Internal School Colloquium

Behrang Noohi: What is derived geometry? 

You may have heard of "derived/higher geometry" and wondered what it is. A brainchild of Alexander Grothendieck, it is more of a "new viewpoint in mathematics" rather than being a specific subject. A recent flurry of activities, aiming to set up the right foundations, has seen the subject transform enormously, resulting in spectacular applications to algebraic geometry, symplectic geometry, topology, physics, computer science, etc. Despite the daunting technical apparatus involved, the main ideas are simple (at least some of them!). In this talk I will try to illustrate some of these ideas using simple examples.

Abhishek Saha: Bounding the heights of peaks of vibration modes on hyperbolic membranes

I will give a gentle introduction to the "sup-norm problem" in a setting where number theory plays a key role. The sup-norm problem asks for non-trivial bounds on the sup-norms of eigenfunctions of the Laplacian on Riemannian manifolds. In the special case when the manifold is a surface of constant negative structure, and is constructed from "quaternion algebras", a famous result of Iwaniec and Sarnak improves upon the trivial bound using number-theoretic techniques. I will explain this result, and then talk about recent progress on an analogous question where the underlying surface is itself allowed to vary.

Mark Jerrum (1:00-1:30)

Title: Developments in rejection sampling

Abstract:  I’ll attempt to ingratiate myself with the maximum number of people by touching on rejection sampling, the Lovász Local Lemma, the Tutte polynomial, etc.  The talk will include joint work with Heng Guo, lately a postdoc in the School.

Justin Ward (1:30-2:00)

Title: Approximation, Hardness, and Proof Theory

Abstract: In this talk, I will provide a brief overview of the main ideas underpinning the modern field of “approximation algorithms.” From a practical perspective, approximation algorithms can be viewed as a way of coping with computationally “hard” problems. More generally, however, the study of so-called “approximability” is about developing a finely-grained notion of what it means for a problem to be “hard.” This notion relies on a collection of fundamental results linking proof theory, computation, and optimisation, which I will briefly describe and discuss.

Martin Benning: What do you meme? Nonlinear scale-space methods in practice. (1:05-1:30)

Abstract: We discuss PDE-based approaches for decomposing signals into components with features at different scales. Starting with linear scale-space and inverse scale-space methods, we see shortcomings when using those to decompose structures with discontinuities. We use this as a motivation for considering nonlinear inverse scale-space methods and their extensions to spectral decomposition methods. Those methods allow to obtain more suitable decompositions of signals with discontinuities, and we show how this can be exploited in a variety of applications in image-processing. We particularly focus on the creation of memes via facial image fusion.

Primoz Skraba: Stability in Applied Topology. (1:30-1:55)

Abstract: The goal of this talk is to give a brief overview of applied topology, introducing persistence and various notions of stability. I will try to explain the type of problems people in the field are working on and how it intersects with other areas of mathematics - including probability, algebra, and algorithms.

Weini Huang: Population dynamics under tradeoffs in an evolving Lotka-Volterra systems (11:05-11:30)

Abstract: Biological systems including cancer are complex systems with interactions of individuals with different traits. Often those traits are linked and the optimization of all traits independently is impossible due to the existence of trade-offs. For example, energy or resources allocated to survival is not available for reproduction. Rather than having a Lotka-Volterra (predator-prey) system with predefined variables, we developed a stochastic model capturing random mutations. We use our model to understand the role of survival-reproduction trade-offs in lab experiments. We found that the shape of trade-offs evolves in experiments as we assumed, and thus could impact the population diversity level as predicted by our model. Indeed, we see first glimpses of these predictions in on-going experimental studies. In addition, the importance of evolving trade-offs may also apply in the evolution of cancer resistance and open an opportunity to control resistant subpopulations with specifically designed evolutionary treatment strategies.   

Silvia Liverani: Clustering with mixture models. (11:30-11:55)

Abstract: I will introduce mixture models and their use for identifying the presence of subpopulations within a population. I will then introduce the basic components of a specific Bayesian mixture model called profile regression and conclude by discussing a few application areas for this type of models. 

Ginestra Bianconi: Emergent Hyperbolic Network Geometry and Dynamics (11:05-11:30)

Abstract: Simplicial complexes naturally describe discrete topological spaces. When their links are assigned a length they describe discrete geometries. As such simplicial complexes have been widely used in quantum gravity approaches that involve a discretization of spacetime. Recently they are becoming increasingly popular to describe complex interacting systems such a brain networks or social networks.

After a brief introduction in this talk we present non-equilibrium statistical mechanics approaches to model large simplicial complexes and we will explore the hyperbolic nature of their emergent geometry.

Moreover we will investigate how the dimension of these simplicial complexes affects their stochastic topology and their dynamics (synchronization and topological percolation/k-connectedness).

Reto Buzano: Mean Curvature Flow and Embedded Spheres (11:30-11:55)

Arick Shao: Control of Wave Equations (13:30-13:55)

Abstract: We discuss the question of whether solutions of a PDE on a finite domain can be controlled, through either its boundary data or its forcing term. Moreover, we focus mainly on wave and hyperbolic equations, where finite speed of propagation puts fundamental constraints on when such control is possible. Here, we give a brief survey of the main techniques and results in this area, and we conclude with some novel results for wave equations on time-dependent domains with moving boundaries.

Boris Khoruzhenko: How many stable equilibria will a large complex system have? (13:05-13:30)

Abstract: In the first part of my talk I will give a bird’s eye view of random matrices to introduce basic concepts. And in the second part of my talk I plan to focus on a recent application of random matrices to the question of stability of large complex systems, extending the analytic study of Robert May (1972) from linear to non-linear systems. This question turns out to be rich on interesting open problems, which I would like to share with you if time permits.

Anna Maltsev: Intracellular calcium signalling and the Ising model (13:30-13:55)

Abstract: Intracellular Ca signals represent a universal mechanism of cell function. Messages carried by Ca are local, rapid, and powerful enough to be delivered over the thermal noise. A higher signal to noise ratio is achieved by a cooperative action of Ca release channels arranged in clusters (release units) containing a few to several hundred release channels. The channels synchronize their openings via Ca-induced-Ca-release, generating high-amplitude local Ca signals known as puffs in neurons and sparks in muscle cells. We construct an exact mapping of such molecular clusters to an Ising model and establish an h-beta phase diagram for signal termination. This is joint work with Prof. Stern's laboratory at the National Institutes of Health.  

Mira Shamis: Applications of discrete Schroedinger equations to the standard map. (1:05-1:30)

Abstract: We shall discuss the Chirikov standard map, an area-preserving map of the torus to itself in which quasi-periodic and chaotic dynamics are believed to coexist. We shall describe how the problem can be related to the spectral properties of a one-dimensional discrete Schroedinger operator, and present a recent result. Based on joint work with T. Spencer.

Stephen Muirhead: The geometry of the level sets of smooth planar Gaussian fields. (1:30-1:55)

Abstract: Gaussian fields are random functions on R^d whose finite-dimensional projections are multivariate Gaussians; they are a widely-used model for spatial noise in physics, astronomy, oceanography, medical imaging etc. One way to analyse a Gaussian field is to consider its level sets (think of the `contour lines' on a map). While `local' geometric functionals of level sets are a classical topic in probability (going back to the work of Kac and Rice in the 1940s on the zeros of Gaussian processes), the study of `non-local' geometric functionals is less well-understood. In this talk I will present an overview of recent work studying two such `non-local' functionals in the case of smooth planar Gaussian fields: (i) the number of connected components of the level sets in large domains, and (ii) the existence of a large `percolating' connected component. Joint work with D. Beliaev, M. McAuley, A. Rivera, H. Vanneuville and I. Wigman.

Christian Beck: Superstatistical methods for complex systems. (1:00-1:30)

Abstract: The superstatistics concept, introduced some 16 years ago in [1], is a useful general method borrowed from statistical physics to describe driven nonequilibrium systems in spatio-temporally inhomogeneous environments that exhibit fluctuations of one or several intensive parameters. The method can be quite generally applied to heterogeneous complex systems if there is time scale separation of the underlying dynamics. After a brief introduction to the basic ideas, I will concentrate onto three examples of useful recent applications, namely acceleration statistics of tracer particles in turbulent flows [2], the measured momentum statistics of cosmic ray particles [3] and the statistics of frequency fluctuations in power grid networks. The fluctuating consumer demand and trading patterns in electricity markets, as measured by tiny frequency deviations from 50 Hz in various European, American and Asian power grids, appear to be well-described by superstatistical models taking into account the growing fraction of renewable energy generation [4].

References

[1] Beck, C., & Cohen, E.G.D. (2003). Superstatistics. Physica A, 322, 267.

[2] Beck, C. (2007). Statistics of 3-dimensional Lagrangian turbulence. Phys. Rev. Lett., 98, 064502.

[3] Yalcin, G.C., & Beck, C. (2018). Generalized statistical mechanics of cosmic rays: Application to positron-electron spectral indices. Scientific Reports, 8, 1764.

[4] Schaefer, B., Beck, C., Aihara, K., Witthaut, D., & Timme, M. (2018). Non-Gaussian power grid frequency fluctuations characterized by Levy-stable laws and superstatistics. Nature Energy, 3, 119.

Vincenzo Nicosia: Dynamic approaches to measure heterogeneity in spatial networks. (1:30-1:55)

Abstract: Spatial networks are often the most natural way to represent spatial information of different kinds. One of the outstanding problems in current spatial network research is to effectively quantify the heterogeneity of the discrete-valued spatial distributions underlying a spatial graph. In this talk we will presentsome recent alternative approaches to estimate heterogeneity in spatial networks based on simple dynamical processes running on them.

Matt Fayers: Iwahori-Hecke algebras of the symmetric group. (1:05-1:30)

Abstract: The Iwahori-Hecke algebra of the symmetric group is a deformation of the group algebra which arises in various ways.  I'll give a brief survey of some of these.

Felipe Rincon: Tropical Ideals. (1:30-1:55)

Abstract: Tropical ideals are combinatorial objects that encode algebraic information in tropical geometry. They can be thought of as combinatorial generalizations of the possible collections of subsets arising as the supports of all polynomials in an ideal. I will introduce and motivate these objects, and talk about recent work studying some of their main properties.

Felix Fischer: Prophet Inequalities from Samples (1:05-1:30)

Abstract: The theory of optimal stopping is concerned with situations where information becomes available over time and irrevocable decisions have to be made based only on partial information. Two well-know stopping problems are the secretary problem and the prophet problem. In the secretary problem we are presented with the elements of a set of arbitrary values in random order and want to maximize the probability of selecting the largest value. We can select only one of the values, and if we choose not to select a particular value it is lost forever. It turns out that we can guarantee a probability of 1/e by discarding an initial 1/e fraction of the values and then stopping at the first value that exceeds all previous values, and this is best possible. In the prophet problem values are drawn independently from known distributions and the goal is to maximize the value selected relative to the maximum value in hindsight. Here it is possible to guarantee half of the maximum value in expectation, and this can be improved to a 0.745 fraction if the values are identically distributed. We ask what happens when values are drawn independently from the same distribution, but we don't know what that distribution is. Based on joint work with José Correa, Paul Dütting, and Kevin Schewior.

Robert Johnson: Voronoi Games in the Hypercube (1:30-1:55)

Voronoi games model a form of facility location problem in which individuals position themselves in competition for some spatially distributed resource. A classical result in this area is the Median Voter Theorem which describes how candidates compete for vote share in a society whose opinions can be expressed by points in a 1-dimensional interval.

We investigate some discrete Voronoi games in which the underlying space is the discrete hypercube. This is a natural context for analogues of the Median Voter Theorem (in an opinion space corresponding to d binary issues). This discrete model has been much less studied than the continuous ones and leads to some appealing problems in the combinatorics of the hypercube. We exhibit some intriguing behaviour, results and open questions.

Joint Work with Nicholas Day

Alex Clark: The dynamics of tilings (1:05)-(1:30)

Abstract: We will examine how one can introduce a topological and dynamical structure on a tiling that reflects some of the important characteristics of the tiling. We will focus on tilings obtained by a substitution rule, such as the Penrose tiling, and explore their connection with attractors of maps of manifolds.

Vito Latora:  Simplicial models of social contagion (1:30-1:55)

Abstract: Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems.

Michael Farber, Topology and automated decision making

Abstract.  I shall explain how algebraic topology and in particular cohomology theory help to analyse algorithms for automated decision making in situations when outcome of the algorithm is a choice made from a continuum of possibilities (rather than from a discrete set of values). This theory applies to algorithms for robot motion planning in engineering, algorithms for coordination of computations in distributed computing and algorithms for aggregation of personal preferences and reaching consensus in social choice theory. 

Mahdi Godazgar, Charges and symmetry in gravity

Abstract.  I will start with Noether's theorem, which relates conserved charges to global symmetries, and explain why it fails in the presence of gravity.  I will explain how these problems can be overcome and finish with recent work of mine on finding new charges.

Moment sequences in combinatorics (Natasha Blitvic)

Take your favorite integer sequence. Is this sequence a sequence of moments of some probability measure on the real line? We will look at a number of interesting examples (some proven, others merely conjectured) of moment sequences in combinatorics. We will consider ways in which this positivity may be expected (or surprising!), the methods of proving it, and the consequences of having it. The problems we will consider will be very simple to formulate, but will take us up to the very edge of current knowledge in combinatorics, ‘classical’ probability, and noncommutative probability.

Numerical relativity and the behaviour of fundamental fields around black holes (Katy Clough)

In this talk I will describe how we can use numerical simulations to model spacetimes in general relativity, and how these simulations can help us learn about the behaviour of fundamental fields around black holes.

Analytic number theory with a perspective of representation theory (Subhajit Jana)
I will talk about certain problems in classical analytic number theory, and how they connect with various topics in e.g. homogeneous dynamics, and Diophantine approximation. I will also discuss how these problems may be approached using various tools in (automorphic) representation theory. Finally, I will speak about some recent progress on these problems.

Automatic sequences in dynamics and number theory (Reem Yassawi)
A finite automaton is a simple model of a computer. An automatic sequence is obtained by feeding the base-q representations of natural numbers into such an automaton. These sequences originate in theoretical computer science, but Cobham’s characterisations give them a dynamical flavour, and as a result a rich research area has developed around them in ergodic theory. At the same time Furstenberg and Christol gave algebraic characterisations of automatic sequences, which have led to applications in number theory. I will discuss some recent results and questions in both settings.

Stochastic dynamics of cancer driven by random partition of extra-chromosal DNA elements (Weini Huang)

Extra-chromosomal DNA (ecDNA) is the product of a random rare event when DNA pieces falling out of chromosomes and forming a circle DNA structure during cell division. Recent studies on the largest tumour databases showed a high prevalence of ecDNA elements across various cancer types. Tumours with ecDNA are on average more aggressive and lead to a worse survival probability of patients. A unique feature of ecDNA is the random partition of ecDNA copies during cell division, which differs from the normal “rules” of chromosomal inheritance. Thus, classical theory on cancer biology such as most population genetics models does not apply to capture ecDNA dynamics. Instead, we built a stochastic model based on the random partition of ecDNA and analysed the evolution of ecDNA copy number distribution over time. Our results showed that random partition leads to a fast evolution and high heterogeneity of ecDNA distribution in cell populations. With collaboration with experimental and clinical groups, we validate our theoretical analysis and computational predictions in tumour cell lines and patient data.

Polygonal billiards (Oscar Bandtlow)

Billiards are a class of dynamical systems given by a point particle moving inside a planar domain with elastic collisions at the boundary of the domain. In this talk I will provide an overview of the case where the domain is a polygon, which, in spite of the fact that it has been studied in considerable detail for a long time, is still not completely understood. In particular, I will report on recent work with Katerina Zahradova, Julia Slipantschuk and Wolfram Just which settles a long-standing open problem in this area.

Speaker 1: Lennart Dabelow

Title: Thermodynamic nature of irreversibility in active matter

Abstract: Active matter is a collective term for systems whose constituents can convert energy from their surroundings into directed motion. Suspensions of such micro- to nanometer-sized active particles -- like bacteria, colloids with catalytic surfaces, or nanorobots -- are inherently out of equilibrium and exhibit highly stochastic dynamics because they are strongly affected by thermal fluctuations. For similar passive, equilibrium systems, the second law of thermodynamics and more recent fluctuation theorems link the dynamical irreversibility to thermodynamic entropy changes, establishing a thermodynamic "arrow of time". We reveal that for large classes of active matter, too, irreversibility is a function of the thermodynamic system parameters. Interestingly, the pertinent thermodynamic counterpart of irreversibility is not entropy, but pressure.

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Speaker 2: Eftychia Solea

Title: Functional graphical modeling

Abstract: Modern applications produce large, complex and highly-structured data such as multivariate functional data. For example, in medical imaging, functional magnetic resonance imaging (fMRI) provides brain signals measured over a time domain for thousands of voxels in the brain, for  each individual in the sample.  A central problem in analysing such data is to discover interrelations among the different functions.  This can be formulated as functional graphical modeling.  In this talk, I will present a brief introduction to functional graphical modeling. I will discuss three set of problems: 1) nonparametric undirected functional graphical modeling, 2) nonparametric functional directed acyclic graphs, and 3) graphical models for heterogeneous functional data. The research has numerous biological and medical applications, including the analyses of brain connectivity networks, gene-regulatory networks, and protein-protein interaction networks. I will also discuss possible extensions for the analysis of samples of univariate probability measures.

Speaker 1: Tarek Anous

Title: Two dimensional gravity as a matrix model

Abstract: In this colloquium, we will review old and recent developments in the field of string theory that link the quantum partition function of two dimensional surfaces to a Random Matrix integral.

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Speaker 2: Shu Sasaki

Title: A perspective on the Langlands program

Abstract: Fermat's Last Theorem was proved almost thirty years ago by A. Wiles. The proof taps into a web of conjectures and questions, often referred to as the Langlands program, that makes connections between a few different areas of mathematics (number theory, geometry, analysis etc.); and it has since instigated a great deal of new mathematics. In this talk (which is aimed at a general audience!), I would like to explain what some of the number theorists in the Langlands program are currently interested in and where they might be heading to, through a few concrete examples.

Speaker 1: Thomas Prellberg

Title: Lattice models for polymer adsorption

Abstract: Polymers tethered to a surface undergo an (ordinary) adsorption transition when the strength of the polymer-surface interaction is increased. If there is also a bulk interaction present, polymers undergo a second order collapse transition, which then changes the nature of the surface adsorption transition. At the critical collapse point this is known as the special adsorption point. On the other hand, collapsed polymers are believed to form a surface-attached globule under adsorption, but this scenario is notoriously hard to see in simulations.

This talk presents a walk through several years of efforts to elucidate this adsorption scenario with the help of computer simulations. The method employed is based on stochastic enumeration techniques combined with uniform sampling techniques. 

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Speaker 2: Qi Hong

Title: Transforming Gravitational Wave Astronomy with Quantum Computing

Abstract: In this seminar, I will first provide an overview of the current landscape in gravitational wave astronomy using interferometers like LIGO, focusing on the detections and analyses of colliding black holes and neutron stars. Subsequently, I will transition to the projected astrophysics advancements and challenges linked to the next-generation detectors. Finally, I will conclude by highlighting our ongoing exploration of quantum computing to address persistent challenges in gravitational wave data analysis, led by my team and collaborators.

Speaker 1: Linus Wunderlich

Title: Efficient valuation of financial derivatives and applications to risk management

Abstract: 

In this talk, I present numerical methods for efficient pricing of financial derivatives and their mathematical background. For low-dimensional problems, we consider Chebyshev methods, while to achieve efficiency in higher dimensions we employ neural networks. One of the key applications addressed in the talk is risk management of counterparty credit risk: the risk that the trading partner (counterparty) in a derivative deal may default and is unable to deliver the underlying asset.  The value lost at default, known as the exposure, is not known in advance but can be estimated using a Monte-Carlo simulation. However, this simulation required frequent valuations of the derivative price, yielding a large computational effort. To reduce this computational effort, I present a recently developed framework on complexity reduction for counterparty credit exposure. Theoretical results highlight that a Chebyshev interpolation meets the requirements to obtain robust and efficient results. 

The work is joint with Kathrin Glau, Domagoj Demeterfi, Ivelina Mladenova and Xiaocheng Wei

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Speaker 2: Navid Nabijou

Title: Enumerative geometry and the joy of degeneration.

Abstract: 

In 218 BC the Carthaginian general Hannibal marched his war elephants across the Alps, on a mission to decapitate the ascendant Roman Republic. At the other end of the Mediterranean, in what is now Turkey, a mathematician called Apollonius was drawing circles in the sand. He noticed that when he fixed three circles, there were precisely eight circles tangent to all three. The field of enumerative geometry was born.

We will tour some highlights of modern enumerative geometry. We will see how simple classical questions give rise to fearsome modern problems, and discuss the deep ideas which have been developed to tackle these. Our journey will take us into the Gromov-Witten theory revolution, which has rewritten the field over the past quarter century.

Speaker 1: Shahn Majid

Title: Quantum gravity on algebras and graphs

Abstract: 

Quantum gravity on a spacetime continuum is a 100 year old unsolved problem due to the non-renormalisable nature of divergences that arise. But what happens if we
replace the coordinate algebra of the manifold by another algebra, maybe even a noncommutative and/or finite dimensional one? I will give a flavour of the general scheme of
noncommutative Riemannian geometry and then show some computations of what Euclidean quantum gravity looks on a triangle and on the algebra of 2 x 2 matrices.

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Speaker 2: Shengwen Wang

Title: The geometry and singularity analysis of low complexity submanifolds

Abstract: The geometric PDEs related to submanifolds are usually highly non-linear, which cause the formation of singularities.  The classification and resolution of such singularities are not only of independent interest as PDE questions, but also of crucial importance for applications in other fields. We will discuss some controls on the geometric and topological properties of submanifolds and the behaviours of singularities when entropy (complexity) is low. 

Speaker 1: Anna Maltsev

Title: Topological eigenfunctions of quantum graphs

Abstract: 

A quantum graph is a set of vertices and edges, like a usual (combinatorial) graph but the edges are taken to be segments of a real line equipped with a Euclidean metric, which allows us to do analysis on graphs. We study the Schrodinger equation on such graphs, with a Kirchoff condition at vertices, which means that outward derivatives at each vertex add up to 0. Quantum graphs not only arise in a multitude of real-world applications (e.g. electrical networks, roads, pipes, neurons, etc.) but also have very interesting mathematical properties. In this talk, which is based on joint work with Evans Harrell, I will discuss eigenfunctions on quantum graphs that reflect graph topology of quantum graphs and illustrate new connections between vertex and edge scattering matrices via a “quantum” Ihara's theorem. 

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Speaker 2: Omer Bobrowski

Title: Universality in Topological Data Analysis

Abstract: Topological Data Analysis (TDA) focuses on using tools based on algebraic topology to extract and exploit information about structure in complex data. One of the most elusive challenges within this area is understanding the statistical properties of topological features. Despite much effort, this is still largely an open problem. In this talk we will present a series of new statements regarding the behavior of persistence diagrams (birth-death diagrams for topological features) arising from random point-clouds. We claim that, viewed in the right way, persistence values obey a universal probability law, that depends on neither the underlying space nor the original distribution of the point-cloud. We will discuss this surprising phenomenon and its potential application.