Past

Ideas from physics have predicted a number of important properties of random constraint satisfaction problems such as the satisfiability threshold and the free energy (the exponential growth rate of the number of solutions). Another prediction is the condensation regime where most of the solutions are contained in a small number of clusters and the overlap of two random solutions is concentrated on two points. We establish this phenomena in the random regular NAESAT model. Joint work with Danny Nam and Youngtak Sohn.

We study expectations of powers and correlations for characteristic polynomials of N x N non-Hermitian random matrices. This problem is related to the analysis of planar models (log-gases) where a Gaussian (or other) background measure is perturbed by a finite number of point charges in the plane. I will discuss the critical asymptotics, for example when a point charge collides with the boundary of the support, or when two point charges collide with each other (coalesce) in the bulk. In many of these situations, we are able to express the results in terms of Painlevé transcendents. The application to certain d-fold rotationally invariant models will be discussed. This is joint work with Alfredo Deaño (University of Kent).

What is the connection between phase transitions in statistical physics and the computational tractability of approximate counting and sampling? There are many fascinating answers to this question but many mysteries remain. I will discuss one particular type of a phase transition: the first-order phase in the Potts model on $\mathbb{Z}^d$ for large $q$, and show how tools used to analyze the phase transition can be turned into efficient algorithms at the critical temperature. In the other direction, I'll discuss how the algorithmic perspective can help us understand phase transitions.

Active matter describes ensembles of self-organizing agents, or particles, interacting with their local environments so that their micro-scale behavior determines macro-scale characteristics of the ensemble. While there has been a surge of activity exploring the physics underlying such systems, less attention has been paid to questions of how to program them to achieve desired outcomes. We will present some recent results designing programmable active matter for specific tasks, including aggregation, dispersion, speciation, and locomotion, building on insights from stochastic algorithms and statistical physics.

In the first part of this lecture, I will discuss the proof of convergence of the Lorentz process, in the Boltzmann-Grad limit, to a random process governed by a generalised linear Boltzmann equation. This will hold for general scatterer configurations, including certain types of quasicrystals, and include the previously known cases of periodic and Poisson random scatterer configurations. The second part of the lecture will focus on quantum transport in the periodic Lorentz gas in a combined short-wavelength/Boltzmann-Grad limit, and I will report on some partial progress in this challenging problem. Based on joint work with Andreas Strombergsson (part I) and Jory Griffin (part II).

We study the countable set of rates of growth of a hyperbolic 
group with respect to all its finite generating sets. We prove that the 
set is well-ordered, and that every real number can be the rate of growth 
of at most finitely many generating sets up to automorphism of the group.

We prove that the ordinal of the set of rates of growth is at least $ω^ω$, 
and in case the group is a limit group (e.g., free and surface groups), it 
is $ω^ω$.

We further study the rates of growth of all the finitely generated 
subgroups of a hyperbolic group with respect to all their finite 
generating sets. This set is proved to be well-ordered as well, and every 
real number can be the rate of growth of at most finitely many isomorphism 
classes of finite generating sets of subgroups of a given hyperbolic 
group. Finally, we strengthen our results to include rates of growth of 
all the finite generating sets of all the subsemigroups of a hyperbolic 
group.

Joint work with Koji Fujiwara.

Calibrated geometry, more specifically Calabi-Yau geometry, occupies a modern, rather sophisticated, cross-roads between Riemannian, symplectic and complex geometry. We will show how, stripping this theory down to its fundamental holomorphic backbone and applying ideas from classical complex analysis, one can generate a family of purely holomorphic invariants on any complex manifold. We will then show how to compute them, and describe various situations in which these invariants encode, in an intrinsic fashion, properties not only of the given manifold but also of moduli spaces.

Interest in these topics, if initially lacking, will arise spontaneously during this informal presentation.

I will talk about a novel idea on the Swampland program that uses consistency of what lives on the string probes in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be calculated by anomaly inflow mechanism and used to provide constraints on the supergravity theories based on unitarity of the worldsheet CFT. I will show some of the theories with 8 or 16 supersymmetries, which are otherwise consistent looking, belong to the Swampland.

Synchronization is a collective phenomenon that pervades the natural systems from neurons to fireflies. In a network, synchronization of the dynamical variables associated to the nodes occurs when nodes are coupled to their neighbours as captured by the Kuramoto model. However many complex systems include also higher-order interactions among more than two nodes and sustain dynamical signals that might be related to higher-order simplices such as nodes of triangles. These dynamical topological signals include for instance fluxes which are dynamical variables associated to links.

In this talk I present a new topological approach [1] to synchronization on simplicial complexes. Here the theory of synchronization is combined with topology (specifically Hodge theory) for formulating the higher-order Kuramoto model that uses the higher-order Laplacians and provides the main synchronization route for topological signals. I will show that the dynamics defined on links can be projected to a dynamics defined on nodes and triangles that undergo a synchronization transition and I will discuss how this procedure can be immediately generalized for topological signals of higher dimension. Interestingly I will show that when the model includes an adaptive coupling of the two projected dynamics, the transition becomes explosive, i.e. synchronization emerges abruptly.

This model can be applied to study synchronization of topological signals in the brain and in biological transport networks as it proposes a new set of topological transformations that can reveal collective synchronization phenomena that could go unnoticed otherwise.

[1] Millán, A.P., Torres, J.J. and Bianconi, G., 2019. Explosive higher-order Kuramoto dynamics on simplicial complexes. Physical Review Letters (in press) arXiv preprint arXiv:1912.04405.

There is a need for a new kind of maths course, to be taught, not to mathematics students, but to biologists with little or no maths background. There have been many recent calls for an upgrade to the mathematical background of biologists: undergraduate biology students need to understand the role of modeling and dynamics in understanding ecological systems, evolutionary dynamics, neuroscience, physiology, epidemiology, and the modeling that underlies the concept of climate change. They also need to understand the importance of feedback, both positive and negative, in creating dynamical systems in biology.

 Such a course is possible. The most important foundational development was the 20^th century replacement of the vague and unhelpful concept of a differential equation by the rigorous geometric concept of a vector field, a function from a multidimensional state space to its tangent space, assigning “change vectors” to every point in state space. This twentieth-century concept is not just more rigorous, but in fact makes for superior pedagogy. We also discuss the key nonlinear behaviors that biological systems display, such as switch-like behavior, robust oscillations and even chaotic behavior.

 This talk will outline such a course. It would have a significant effect on the conduct of biological research and teaching, and bring the usefulness of mathematical modeling to a wide audience.

Maths-Whizz is an online, virtual maths tutor for 5-13 year-olds that is designed to behave like a human tutor. Using adaptive assessment and decision-tree algorithms, the virtual tutor guides each student along a personalised learning journey tailored to their needs. As students interact with the tutor, the system captures a range of learning analytics as an automatic by-product. These analytics, collected on a per-lesson and per-question basis, then inform a range of research projects centred on students' learning patterns. This workshop will introduce the mechanics of the Maths-Whizz tutor, as well as its related learning analytics. We will summarise the research behind four InfoMM mini-projects and present open questions we are currently grappling with. Maths-Whizz has supported over a million children and thousands of schools worldwide, from the UK and US to rural Kenya, the DRC and Mexico. In a world of social distancing and widespread school closures, the need for virtual tutoring has never been more paramount to children's learning - and nor has your data analytical expertise!

The Cuntz semigroup is a geometric refinement of K-theory that plays an important role in the structure theory of C*-algebras. It is defined analogously to the Murray-von Neumann semigroup by using equivalence classes of positive elements instead of projections.
Starting with the definition of the Cuntz semigroup of a C*-algebra, we will look at some of its classical applications. I will then talk about the recent breakthroughs in the structure theory of Cuntz semigroups and some of the consequences.

Multi-agent reinforcement learning (MARL) has enjoyed substantial successes in many applications including the game of Go, online Ad bidding systems, realtime resource allocation, and autonomous driving. Despite the empirical success of MARL, general theories behind MARL algorithms are less developed due to the intractability of interactions, complex information structure, and the curse of dimensionality. Instead of directly analyzing the multi-agent games, mean-field theory provides a powerful approach to approximate the games under various notions of equilibria. Moreover, the analytical feasible framework of mean-field theory leads to learning algorithms with theoretical guarantees. In this talk, we will demonstrate how mean-field theory can contribute to the simultaneous-learning-and-decision-making problems with unknown rewards and dynamics. 

To approximate Nash equilibrium, we first formulate a generalized mean-field game (MFG) and establish the existence and uniqueness of the MFG solution. Next we show the lack of stability in naive combination of the Q-learning algorithm and the three-step fixed-point approach in classical MFGs. We then propose both value-based and policy-based algorithms with smoothing and stabilizing techniques, and establish their convergence and complexity results. The numerical performance shows superior computational efficiency. This is based on joint work with Xin Guo (UC Berkeley), Anran Hu (UC Berkeley), and Junzi Zhang (Stanford).

If time allows, we will also discuss learning algorithms for multi-agent collaborative games using mean-field control. The key idea is to establish the time consistent property, i.e., the dynamic programming principle (DPP) on the lifted probability measure space. We then propose a kernel-based Q-learning algorithm. The convergence and complexity results are carried out accordingly. This is based on joint work with Haotian Gu, Xin Guo, and Xiaoli Wei (UC Berkeley).

After recalling some motivation for studying highly-connected graphs in the context of operator algebras and large-scale geometry, we will introduce the notion of "asymptotic expansion" recently defined by Li, Nowak, Spakula and Zhang. We will explore some applications of this definition, hopefully culminating in joint work with Li, Vigolo and Zhang.

Abstract

This talk will provide an overview of mathematical modelling applied to the behaviour of ice sheets and their role in the climate system.  I’ll provide some motivation and background, describe simple approaches to modelling the evolution of the ice sheets as a fluid-flow problem, and discuss some particular aspects of the problem that are active areas of current research.  The talk will involve a variety of interesting continuum-mechanical models and approximations that have analogues in other areas of applied mathematics.

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The heart of modern machine learning (ML) is the approximation of high dimensional functions. Traditional approaches, such as approximation by piecewise polynomials, wavelets, or other linear combinations of fixed basis functions, suffer from the curse of dimensionality (CoD). We will present a mathematical perspective of ML, focusing on the issue of CoD. We will discuss three major issues: approximation theory and error analysis of modern ML models, dynamics and qualitative behavior of gradient descent algorithms, and ML from a continuous viewpoint. We will see that at the continuous level, ML can be formulated as a series of reasonably nice variational and PDE-like problems. Modern ML models/algorithms, such as the random feature and two-layer and residual neural network models, can all be viewed as special discretizations of such continuous problems. We will also present a framework that is suited for analyzing ML models and algorithms in high dimension, and present results that are free of CoD. Finally, we will discuss the fundamental reasons that are responsible for the success of modern ML, as well as the subtleties and mysteries that still remain to be understood.

The boundary integral equation method is a popular method for solving elliptic PDEs with constant coefficients, and systems of such PDEs, in bounded and unbounded domains. An attraction of the method is that it reduces solution of the PDE in the domain to solution of a boundary integral equation on the boundary of the domain, reducing the dimensionality of the problem. Second kind integral equations, featuring the double-layer potential operator, have a long history in analysis and numerical analysis. They provided, through C. Neumann, the first existence proof to the Laplace Dirichlet problem in 3D, have been an important analysis tool for PDEs through the 20^th century, and are popular computationally because of their excellent conditioning and convergence properties for large classes of domains. A standard numerical method, in particular for boundary integral equations, is the Galerkin method, and the standard convergence analysis starts with a proof that the relevant operator is coercive, or a compact perturbation of a coercive operator, in the relevant function space. A long-standing open problem is whether this property holds for classical second kind boundary integral equations on general non-smooth domains. In this talk we give an overview of the various concepts and methods involved, reformulating the problem as a question about numerical ranges. We solve this open problem through counterexamples, presenting examples of 2D Lipschitz domains and 3D Lipschitz polyhedra for which coercivity does not hold. This is joint work with Prof Euan Spence, Bath.

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https://arxiv.org/pdf/2004.06929.pdf

Motivation for the study of fusion categories is twofold: Fusion categories arise in wide array of mathematical subjects, and provide the necessary input for some fascinating topological constructions. We will carefully define what fusion categories are, and give representation theoretic examples. Then, we will explain how fusion categories are inherently finite combinatorial objects. We proceed to construct an example that does not come from group theory. Time permitting, we will go some way towards introducing so-called modular tensor categories.

Let $X_1, \ldots$ be i.i.d. copies of some real random variable $X$. For any $\varepsilon_2, \varepsilon_3, \ldots$ in $\{0,1\}$, a basic algorithm introduced by H.A. Simon yields a reinforced sequence $\hat{X}_1, \hat{X}_2, \ldots$ as follows. If $\varepsilon_n=0$, then $\hat{X}_n$ is a uniform random sample from $\hat{X}_1, …, \hat{X}_{n-1}$; otherwise $\hat{X}_n$ is a new independent copy of $X$. The purpose of this talk is to compare the scaling exponent of the usual random walk $S(n)=X_1 +\ldots + X_n$ with that of its step reinforced version $\hat{S}(n)=\hat{X}_1+\ldots + \hat{X}_n$. Depending on the tail of $X$ and on asymptotic behavior of the sequence $\varepsilon_j$, we show that step reinforcement may speed up the walk, or at the contrary slow it down, or also does not affect the scaling exponent at all. Our motivation partly stems from the study of random walks with memory, notably the so-called elephant random walk and its variations.

Fyodorov-Hiary-Keating established a series of conjectures concerning the large values of the Riemann zeta function in a random short interval. After reviewing the origins of these predictions through the random matrix analogy, I will explain recent work with Louis-Pierre Arguin and Maksym Radziwill, which proves a strong form of the upper bound for the maximum.

We say that a graph $G$ is $H$-free if $G$ does not contain $H$ as a (not necessarily induced) subgraph. For a positive integer $n$, denote by $\text{ex}(n,H)$ the largest number of edges in an $H$-free graph with $n$ vertices (the Turán number of $H$). The classical theorem of Erdős, Kleitman, and Rothschild states that, for every $r\geq3$, there are $2^{\text{ex}(n,H)+o(n2)}$ many $K_r$-free graphs with vertex set $\{1,…, n\}$. There exist (at least) three different derivations of this estimate in the literature: an inductive argument based on the Kővári-Sós-Turán theorem (and its generalisation to hypergraphs due to Erdős), a proof based on Szemerédi's regularity lemma, and an argument based on the hypergraph container theorems. In this talk, we present yet another proof of this bound that exploits connections between entropy and independence. This argument is an adaptation of a method developed in a joint work with Gady Kozma, Tom Meyerovitch, and Ron Peled that studied random metric spaces.