Time: Tuesday at 3pm (except special events)
Location: Mathematics Building, Room MB-503
Organisers: Arick Shao and Huy Nguyen. Please e-mail us in case you have any questions or want to give a talk yourself.
Abstract: Taking cue from the group of automorphisms of the open unit disk, Sz.-Nagy and Foias constructed a complete unitary invariant for a contractive operator. The relation was that a contractive operator (henceforth a contraction) on a Hilbert space has its spectrum in the closed unit disk. Using this fact, they constructed a function with its values in a certain Banach space. This function turned out to be the inavariant for a certain class of contractions (not for all contractions, for obvious reasons which will be explained in the talk).
In recent times, there has been a great deal of activity in domains more general than the disk, the unit ball in the d-dimensional complex space for example, or the polydisk. This involves tuples of operators rather than a single contraction. The connection with multivariable complex analysis is fascinating. An old theorem of Schur comes in naturally.
So, in a general domain, one could consider any positive definite kernel and a tuple of operators suited to the kernel. Is there a complete unitary invariant? If so, for which class of tuples of operators? Such are the questions which will be addressed in this talk.
The talk will be self contained with no prerequisite except basic knowledge of Hilbert space operators.
Rado introduced the following `lion and man' game in the 1930's: two
players (the lion and the man) are in the closed unit disc and they
can each run at the same speed. The lion would like to catch the man
and the man would like to avoid being captured.
This problem had a chequered history with several false proofs before
Besicovitch finally gave a correct proof.
We ask the surprising question: can both players win?
Professor Chatterji was a colleague of de Rham in Laussane and has kindly made available his article on de Rham at the website http://www.maths.qmul.ac.uk/~cchu
The talk will be an anlytic introduction to (very) basic Lie theory. I will focus on just matrix groups, using the matrix exponential to formulate the idea of Lie algebras and then to build up a statement of the Baker-Campbell-Hausdorff formula.
Non-associative Banach algebras play an important role in many areas of modern mathematics and science, e.g. Biology, Physics,
Cellular Automata and Cryptography. Nevertheless, a non-associative spectral theory has yet to be developed. Indeed, there is no clear idea what the spectrum of an element in a non-associative Banach algebra should mean.
The aim of this talk is to provide an overview of the situation and also to show a way to extend the classical spectral theory to the non-associative setting. We also discuss applications of such a theory to the problem of automatic continuity of homomorphisms.
In this talk we will discuss the following long standing and fundamental problem: Given an operator on a separable Hilbert space (with an orthonormal basis), can one compute/construct its spectrum from its matrix elements. As we want such a construction to be useful in application (i.e. implementable on a computer), we restrict ourselves to only allowing the use of arithmetic operations and radicals of the matrix elements and taking limits. We will give an affirmative answer to the question, and also introduce a classification tool for the complexity of different computational spectral problems, namely, the Solvability Complexity Index.
In the Euclidean plane, it is easy to determine the smooth or Schwartz functions which are invariant under rotations, using the Fourier transform, which also gives a characterisation of the multipliers in the Laplace operator. Similar characterisations are valid for any action of a compact group on a Euclidean space by the G. Schwarz Theorem.
In this talk, I will present a study of this question in another
setting: the Heisenberg group under the action of the unitary group as
well as more general nilpotent Gelfand pairs. This is a joint work
with Fulvio Ricci and Oksana Yakimova.
We shall use a theorem of probability to prove a geometrical result, which when applied in an analytical context yields an interesting and surprisingly strong result in combinatorics on the existence of long arithmetic progressions in sums of two sets of integers.This is joint work with Ernie Croot and Izabella Laba.
Pseudo-differential operators (PDO's) are primarily defined in the familiar setting of the Euclidean space. For four decades, they have been standard tools in the study of PDE's and it is natural to attempt defining PDO's in other settings. In this talk, after discussing the concept of PDO's on the Euclidean space and on the torus, I will present some recent results and outline future work regarding PDO's on Lie groups as well as some of the applications to PDE's. This will be a joint work with Michael Ruzhansky (Imperial College London).
I will discuss a problem of Kolmogorov concerned with
the epsilon-entropy of classes of analytic functions.
In one complex variable, this problem was solved
in the 1960s using classical potential theory. In several
complex variables it was shown in the 1980s that this problem is
equivalent to a certain problem in pluripotential theory, now called
Zahariuta's conjecture.
In this talk I will discuss this conjecture and outline a strategy of proof.
Unital operator algebras are characterized up to complete isometry using only the holomorphic structures of the associated Banach spaces. This is a joint work with Matthew Neal.
This talk concerns the applications of Hilbert's metric in operator theory and in dynamics of nonlinear operators.
We shall present some recent solutions to problems which have
been open for over twenty ve years. We refer to the problems of
describing the norm-closed faces of the (closed) unit ball of a JB-triple
E and the weak-closed faces of the closed unit ball of E. Around
twenty three years ago, C. Akemann and G.K. Pedersen described the
structure of norm-closed faces of the unit ball of a C-algebra A, and
the weak-closed faces of the unit ball of A, in terms of the \compact"
partial isometries in A. Three years earlier, C.M. Edwards and G.T.
Ruttimann gave a complete description of the weak-closed faces of
the unit ball of a JBW-triple, and in particular, in a von Neumann
algebra. However, the question whether the norm-closed (respectively,
weak-closed) faces of the unit ball in a JB-triple E (respectively, E)
are determined by the \compact" tripotents in E has remained open.
We shall survey the positive answers established by C.M. Edwards, F.J.
Fernndez-Polo, C. Hoskin and the author of this talk in recent papers.
We introduce a data-based approach to estimating key quantities
which arise in the study of nonlinear control systems and random nonlinear
dynamical systems. Our approach hinges on the observation that much of the
existing linear theory may be readily extended to nonlinear systems - with
a reasonable expectation of success - once the nonlinear system has been
mapped into a high or infinite dimensional Reproducing Kernel Hilbert
Space. In particular, we develop computable, non-parametric estimators
approximating controllability and observability energy functions for
nonlinear systems, and study the ellipsoids they induce. It is then shown
that the controllability energy estimator provides a key means for
approximating the invariant measure of an ergodic, stochastically forced
nonlinear system. We also apply this approach to the problem of model
reduction of nonlinear control systems. In all cases the relevant
quantities are estimated from simulated or observed data. These results
collectively argue that there is a reasonable passage from linear dynamical
systems theory to a data-based nonlinear dynamical systems theory through
reproducing kernel Hilbert spaces. This is joint work with J. Bouvrie (MIT).
In a talk in the first semester entitled "Matings and discreteness
in holomorphic dynamics" I discussed various examples of "matings"
but did not have time to address the second topic. Today I will
talk about what we might mean by "discreteness" in the context of
actions of holomorphic systems on the Riemann sphere, and investigate
the "discreteness locus" for certain parameterised families of
Kleinian groups and holomorphic correspondences.
When looking at classification results of Jordan algebras and superalgebras, one would notice that "new examples" appear in simple Jordan superalgebras that do not have a counterpart in algebras. Of special interest is the case of prime characteristic and non-semisimple even part. There are also several important differences between the representation theory of Jordan algebras and that of Jordan superalgebras.
The aim of this talk is to offer a general view of Jordan superalgebras, the
classificantion results and representation theory, emphasizing similarities and differences in the behaviour of algebras and superalgebras.
Amenability of a Banach algebra may be thought of as an
infinite-dimensional replacement for certain splitting properties that are fundamental to the study of finite-dimensional algebras. If H is a Hilbert space, then by deep work of several authors, we know exactly which self-adjoint subalgebras of B(H) are amenable. In particular, all commutative self-adjoint subalgebras are amenable.
This last statement is false if we drop the words "self-adjoint", and it has been an open problem for many years now to characterize the commutative amenable subalgebras of B(H). In this talk, I will present some of the background to this problem, and try to give an overview of the known results to date, obtained in papers of Sheinberg, Curtis, Loy, Willis, Gifford, Marcoux, and myself.
In this talk, we will explain how Perelman's entropy functional for the Ricci Flow can be used
to give a proof of Hamilton's conjecture, stating that so-called "Type I singularity models" are
gradient shrinking solitons. Our proof, obtained in joint work with Carlo Mantegazza, combines
geometric ideas with new analytic estimates such as new Gaussian heat kernel bounds on evolving
manifolds. While this is formally a continuation of our more introductory talk in the Pure Maths
Colloquium on Monday, October 14, we attempt to make this lecture completely self-contained.
We consider surfaces conformally immersed in R^3 with L^2 bounds on the norm of the second
fundamental form. In particular we will study the Liouville equation for such surfaces and give an extension
of the Classical Gauss-Bonnet formula for surfaces and study its behaviour under conformal transformations
of Euclidean space.
We will then classify certain limit cases of these bounds, for example we will suitably generalise Osserman’s
classification of complete non-compact minimal surfaces with total curvature equal to 8\pi to the case of
complete non-compact surfaces with total bounded curvature.
We introduce the class of n-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the
open unit disc into the symmetrised bidisc . We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater.
We introduce a sequence $C_n (n \geq 0)$ ; of necessary conditions for solvability, prove that they are of strictly increasing strength and show that $C_{n-3}$ is insufficient for the solvability of an n-point problem for $ n \geq 3$.
We introduce a classification of rational $\Gamma$-inner functions, that is, analytic functions from the disc into $\Gamma$ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of $\Gamma$. The classes are related to n-extremality and the conditions $C_\nu$; we present numerous strict inclusions
between the classes. The talk is based on a joint work with Jim Agler and N. J. Young.
Seminar series:
Geometry and Analysis
In this talk I will first motivate and explain the definition of quantum automorphisms of finite dimensional
C*-algebras, leading to compact quantum groups in the sense of Woronowicz. In the second part of the talk I will explain a general strategy how to compute their K-theory using methods from the Baum-Connes
conjecture.
In this talk we focus on the fact that the map induced by a cpc order zero map in the category Cu does not preserve the compactly containment relation. In particular, these kinds of maps are not in the category Cu, so that in general, they may not be used in the classification of C*-algebras via the Cuntz Semigroup. Nevertheless, there is a subclass of these maps which preserves the relation, and so they can be used in the above mentioned classification. Our main result characterizes these maps via the positive element induced by the description of cpc order zero maps shown by Winter and Zacharias.
In this talk I will give a brief overview of methods for the analysis of global solutions to the equations of General Relativity -- the Einstein field equations. In particular, I will discuss how the notion of conformal transformations can be used to rephrase questions about global existence into questions of local existence of solutions to the Einstein field equations. I will exemplify this method with the proof of the non-linear stability of the de Sitter spacetime. This talk is aimed at non-specialists.
Transfer operators play an important role in the study of chaotic dynamical systems. Spectral properties of these operators yield insight into dynamical and geometric invariants of the underlying system. In this talk I will focus on transfer operators associated with dynamical systems with holes and will discuss a recent result with H.H. Rugh on the regularity of the leading eigenvalue as a function of hole size and position.
Mean curvature flow (MCF) is a deformation of the area of hypersurfaces in the steepest way. The entropy of a hypersurface is the supremum of the Gaussian surface area of all translates and scalings of the hypersurface. It is monotone decreasing under MCF and so indicates important information about the dynamics of the flow. In this talk, we will use weak MCF to show that the round sphere uniquely minimizes the entropy of closed hypersurfaces up to dimension six. This is joint work with Jacob Bernstein.
After a short introduction to normal form problems of analytic vector fields we will give an answer to the following problem. Let S be a homogeneous polynomial vector field and let X be an analytic perturbation of S by higher order terms. If X is formally conjugated to S, is it also analytically conjugated to it? When S is linear (and "diagonal"), the answer is due to Siegel and involves the analysis of the so called "small divisors".
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.
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)
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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).
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.
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.
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.
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.
AttachmentSizeAbstract [PDF 28KB]28.83 KB
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?
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.
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.
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.
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 $0 < \lambda \leq 1$.
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.
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.
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.
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.
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 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.
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.
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.
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 present a new (2016) local result for the Ricci flow and explain the proof thereof. Joint work with Peter Topping.
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.
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.
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.
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.
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.
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.
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.
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 a sharp one-sided curvature estimate for the mean curvature flow and some applications, in particular to ancient solutions of the flow.
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.
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).
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).
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
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) .
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.
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.
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.
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 percisely 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.
1
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
(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.
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.
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.
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.
The construction of initial data for the Cauchy problem in General Relativity is an interesting problem from both the mathematical and phys- ical points of view. As such, there have been numerous methods studied in the literature —the “Conformal Method” of Lichnerowicz–Choquet- Bruhat–York and the “gluing” method of Corvino–Schoen being perhaps the best-explored. In this talk I will describe an alternative, perturbative, approach proposed by A. Butscher and H. Friedrich, and show how it can be used to construct non-linear perturbations of initial data for spatially- closed analogues of the “k = −1” FLRW spacetime. Time permitting, I will discuss possible refinements/extensions of the method, along with its generalisation to the full Conformal Constraint Equations of H. Friedrich.
This is a two-day mini-conference on geometric analysis and mathematical relativity. See here for details.
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.
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.
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)
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.
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.
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.
(This seminar is held jointly with the Relativity and Cosmology Seminar)
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).
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.
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)
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.
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.
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.
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 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].
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:
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.
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: 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).
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)
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 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.
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.
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.)
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.)
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.
(This seminar is held jointly with the Relativity and Cosmology Seminar)
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.
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
TBA
Mini-workshop website: http://www.maths.qmul.ac.uk/~shao/events/ws2020
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.
Inversions of complete minimal surfaces with finite total curvature in three-space are known to be critical points of the Willmore energy, or of the integral of mean curvature squared, a conformal invariant first studied by Poisson and Sophie Germain in the beginning of the 19th century. Furthermore, Bryant showed that Willmore immersions of genus 0, also called Willmore spheres, are all inversions of minimal surfaces. More generally, we extended Bryant’s result and showed in particular with Tristan Rivière that branched Willmore spheres arising as weak limits of bubbles of immersions are conformally minimal. We will show that the Morse index of conformally minimal Willmore surfaces in three-space is equal to the index of a canonically associated matrix whose dimension is equal to the number of ends of the dual minimal surface.
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.
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.
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.
(This Geometry and Analysis Webinar is held online on MS Teams.)
(This Geometry and Analysis Webinar is held online on MS Teams.)
(This seminar is held jointly with the Relativity and Cosmology Seminar)