth order additive term, we show every induced boundary operator is a bi-sectorial operator via the ellipticity of the interior operator.
An essential tool at this level of generality is the bounded holomorphic functional calculus, coupled with pseudo-differential operator theory, semi-group theory as well as methods arising from the resolution of the Kato square root problem.
This perspective also paves way for studying non-compact boundary, Lipschitz boundary, as well as boundary value problems in the L^p setting.

We will discuss a fully nonlinear geometric flow of three-convex hypersurfaces, where the normal speed at each point of the solution is given by a concave function of the second fundamental form. Three-convexity means that at each point, the sum of the smallest three principal curvatures is positive. The flow smoothly deforms compact three-convex initial hypersurfaces until their curvature becomes unbounded. Our main result is a convexity estimate, which says that where the curvature is very large, the second fundamental form is approximately nonnegative. Such an estimate is known to hold for mean-convex mean curvature flow, and for a large class of fully nonlinear flows where the speed function is convex. For concave speeds, previous results of this kind assume two-convexity.

(This seminar is held jointly with the Relativity and Cosmology Seminar)

Abstract: The parabolic Allen-Cahn equations is the gradient flow of phase transition energy and can be viewed as a diffused version of mean curvature flows of hypersurfaces. It has been known by the works of Ilmanen and Tonegawa that the energy densities of the Allen-Cahn flows converges to mean curvature flows in the sense of varifold and the limit varifold is integer rectifiable. It is not known in general whether the transition layers have higher regularity of convergence yet. In this talk, I will report on a joint work with Huy Nguyen that under the low entropy condition, the convergence of transition layers can be upgraded to C^{2,\alpha} sense. This is motivated by the work of Wang-Wei and Chodosh-Mantoulidis in elliptic case that under the condition of stability, one can upgrade the regularity of convergence.

Kazhdan's Property (T) for topological groups has found applications in domains as diverse as group theory, differential geometry, potential theory, operator algebras, combinatorics and computer science. The aim of my talk, which is intended for a wide audience, is to introduce the notions of Kazhdan sets and Kazhdan constants, and to present some unconventional applications in harmonic analysis, ergodic theory and dynamical systems. A problem related to Furstenberg's $\times 2$-$\times 3$ conjecture will be also discussed.

Mini-workshop website: http://www.maths.qmul.ac.uk/~shao/events/ws2020

In this talk we introduce a geometrical model of continued fractions
and discuss its appearance in rather distant research areas:
-- values of quadratic forms (Perron Identity for Markov spectrum)
-- the 2nd Kepler law on planetary motion
-- Global relation on singularities of toric varieties

A well-known theorem of Choquet-Bruhat and Geroch states that for given smooth initial data for the Einstein equations there exists a unique maximal globally hyperbolic development. In particular, time evolution of globally hyperbolic solutions is unique. This talk investigates whether the same result holds for quasilinear wave equations defined on a fixed background. After recalling the notion of global hyperbolicity, we first present an example of a quasilinear wave equation for which unique time evolution in fact fails and contrast this with the Einstein equations. We then proceed by presenting conditions on quasilinear wave equations which ensure uniqueness. This talk is based on joint work with Harvey Reall and Felicity Eperon.

Abstract
Given a three-dimensional Riemannian manifold M with boundary, free
boundary minimal surfaces (FBMS) in M are critical points of the area
functional with respect to variations that constrain their boundary to
the boundary of M.

In recent years several different examples of FBMS have been discovered,
opening the problem of classifying the rich variety of all such surfaces
in a given ambient manifold.
Towards this purpose, we begin by studying how different properties of a
FBMS relate to each other. In particular we focus on the topology, the
area and the Morse index of a FBMS and we give a complete description of
the connections among these "complexity criteria".

This is joint work with Alessandro Carlotto.

TBA

The origins of this talk go back to the fundamental theorem of Loewner in 1934 on operator monotone real functions and also to
the hyperbolic geometry of positive matrices. Among others it lead to the Kubo-Ando theory of two-variable operator means
of positive operators in 1980. One of the nontrivial means of the Kubo-Ando theory is the non-commutative generalization of the  
geometric mean which is intimately related to the hyperbolic, non-positively curved Riemannian structure of positive matrices.
This geometry provides a key tool to define multivariable generalizations of two-variable operator means. Arguably the most important
example of them all is the Karcher mean which is the center of mass on this manifold. This formulation defines this mean
for probability meaures on the cone of positive definite matrices extending further the multivariable case. Even the infinite dimensional
case of positive operators is tractable by abandoning the Riemannian structure in favor of a Banach-Finsler structure provided by
Thompson's part metric on the cone of positive definite operators. This metric enables us to develop a general theory of means of
probability measures defined as unique solutions of nonlinear operator equations on the cone, with the help of contractive semigroups
of nonlinear operators.

Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also briefly indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.

(Note the special time of the seminar.)

Symbolic dynamical systems have generated a wide a rich class of C*-algebras with interesting properties. Recently, we have learned that these C*-algebras remember a surprising amount of the information about the underlying dynamical system if we include the finer structure of the algebra as part of the data. This finer structure could be commutative subalgebras or certain circle actions. There will be an emphasis on open problems.

(Note the special time of the seminar.)

Abstract: A CMC surface in 3-space is constrained Willmore and isothermic. It is well known that these 3 surface classes are each determined by a family of flat connections. In this talk we discuss links between the corresponding families of flat connections: we show that parallel sections of the associated family of flat connections of the harmonic Gauss map give algebraically the parallel sections of the other families. In particular, we obtain links between transformations of CMC surfaces, isothermic surfaces and constrained Willmore surfaces which are given by parallel sections, such as the associated family, the simple factor dressing and the Darboux transformation.

There has been much interest in whether or not strong cosmic
censorship holds in de Sitter spacetimes. I will give an overview of
the strong cosmic censorship conjecture in general and more
specifically for Reissner-Nordstrom-de Sitter and Kerr-de Sitter black
holes. For Kerr-de Sitter, I will explain how the quasinormal modes
can be used to show that strong cosmic censorship is indeed respected.

(This seminar is held jointly with the Relativity and Cosmology Seminar)

Abstract: The Allen-Cahn functional provides a way to construct min-max minimal hypersurfaces, in an ambient closed Riemannian manifold, as limiting interfaces in a phase transition between two different states. A naive attempt to carry out a similar theory in codimension two would be to use the Ginzburg-Landau functional.
After briefly describing the difficulties arising with this functional, we will see how they disappear looking instead at the Yang-Mills-Higgs action, whose features are strikingly similar to those of Allen-Cahn. We will also glimpse a min-max construction of codimension two min-max integer stationary varifolds using this approach. This is joint work with Daniel Stern (Princeton University - University of Toronto).

We consider co-rotational wave maps from the (1 + d)-dimensional Minkowski space into the d-sphere for d ≥ 3 odd. This is an energy-supercritical model which is known
to exhibit finite-time blowup via self-similar solutions. Based on a method developed by Donninger and Schörkhuber, we prove the asymptotic nonlinear stability of the
“ground-state” self-similar solution.

 Abstract:
 
 A Ricci iteration is a sequence of Riemannian metrics on a manifold such that every metric
 in the sequence is equal to the Ricci curvature of the next metric. These sequences of metrics
 were introduced by Rubinstein to provide a discretisation of the Ricci flow. In this talk, I will
 discuss the relationship between the Ricci iteration and the Ricci flow. I will also describe
 a recent result concerning the existence and convergence of Ricci iterations close to certain
 Einstein metrics. (Joint work with Max Hallgren)

Abstract:

Yamabe flows are solutions to a geometric evolution equation which tends
to evolve a given Riemannian metric towards a conformal metric of
constant scalar curvature. On any closed Riemannian manifold the
existence and uniqueness of a Yamabe flow is known. The question of
well-posedness is more delicate if the initial manifold is noncompact
and uniqueness fails in general. We examine the existence and uniqueness
of instantaneously complete Yamabe flows starting from noncompact,
possibly incomplete Riemannian manifolds of arbitrary dimension. In this
context, we find a sharp condition on the dimension of a submanifold
which implies that its removability as a singularity is necessarily
preserved along the Yamabe flow.

In this talk, we will first review some classical results on the so-called
’spectral inequalities’, which yield a sharp quantification of the unique
continuation of the spectral family associated with the Laplace-Beltrami
operator in a compact manifold. In a second part, we will discuss how to
obtain the spectral inequalities associated to the Schrodinger operator
$-\Delta_x + V(x)$, in R^d, in any dimension $d \geq 1$, where
$V=V(x)$ is a real analytic potential. In particular, we can handle some long-
range potentials. This is a joint work with Prof G. Lebeau (Université de
Nice-Côte d'Azur, France).

Abstract:
Recent studies of gauge and gravity theories on spacetimes with
boundaries revealed an interesting interplay between gauge symmetries
and boundaries. Quite surprisingly, one finds new physical degrees of
freedom, called edge modes, that live only on the boundary and play a
crucial role in physical applications such as entanglement entropy
calculations and topological insulators. In this talk, I will provide a
conceptual and mathematical interpretation of such edge mode phenomena
in terms of basic homological algebra and show that this naturally
explains the concrete models proposed by Donnelly and Freidel. This talk
is based on a joint work with P. Mathieu, N. Teh and L. Wells
[arXiv:1907.10651].

Topological entropy is a numerical invariant for group actions on compact spaces. Whether the entropy is positive or not makes a fundamental difference for the dynamical behavior. When a group G acts on a compact space X, a subset H of G is called an independence set for a finite family W of subsets of X if for any finite subset M of H and any map f from M to W, there is a point x of X with sx in f(s) for all s in M. I will discuss how positivity of entropy can be described in terms of density of independence sets, and give a few applications including the relation between positive entropy and Li-Yorke chaos. The talk is based on various joint works with David Kerr and Zheng Rong.

Abstract: Recent experiments have, for the first time, directly measured gravitational waves created by colliding black holes. An important part of the signal from such events is the `ringdown’ phase where a distorted black hole emits radiation at certain fixed (complex) frequencies called the quasinormal frequencies. To mathematically model this phenomenon, one should study geometric wave equations on a class of open geometries. I will discuss how the quasinormal frequencies can be realised as eigenvalues of a (non-standard) spectral problem, with connections to scattering resonances on asymptotically hyperbolic manifolds. If time permits I will also discuss recent work with Gajic on the asymptotically flat case.

Abstract: The problem of quasi-local mass in general relativity is the problem of assigning some meaningful notion of the total mass (or energy) contained in a bounded domain. We first will give an introduction to the problem and review several important proposed definitions of quasi-local mass, before turning to discuss some recent progress on understanding the definition due to Bartnik. We will discuss a method to obtain estimates of the Bartnik mass in the CMC case, and briefly mention how these estimates can be used to obtain estimates outside of the CMC case. If time permits we will then present a new formula for the evolution of the full spacetime Bartnik mass, under some strict assumptions. The work presented here include results obtained in collaboration with Armando Cabrera Pacheco, Carla Cederbaum, and Pengzi Miao.

(This seminar is held jointly with the Relativity and Cosmology Seminar)

Abstract: In this talk I will discuss recent work with P. Daskalopoulos on sufficient conditions to prove uniqueness of complete graphs evolving by mean curvature flow. It is interesting to remark that the behaviour of solutions to mean curvature flow differs from the heat equation, where non-uniqueness may occur even for smooth initial conditions if the behaviour at infinity is not prescribed for all times. 

Abstract:
I will discuss some applications to smooth Riemannian manifolds of the optimal transport formulation of Ricci curvature lower bounds, leading to the theory of possibly non-smooth CD spaces by Lott-Sturm-Villani.

These include: rigidity/stability of the Levy-Gromov inequality and an
almost euclidean isoperimetric inequality motivated by the celebrated
Perelman's Pseudo-Locality Theorem for Ricci flow. (joint work with Fabio Cavalletti, SISSA).

A key question in general relativity is whether solutions to the Einstein equations, viewed as an initial value problem, are stable to small perturbations of the initial data. For example, previous results have shown that the Milne spacetime, which represents an expanding universe emanating from a big bang singularity with a linear scale factor, is a stable solution to the Einstein equations. With such a slow expansion rate, particularly compared to related isotropically expanding models (such as the exponentially expanding de Sitter spacetime observed in our universe), there are interesting questions one can ask about stability of this spacetime. Previous results have shown that the Milne model is a stable solution to the vacuum Einstein, Einstein-Klein-Gordon and Einstein-Vlasov systems. Motivated by techniques from the last result, I will present a new proof of the stability of the Milne model to the Einstein-Klein-Gordon system and compare our method to a recent result of J. Wang.  This is joint work with D. Fajman.

In joint work with Peter Topping we introduce local pyramid Ricci flows, existing on uniform regions of spacetime, that are inspired by Hochards partial Ricci flows. As an application of pyramid Ricci flows, we construct a global homeomorphism from any 3D Ricci limit space to a smooth manifold that is bi-Holder once restricted to any compact subset. This extends the recent work of Miles Simon and Peter Topping, and builds upon their techniques.

The Vlasov-Maxwell system is a classical model in plasma physics. Glassey and Strauss proved global existence for the small data solutions of this system under a compact support assumption on the initial data. I will present how vector field methods can be applied to revisit this problem. In particular, it allows to remove all compact support assumptions on the initial data and obtain sharp asymptotics on the solutions. We will also discuss the null structure of the system which constitutes a crucial element of the proof.

11:30 Giovanni Cation (Milano) - TBA
14:00 Gilles Caron (Nantes) - TBA
15:30 Yuxin Ge (Toulouse) - TBA

(This seminar is held jointly with the Relativity and Cosmology Seminar)

We are interested in the following unique continuation question: does the intensity of waves observed from a subdomain during a time interval determine their total energy? What is the intensity of waves in the shadow of an obstacle? In this talk, we shall give a stability estimate answering to these questions in a quantitative way.

This is joint work with Camille Laurent.

Abstract: We will discuss the qualitative change of behavior for solutions to certain classes on PDEs in various Holder and Sobolev regularity regimes, with a focus on the Monge-Ampere equations. We will discuss various techniques, from theory of mappings with finite distortion and geometric measure theory to convex integration, which are used to obtain a panorama of -yet- incomplete results.

Similar methods have been used to construct models of rapidly rotating or binary stars, in Newtonian and relativistic contexts. The choice of method has been based on numerical experiments, which indicate that particular methods converge quickly to a solution, while others diverge. The theory underlying these differences, however, has not been understood. In an attempt to provide a better theoretical understanding, we analytically examine the behavior of different iterative schemes near an exact static solution. We find the spectrum of the linearized iteration operator and show for self-consistent field methods that iterative instability corresponds to unstable modes of this operator. Minimizing the maximum eigenvalue accelerates convergence and allows computation of highly compact configurations that were previously inaccessible via self-consistent field methods.

(This seminar is held jointly with the Relativity and Cosmology Seminar)

In the first part of the talk we will give an overview of the inverse problem of recovering an unknown coefficient for an ellipitic PDE from boundary measurements. In particular we will derive a uniqueness result for determining a potential function for the Schrodinger operator in a Riemannian manifold from the knowledge of the local Dirichlet to Neumann map using the machinery of complex geometric optics and Carleman estimates. In the second part of the talk we will discuss a hyperbolic analogue of the same inverse problem.

Abstract: I will test drive some ideas, expanded in the PDE direction, from the second edition of 'Back-Of-The-Envelope Quantum Mechanics: With Extensions To Many-Body Systems And Integrable PDEs.’  The talk will be accessible to researchers and postgraduates in both mathematics and physics.

In this talk I will discuss a conjecture of Ehrnström and Wahlén on the profile of travelling wave solutions of extreme form to Whitham's non-local dispersive equation. We will see that there exists a highest, cusped and periodic solution that is convex between consecutive crests, at which C^1/2-regularity has been shown to be optimal. The talk is based on joint work with A. Enciso and J. Gómez-Serrano.

In this talk we will be interested in the evolution under the Euler and Navier-Stokes equations of several geometric structures defined by the vorticity of the fluid. First we will see how vortex lines and vortex tubes of complicated topologies are created and destroyed in the 3D Navier-Stokes equations. Next we will consider the emergence of non-smooth interfaces of surprising geometry in the free boundary Euler equations. The talk is based on joint work with D. Córdoba, C. Fefferman, N. Grubic, R. Lucà and D. Peralta-Salas.

Abstract : We consider entire higher codimensional mean curvature flow in $R^{n,m}$ of $n$-dimensional spacelike manifolds and prove a long time existence theorem starting from arbitrary spacelike initial data. We will see that the key to the proof is to demonstrate local spacelike gradient estimates, and to get around difficulties with cutoff functions in $R^{n,m}$. Surprisingly, this theorem leads to some new long time existence results for the $G_2$-Laplacian flow.

We will present a classification theorem for amenable simple stably projectionless C*-algebras with generalized tracial rank one.

With many decades' work, unital separable simple amenable Z-stable C*-algebras in the UCT class have been classified by the Elliott invariant. Non-unital case can be easily reduced to the unital case if the stabilized C*-algebras have a non-zero projection.

However, there are many non-unital separable simple amenable C*-algebras which are stably projectionless. In other words, K_0(A)_+ = {0}.

One of these simple C*-algebras is what we called Z_0. This C*-algebra can be constructed as an inductive limit of so-called non-commutative finite CW complexes. It has exactly one tracial state and has the properties that K_0(Z_0) = Z, K_0(Z_0)_+ = {0} and K_1(Z_0) = {0}.

We will show that there is exactly one Z_0 in the class of simple separable C*-algebras with finite nuclear dimension and satisfying the UCT (up to isomorphism).

Let A and B be two separable simple C*-algebras satisfying the UCT and have finite nuclear dimension.

We show that A \otimes Z_0 \cong B \otimes Z_0 if and only if Ell(B \otimes Z_0) = Ell(A \otimes Z_0).

A class of simple separable C*-algebras which are approximately sub-homogeneous whose spectra having bounded dimension is shown to exhaust all possible Elliott invariant for C*-algebras of the form A \otimes Z_0, where A is any finite separable simple amenable C*-algebra.

Suppose that  A and B are two finite separable simple C*-algebras with finite nuclear dimension satisfying the UCT such that both K_0(A) and K_0(B) are torsion (but arbitrary K_1). 

One consequence of the main results in this situation is that A \cong B if and only if A and B have isomorphic Elliott invariants.

In this talk, we give a result of the complete classification of Bost-Connes systems. For a number field K, there is a semigroup dynamical system attached to K, which is so called the Bost-Connes semigroup dynamical systems. By taking the crossed product, we obtain the Bost-Connes C*-algebra for K. We show that Bost-Connes C*-algebras for two number fields are isomorphic if and only if the Bost-Connes semigroup actions are conjugate. Together with the reconstruction results in number theory by Cornelissen-de Smit-Li-Marcolli-Smit, we conclude that two Bost-Connes C*-algebras are isomorphic if and only if the original number fields are isomorphic.

We will introduce L^p improving inequalities for discrete spherical averages and their generalizations. Subsequently we will give a new proof of a recent theorem of Kesler on sparse bounds for such averages. The latter is joint with Tess Anderson. Throughout we will pay attention to motivation and discuss a couple principles that influence the area.

(This seminar is held jointly with the Relativity and Cosmology Seminar)

In this talk we shall discuss our recent work which establishes that the Schwarzschild family of black holes are linearly stable as a family of solutions to the Einstein vacuum equations when expressed in a generalised wave gauge. The result therefore provides an important step towards a resolution of the black hole stability problem in general relativity and thus complements the recent work of Dafermos—Holzegel—Rodnianski in a similar vein as to how the work of Lindblad—Rodnianski complemented that of Christodoulou—Klainerman in establishing the nonlinear stability of the Minkowski space.

This is a two-day mini-conference on geometric analysis and mathematical relativity. See here for details.

In this talk I explain how my colleague Michael Kiessling and I used the ground-breaking work of Marcel Riesz on the analysis of Clifford-algebra-valued wave equations, and combined it with a key observation of Harish-Chandra --made while he was Dirac's student in Cambridge--  to obtain a relativistic quantum-mechanical wave equation for a photon (the quantum of light) in position-space representation, a task that has been declared "impossible" by many prominent physicists. I will also show that this wave equation has all the properties needed in order to treat the photon just like an electron, i.e., a point-particle whose motion is guided by a wave function defined on its configuration space.  As an application, I will  present some recent results we have, in collaboration with Matthias Lienert, concerning a fully-relativistic,  two-body photon-electron system in one space dimension, thereby paving the way for a rigorous geometric study of quantum effects in the interactions of radiation with matter.

I consider systems of first-order PDEs, which are weakly hyperbolic: the spectrum of the principal symbol is real but eigenvalues may cross. Close to one of those crossing eigenvalues,  lower order linear terms may induce a typical Gevrey growth in frequency. I will present an energy estimate in Gevrey regularity, using an approximate symmetrizer of the principal symbol. The symbol of such an approximate symmetrizer is in a special class of symbols, related to a specific metric in phase space. For such symbols, composition of associated operators lead to error terms that only can be handle thanks to the Gevrey energy.

Let $(M,g)$ be a compact Riemannian surface without boundary.  Consider the corresponding $L^2$-normalized Laplace-Beltrami eigenfunctions.  Eigenfunctions of this type arise in physics as modes of periodic vibration of drums and membranes. They also represent stationary states of a free quantum particle on a Riemannian manifold.  In the first part of the lecture, I will give a survey of results which demonstrate how the geometry of $M$ affects the behaviour of these special functions, particularly their “size” which can be quantified by estimating $L^p$ norms.

In joint work with Malabika Pramanik (U. British Columbia), I will present in the second part of my lecture a result on the $L^p$ restriction of these eigenfunctions to random Cantor-type subsets of $M$.  This, in some sense, is complementary to the smooth submanifold $L^p$ restriction results of Burq-Gérard-Tzetkov ’06 (and later work of other authors).  Our method includes concentration inequalities from probability theory in addition to the analysis of singular Fourier integral operators on fractals.

See http://geometry.ulb.ac.be/brussels-london/ for additional details.

Recently Austin showed that, for free probability-measure-preserving actions of
countable infinite amenable groups, entropy is preserved under bounded and L^1
orbit equivalence, and more generally that an entropy scaling formula holds
for stable versions of these equivalences. I will explain how Austin's approach
translates into the realm of topological dynamics and then speculate on how it
might extend beyond the amenable setting.

All known singularity models in Ricci flow are Ricci solitons. In this talk we will construct new steady and expanding Ricci solitons of cohomogeneity one. The solitons are defined on complex line bundles over products of Fano manifolds or HP^{m} \setminus \{ point \} amongst others. The main tool is a general estimate on the growth of the soliton potential.

The spinorial energy functional is a functional on the space of metrics whose critical points are special holonomy metrics in dimension 3 and higher. The spinor flow is its gradient flow. On surfaces the functional has a different geometric interpretation which will be explained in the talk. After that I will report on recent work concerning the formation of singularities, based on a decomposition of the flow into the evolution of a conformal factor and a movement of constant curvature metrics, which has been introduced by Buzano and Rupflin for the Ricci-harmonic flow.

A $f$-extremal domain in a manifold $M$ is a domain $\Omega$ which admits a positive solution $u$ to the equation $\Delta u+f(u)=0$ with $0$ Dirichlet boundary data and constant Neuman boundary data. Thanks to a result of Serrin, it is known that in $\mathbb R^n$ such a $f$-extremal domain has to be a round ball. In this talk, we will prove that a $f$-extremal domain in $\mathbb S^2$ which is a topological disk is a geodesic disk under some asumption on $f$. This is a joint work with J.M. Espinar.

For a given finite subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we prove the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of $S$ corresponds to an asymptotically flat end and that the Schouten tensor of the new conformal metric belongs to the boundary of the given cone. Joint work with Yanyan Li.

Let f be an orientation preserving branched covering of the two dimensional sphere. Is f realized (up to homotopy) by a rational function of the sphere? If yes, is the corresponding rational function unique up to the Mobius transformations (the rigidity)? These questions amount to the existence and uniqueness of a complex structure that is invariant under the action of (the homotopy class of) f. The geometric and topological structure of "the orbits of the branched points”, play a key role in these problems. When this set has finite cardinality, a classical result of W. Thurston provides a complete topological characterisation of the branched coverings that are realised by rational functions (and the uniqueness). On the other hand, when the orbits of branched points forms a more complicated set of points, say a Cantor set, the questions have been extensively studied over the last three decades. In this talk we survey the main results of these studies, and describe a recent advance made on the uniqueness part using a renormalization technique.

When a minimal submanifold with boundary on a given Riemannian manifold meets another hypersurface orthogonally, it is said to be a free boundary minimal submanifold. This constraint is very natural from a variational point of view. We will talk about some recent progress on the understanding of compact free boundary minimal hypersurfaces in various ambient domains. In particular, we will discuss our work on classification of free boundary minimal surfaces in the unit ball of the Euclidean space (joint with I Nunes, UFMA), on some general index estimates, and on convergence of sequences of free boundary minimal hypersurfaces under various assumptions (joint with A. Carlotto, ETH, and B. Sharp, Warwick).

In this talk I will review recent results on the structure of the linearized gravity equations. The results apply to yield new symmetry operators and conservation laws for linearized gravity on the Kerr spacetime, as well as new hyperbolic systems governing the linearized gravitational field.

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (AdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate a correspondence statement as a unique continuation problem for PDEs: Is an asymptotically AdS solution of the Einstein equations uniquely determined by its data on its conformal boundary at infinity?

In this presentation, we establish a key step in toward a positive result; we prove an analogous unique continuation result for linear and nonlinear wave equations on fixed asymptotically AdS spacetimes satisfying a positivity condition at infinity. We show, roughly, that if a wave ϕ on this spacetime vanishes on a sufficiently large but finite portion of its conformal boundary, then ϕ must also vanish in a neighbourhood of the boundary. In particular, we highlight the analytic and geometric features of AdS spacetimes which enable this uniqueness result, as well as obstacles preventing such a result from holding in other cases.

This is joint work with Gustav Holzegel.

Using the technique of holomorphic motions, we study the regularity of the limit set of the one-parameter family of holomorphic correspondences (w-c)^q=z^p, outlining some of the main contributions in the field in the last decades. This family is the simplest generalisation of the quadratic family z^2+c. In the quasi post-critically finite case, the limit set splits into a repeller and an attractor: the usual Julia set (closure of repelling periodic points) and the dual Julia set (closure of attracting periodic points). Conformal iterated function systems hidden in the dynamics of this correspondence appear naturally in the form of dual Julia sets. We also estimate the Hausdorff dimension of the Julia set using the formalism of Gibbs states.

Abstract: We consider layer potentials for second-order divergence form elliptic operators with bounded measurable coefficients on Lipschitz domains. A ''Calderón-Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that null solutions satisfy interior de Giorgi-Nash-Moser type estimates. In particular, we prove that $L^2$-estimates for layer potentials imply sharp $L^p$- and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.

In this talk I will give an overview of Friedrich’s construction of a regular asymptotic initial value problem at spatial
infinity and the open questions related to it. In particular, I will show how this framework can be used to identify initial data sets for the vacuum Einstein field equations which should lead to spacetimes not satisfying the peeling behaviour. This is research in collaboration with Edgar Gasperin.

We will discuss stability issues of a Schwarzschild singularity. I will describe past and recent work on the backward and forward initial value problem for the Einstein vacuum equations with near Schwarzschild configurations close to the singularity.

This is a talk of the Brussels-London Geometry seminar. For more information on this event, please visit the website at http://geometry.ulb.ac.be/brussels-london/(link is external) .

This is a talk of the Brussels-London Geometry seminar. For more information on this event, please visit the website at http://geometry.ulb.ac.be/brussels-london/(link is external) .

This is a talk of the Brussels-London Geometry seminar. For more information on this event, please visit the website at http://geometry.ulb.ac.be/brussels-london/(link is external) .

This talk is part of the QMUL Geometric Analysis Day. For more information and to register, please visit the event website at http://www.maths.qmul.ac.uk/~buzano/geometricanalysis.html

This talk is part of the QMUL Geometric Analysis Day. For more information and to register, please visit the event website at http://www.maths.qmul.ac.uk/~buzano/geometricanalysis.html

This talk is part of the QMUL Geometric Analysis Day. For more information and to register, please visit the event website at http://www.maths.qmul.ac.uk/~buzano/geometricanalysis.html

This talk is part of the QMUL Geometric Analysis Day. For more information and to register, please visit the event website at http://www.maths.qmul.ac.uk/~buzano/geometricanalysis.html

We consider the inverse boundary value problem for the wave equation in a geometric setting. This problem gives, for example, an idealized model of seismic imaging when the speed of sound is anisotropic but time-independent. We present two recent results: one related to the case where the speed of sound is time-dependent (joint work with Y. Kian, arXiv:1606.07243) and the other to the case where the wave equation is vector valued (joint work with Y. Kurylev and G. Paternain, arXiv:1509.02645).

In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the free boundary isentropic compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time. We show that if the adiabatic exponent gamma belongs to the interval (1, 5/3] then these affine motions are nonlinearly stable; small perturbations lead to global-in-time solutions that remain "close" to the moduli space of affine solutions and no shocks are formed in the process. Our strategy relies on two key ingredients: a new interpretation of the affine motions using an (almost) invariant action of GL(3) on the compressible Euler system and the use of Lagrangian coordinates. The former suggests a particular rescaling of time and a change of variables that elucidates a stabilisation mechanism, while the latter requires new ideas with respect to the existing well-posedness theory for vacuum free boundary fluid equations. This is joint work with Juhi Jang (USC).

I will discuss the global evolution problem for self-gravitating massive matter in the context of Einstein's theory and, more generally, of the f(R)-theory of gravity. In collaboration with Yue Ma (Xian), I have investigated the global existence problem for the Einstein-Klein-Gordon system and established that Minkowski spacetime is globally nonlinearly stable in presence of massive fields. The original method proposed by Christodoulou and Klainerman as well as the proof in wave gauge by Lindblad and Rodnianski cover vacuum spacetimes or massless fields only. Analyzing the time decay of massive waves requires a completely new approach, the Hyperboloidal Foliation Method, which is based on a foliation by asymptotically hyperboloidal hypersurfaces and on investigating the algebraic structure of the Einstein-Klein-Gordon system.

We will present a sharp one-sided curvature estimate for the mean curvature flow and some applications, in particular to ancient solutions of the flow.

Motivated by Gromov’s minimal volume problem, we introduce the class of noncompact graph 3-manifolds. We show that some of the structure theory of compact graph manifolds, due to Waldhausen in the late 60s, goes through. However, some results do not; we will present examples to that effect.

Part of this is still work in progress.

We will present some recent results which relate the Morse index of a minimal hypersurface with its first Betti number. The Morse index of a minimal hypersurface measures the number of different ways in which one can reduce area (up to second order). In the presence of positive curvature it is expected that the index controls the topology of such objects. We will state and prove some special cases of this phenomenon, in particular we show that in a variety of cases the first Betti number is linearly bounded from above by the index. In particular we will present separate joint works with Reto Buzano, Alessandro Carlotto and Lucas Ambrozio.

I will present the problem of the motion by curvature of a network of curves in the plane and I will discuss the state-of-the-art of the subject, in particular, about existence, uniqueness, singularity formation and asymptotic behavior of the flow.

Bost-Connes C*-algebra is a C*-algebra attached to number fields. In my series of work, Bost-Connes C*-algebras are shown to remember some number theoretic invariants. The next step we are interested in is to reconstruct C*-algebraic structures from invariants. We are conjecturing that all information is concentrated on K-groups of simple composition factors. Toward this, we are at first trying to give an isomorphism between Bost-Connes C*-algebras after trivializing K-theory. We give a partial result on this direction and explain what the remaining problem is.
This work is in progress. This is a joint work with Y. Kubota at the Univ. of Tokyo.

We consider solutions to the Klein-Gordon equation in the black hole exterior of Kerr-AdS spacetimes. It is known that, if the spacetime parameters satisfy the Hawking-Reall bound, solutions (with Dirichlet boundary conditions at infinity) decay logarithmically. We shall present our recent result of the existence of exponentially growing mode solutions in the parameter range where the Hawking-Reall bound is violated. We will discuss both Dirichlet and Neumann boundary conditions.

Starting with a substitution tiling, such as the Penrose tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles typically have fractal boundary. As an application of fractal tilings, we construct an odd spectral triple on a C*-algebra associated with an aperiodic substitution tiling. Even though spectral triples on substitution tilings have been extremely well studied in the last 25 years, our construction produces the first truly noncommutative spectral triple associated with a tiling. My work on fractal substitution tilings is joint with Natalie Frank and Sam Webster, and my work on spectral triples is joint with Michael Mampusti.

A theorem by J.D.S. Jones from 1987 identifies the cohomology of the free loop space of a simply connected space with the Hochschild homology of the singular cochain algebra of this space. There are very strong relations between the Floer homology of cotangent bundles in symplectic geometry and the homology of free loop spaces of closed manifolds. In the light of these connections, one wants to have a geometric and Morse-theoretic identification of free loop space cohomology and the Hochschild homology of Morse cochain algebras in order to establish relations between Floer homology and Hochschild homology. After describing the underlying Morse-theoretic constructions and especially the Hochschild homology of Morse cochains, I will sketch a purely Morse-theoretic version of Jones' map and discuss its most important properties.
If time permits, I will further discuss compatibility results with product structures like the Chas-Sullivan loop product and give explicit Morse-theoretic descriptions of products in Hochschild cohomology in terms of gradient flow trees.

In general relativity, the Kerr de Sitter spacetimes are models of black holes in an expanding universe. In this talk I will discuss my current understanding of the dynamics of nearby solutions to the Einstein equations with positive cosmological constant, and show in particular that in the cosmological region the conformal Weyl curvature decays in a sufficiently general setting. I will relate my work to recent results of Hintz and Vasy, and early work on the stability of de Sitter by Friedrich.

We present a new (2016) local result for the Ricci flow and explain the proof thereof. Joint work with Peter Topping.

A C*-algebra is a closed *-subalgebra of the algebra of bounded linear operators on some Hilbert space.
Originally considered for the purpose of a mathematical description of quantum mechanics, C*-algebras
in their own right have been studied extensively, especially since their abstract characterization by
Gelfand and Naimark in 1943. Nuclear C*-algebras form a prominent subclass, characterized either in terms of
a certain finite dimensional approximation property, or equivalently, as those C*-algebras that are
amenable as Banach algebras. Very recently, by work of many hands over several years, a big class of
separable, simple, nuclear C*-algebras satisfying further technical regularity properties has been classified
successfully in terms of K-theoretical data. In this talk, I will outline these results and point out the probably
most mysterious of these regularity properties: the universal coefficient theorem (UCT) by Rosenberg
and Schochet. I will then present recent joint work with Xin Li on the question which nuclear C*-algebras satisfy
the UCT.

We investigate the topology of the space of smoothly embedded n-spheres in R^{n+1}. By Smale’s theorem, this space is contractible for n=1 and by Hatcher’s proof of the Smale conjecture, it is also contractible for n=2. These results are of great importance, generalising in particular the Schoenflies theorem and Cerf’s theorem. In this talk, I will explain how geometric analysis can be used to study a higher-dimensional variant of these results. The main theorem (joint with Robert Haslhofer and Or Hershkovits) states that the space of 2-convex embedded spheres is path-connected in every dimension n. The proof uses mean curvature flow with surgery.

Massless collisionless matter is described in general relativity by the massless Einstein-Vlasov system. I will present key steps in a proof that, for asymptotically flat Cauchy data for this system, sufficiently close to that of the trivial solution, Minkowski space, the resulting maximal development of the data exists globally in time and asymptotically decays appropriately. By appealing to the corresponding result for the vacuum Einstein equations, a monumental result first obtained by Christodoulou-Klainerman in the early '90s, the proof reduces to a semi-global problem. A key step is to gain a priori control over certain Jacobi fields on the mass shell, a submanifold of the tangent bundle of the spacetime endowed with the Sasaki metric.

In 1942 M. H. A. Newman formulated and proved a simple lemma of great importance for various fields of mathematics, including algebra and the theory of Groebner–Shirshov bases. Later it was called the Diamond Lemma, since its key construction was illustrated by a diamond-shaped diagram. In the talk we will describe a new version of this lemma suitable for topological applications. Using it, we prove several results on the existence and uniqueness of prime decompositions of various topological objects: three-dimensional manifolds, knots in thickened surfaces, knotted graphs, three-dimensional orbifolds, knotted theta-curves in 3-manifolds. As it turned out, all these topological objects admit a prime decomposition, although in some cases it is not unique.

Bratteli diagrams are closely related to AF algebras and provide a convenient description of privileged states such as traces and Gibbs states. In view of applications to random walks on locally compact groups and groupoids, I shall define topological Bratteli diagrams. Markov measures will be identified as a class of quasi-invariant measures. The Poisson boundary of a random walk can be studied in this context. This is a work in progress with T. Giordano.

In this talk, I will present some recent developments in the theory of pseudo-differential operators on Lie groups. I will discuss first the case of R^n and the torus and then give a brief overview of the analysis in the context of Lie groups. I will conclude with some recent works developing pseudo-differential calculi on certain classes of Lie groups.

Marginally stable circular orbits, or MSCOs, play an important role in our understanding of astrophysical phenomena (e.g., matter configurations in accretion, motion around neutron stars). We derive a necessary condition for the existence of MSCOs for stationary axisymmetric spacetimes using, unexpectedly, a tool from algebraic geometry: resultants. This yields a concrete algorithm for determining MSCOs, which we demonstrate using several examples of physical interest. No prior knowledge of astrophysics or algebraic geometry is assumed and we shall provide definitions and discussion along the way.

Using the technique of holomorphic motions, we study the regularity of the limit set of the one-parameter family of holomorphic correspondences (w-c)^q=z^p, outlining some of the main contributions in the field in the last decades. This family is the simplest generalisation of the quadratic family z^2+c. In the quasi post-critically finite case, the limit set splits into a repeller and an attractor: the usual Julia set (closure of repelling periodic points) and the dual Julia set (closure of attracting periodic points). Conformal iterated function systems hidden in the dynamics of this correspondence appear naturally in the form of dual Julia sets. We also estimate the Hausdorff dimension of the Julia set using the formalism of Gibbs states.

Delay differential equations provide well-known models for many processes in
nature and technology. While models with constant delay are a reasonable and
much investigated starting point for many applications, it is also clear
that in reality delays fluctuate naturally and often can be influenced to
vary systematically. Despite its high, practical relevance, the consequences
of such time-varying delays is still poorly understood. I first introduce
into the general topic of delay systems and subsequently elaborate the
relevant developments in machining applications, such as turning and
milling. Finally I report on our own recent results, which reach from
applications in machining to fundamental aspects of systems with
time-varying delay.

I will report on some recent joint work with Yusuke Isono in which we investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras arising from arbitrary actions of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. I will first explain a general spectral gap rigidity result inside such crossed product von Neumann algebras. I will then show that group measure space factors arising from strongly ergodic essentially free nonsingular actions of bi-exact discrete groups on standard probability spaces are full, that is, they have no nontrivial central sequence. I will finally explain how to use recent results of Boutonnet-Ioana-Salehi Golsefidy (2015) to construct examples of group measure space type III factors which are full and of any possible type III$_\lambda$ with $0 < \lambda \leq 1$.

A deep theorem of Kirchberg showed that any separable exact C*-algebra admits an ambient nuclear C*-algebra.
In this talk, we investigate how can an ambient nuclear C*-algebra of a given C*-algebra be "tight".
For certain group C*-algebras, we construct surprisingly tight ambient nuclear C*-algebras.
This in particular gives the first examples of minimal ambient nuclear C*-algebras of non-nuclear C*-algebras.
For this purpose, we study generic behaviors of Cantor systems.

We will discuss the correspondence between certain difference algebras and subshifts of finite type as studied in symbolic dynamics. We will relate the difference algebra of a subshift of finite type to its C*-algebra and pose a few questions in this context.

Given a sequence of closed minimal hypersurfaces of bounded area and index, we
prove that the total curvature along the sequence is quantised in terms of the total curvature
of some limit hypersurface, plus a sum of total curvatures of complete properly embedded
minimal hypersurfaces in Euclidean space. This yields qualitative control on the geometry
and the topology of the hypersurfaces and thus for the class of all minimal hypersurfaces with
bounded index and area. This is joint work with Ben Sharp.

I will describe a joint work with Gabirele Mondello, where we study the following question: what are possible conical angles of a curvature one metric with conical singularities on S^2?

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In two-dimensional Minkowski space, integrable quantum field
theories can be constructed from a scattering function and a
corresponding inclusion of von Neumann algebras, related to quantum
fields localized in Rindler wedges. In this setting, the solution of the
inverse scattering problem (i.e. the construction of the field theory
from its scattering data) is intimately connected with the analysis of
the relative commutant of this inclusion.

This talk will focus on an explanation how this can be done with the
help of complex analysis - more precisely, by studying the decay rate of
the singular values of certain composition and restriction operators on
Hardy spaces over tube domains.

The theory of K-stability provides a means of studying the canonical metric problem on polarised varieties using methods of algebraic geometry. We give a short introduction of the theory and examples of its use. Test configurations are a central concept: A projective manifold is said to be K-stable if there does not exist a destabilising test configuration. We also introduce the notions of K-stability relative to a base variety, which recovers several known examples using elementary constructions. Finally, we define a convex combination operation on the set of test configurations.

Robin Forman’s discrete Morse theory is a powerful technique (at least as powerful as the smooth Morse is): it allows to compute homologies, cup-product, Novikov homologies, develop Witten’s deformation of the Laplacian, etc. In the talk we demonstrate how it works: we build a perfect discrete Morse function on the configuration space of a flexible polygon. The starting point of our construction is a cellulation of the moduli space of a planar polygonal n-linkage.

In the presence of such a space-like translation Killing field, the 3 + 1 vacuum Einstein equations reduce to the 2 + 1 Einstein equations with a scalar field. In generalized wave coordinates, Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty to prove global existence of solutions is due to the decay of free solutions to the wave equation in 2+1 dimensions which is weaker than in 3+1 dimensions. As in the work of Lindblad and Rodnianski, we have to rely on the particular structure of Einstein equations in wave coordinates. We also have to carefully chose an approximate solution with a non trivial behaviour at space-like infinity, and a well-suited wave gauge, to enforce convergence to Minkowski space-time at time-like infinity.

I will review some recent progress in the black hole stability problem including a proof of the linear stability of the Schwarzschild spacetime under gravitational perturbations (joint work with Dafermos and Rodnianski).

There are several attempts to construct C*-algebras from number field, and those constructions give an interesting family of non-simple C*-algebras.
Bost-Connes C*-algebra is the origin of those attempts, which is constructed using class field theory.
It turned out that the structure of primitive ideals of Bost-Connes C*-algebras is related to primes of original number field.
In this talk, I would like to explain the relation and an application to classify those C*-algebras.

The construction of operator algebras from groups goes back to the foundational work of Murray and von Neumann. Rigidity asks how much of the group is remembered by the operator algebra. The last 5 years have seen dramatic progress in the setting of von Neumann algebras with the first von Neumann rigid groups being constructed by Ioana, Popa and Vaes. I'll review these results, and then discuss the setting of C*-algebras, giving examples of non-abelian torsion free C*-rigid groups. This is joint work with Søren Kundby and Hannes Thiel.

This talk is about Cartan subalgebras in C*-algebras, and continuous orbit equivalence for topological dynamical systems. These two notions build bridges between operator algebras, topological dynamics, and geometric group theory. Moreover, we explore rigidity phenomena for continuous orbit equivalence. Along the way, we discuss continuous cocycle rigidity for topological dynamics.

I will explain a recent joint result with Aaron Tikuisis and Stuart White, saying that faithful traces on separable nuclear C*-algebras which satisfy the universal coefficient theorem are quasidiagonal. This confirms Rosenberg’s conjecture that discrete amenable groups have quasidiagonal C*-algebras. It also resolves the Blackadar-Kirchberg problem in the simple UCT case. Moreover, there are several consequences for Elliott’s classification programme; in particular, the classification of separable, simple, unital, nuclear, Z-stable C*-algebras with at most one trace and satisfying the UCT is now complete; the invariant in this case is ordered K-theory.

In many cases the construction of a C*-algebra from an associated algebraic or geometric object involves making an arbitrary choice of Hilbert space (satisfying certain criteria) and considering operators on that Hilbert space possessing properties determined by the algebraic or geometric structure.

A C*-category is a generalisation of a C*-algebra in the same way that a groupoid is a generalisation of a group. An extension of the GNS-construction and the associated Gelfand-Naimark Theorem tells us that they are precisely the norm-closed, *-closed subcategories of the category of all Hilbert spaces and bounded linear maps between them. In cases such as outlined above, it is more natural to construct a C*-category rather than a C*-algebra, which amounts to considering all suitable Hilbert spaces at once.

This talk is meant as an introduction to C*-categories and an overview of the basic theory. I will demonstrate how C*-categories form the bridges described in the title, using as examples groupoids - both discrete (algebra) and topological (geometry). I will also say a little about how we can use Banach bundles to provide a formal characterisation of "continuous C*-category" and describe how this relates to Fell bundles over topological groupoids. Time permitting, I will also say something about the construction of K-theory for C*-categories.

I will report on joint work with Luna Lomonaco (Sao Paolo). The classical Mandelbrot set M is the subset of parameter space for which the Julia set of the quadratic polynomial z^2 + c is connected. Two analogous connectivity loci are M_1 for the family of rational maps of the form z+1/z+A (containing the matings of z^2+c with z^2+1/4) and M_corr for the family of quadratic holomorphic correspondences which are matings between polynomials z^2+c and the modular group PSL(2,Z).

Theorem 1 (SB+LL, 2015): M_corr is homeomorphic to M_1.

In our 1994 article introducing the matings between z^2+c and the modular group, Chris Penrose and I conjectured that M_corr is homeomorphic to the classical Mandelbrot set M. By Theorem 1 this becomes equivalent to the well-supported conjecture that M_1 is homeomorphic to M.

In the talk I will outline the main steps in the proof of Theorem 1, focussing in particular on a new Yoccoz inequality for the family of correspondences.

We sketch a very short proof for the index theorem by Baum-Douglas-Taylor and explain how this implies the index theorems by Kasparov and Atiyah-Singer.

In many areas of geometry and physics we often require that the manifolds we work with carry a spin structure, that is a lift of the structure group of the tangent bundle from SO(n) to its simply connected cover Spin(n). In string theory and in higher geometry the analogue is to ask for a string structure; this is a further lift of the structure group to the 3-connected group String(n). Waldorf has given a way to describe string structures in terms of bundle gerbes (which are the abelian objects in higher geometry—a sort of categorification of a line bundle). Unfortunately, explicit examples are lacking. In this talk I will explain how all this works and give some examples of such structures. I will also explain some current work in progress on the geometry of string structures. This is joint work with David Roberts.

Teichmüller harmonic map flow is a gradient flow of the Dirichlet energy which is designed to evolve parametrised surfaces towards critical points of the Area.
In this talk we will discuss how to flow cylindrical surfaces in Euclidian with given boundary curves to a solution of the Douglas-Plateau-Problem of finding a minimal surface that spans the two given boundary curves.

A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy The Nguyen.

Integrable harmonic maps have provided deep insight into important mathematical and physical problems, appearing in settings ranging from classical complex analysis to supergravity. In this talk, we shall answer the question "when is a harmonic map integrable?" by providing a theorem for a class of harmonic maps having noncompact symmetric space targets. An immediate corollary recovers classical results of Zakharov and Belinski/Alekseev for the stationary, axisymmetric Einstein vacuum/Einstein-Maxwell equations.

In conjunction with previous work on compact targets by Uhlenbeck and Terng, the techniques may illuminate our understanding of other geometric field theories as well as suggest the first inroads in answering the Xanthopoulos conjecture.

This is joint work with S. Tahvildar-Zadeh.

We know from the recent results of Kahn and Markovic that every compact hyperbolic 3-manifold
has a finite cover with a very special structure, fibering over the circle with fibre a compact surface of genus at least 2. I will discuss these manifolds and explain why that their automorphism groups are a very restricted class of finite group. This contrasts strongly with the situation for hyperbolic surfaces.

I will present a new compactness theorem for minimal hypersurfaces embedded in a closed Riemannian manifold N^{n+1} with n<7. When n=2 and N has positive Ricci curvature, Choi and Schoen proved that a sequence of minimal hypersurfaces with bounded genus converges smoothly and graphically to some minimal limit. A corollary of our main theorem recovers the result of Choi-Schoen and extends this appropriately for n<7.

Various noncommutative generalisations of dimension have been considered and studied in the past decades. In recent years certain new dimension concepts for noncommutative C*-algebras, called nuclear dimension and a related dimension concept for dynamical systems, called Rokhlin dimension have been defined and studied. They play an important role in the classification programme. The theory is geared towards the class of nuclear C*-algebras and generalises the concept of covering dimension, in case of dynamical systems a type of equivariant covering dimension of topological spaces with a group action. There are interesting connections between coarse geometry and Rokhlin dimension. We will give an introduction to these concepts and survey some applications and connections between them.

(in collaboration with Hirshberg, Szabo, Winter, Wu)

In the past four decades, the theory of integrable partial differential equations has had a rich and varied impact on both mathematics and physics. We shall survey the broad reach of integrability techniques into other mathematical disciplines through concrete examples, ranging from construction of singular solutions to Einstein’s Equations (using harmonic maps into symmetric spaces) to description of shallow-water wave interactions by way of combinatorial structures (such as Grassmann necklaces and Young diagrams). Recent joint work and open questions will be toured along the way.

This talk is intended for non-specialists and graduate students are also welcome.

Seminar series: 
Geometry and Analysis