We discuss the behavior of geodesics in the continuous models of random geometry known as the Brownian map and the Brownian plane. We say that a point $x$ is a geodesic star with $m$ arms if $x$ is the endpoint of $m$ disjoint geodesics. We prove that the set of all geodesic stars with $m$ arms has dimension $5-m$, for $m=1,2,3,4$. This complements recents results of Miller and Qian, who derived upper bounds for these dimensions.

How much does the chromatic number of the random graph $G(n, 1/2)$ vary? Shamir and Spencer proved that it is contained in some sequence of intervals of length about $n^{1/2}$. Alon improved this slightly to $n^{1/2} / \log n$. Until recently, however, no lower bounds on the fluctuations of the chromatic number of $G(n, 1/2)$ were known, even though the question was raised by Bollobás many years ago. I will talk about the main ideas needed to prove that, at least for infinitely many $n$, the chromatic number of $G(n, 1/2)$ is not concentrated on fewer than $n^{1/2-o(1)}$ consecutive values.
I will also discuss the Zigzag Conjecture, made recently by Bollobás, Heckel, Morris, Panagiotou, Riordan and Smith: this proposes that the correct concentration interval length 'zigzags' between $n^{1/4+o(1)}$ and $n^{1/2+o(1)}$, depending on $n$.
Joint work with Oliver Riordan.

The title of the talk corresponds to a family of interesting random processes, which includes lazy random walks on graphs and much beyond them. Often, a key step in analysing such processes is to estimate their spectral gaps (ie. the difference between two largest eigenvalues). It is thus of interest to understand what else about the chain we can know from the spectral gap. We will present a simple comparison idea that often gives us the best possible estimates. In particular, we re-obtain and improve upon several known results on hitting, meeting, and intersection times; return probabilities; and concentration inequalities for time averages. We then specialize to the graph setting, and obtain sharp inequalities in that setting. This talk is based on work that has been in progress for far too long with Yuval Peres.

In this talk we will consider the extremal values of the stationary distribution of the sparse directed configuration model. Under the assumption of linear $(2+\eta)$-moments on the in-degrees and of bounded out-degrees, we obtain tight comparisons between the maximum value of the stationary distribution and the maximum in-degree. Under the further assumption that the order statistics of the in-degrees have power-law behavior, we show that the upper tail of the stationary distribution also has power-law behavior with the same index. Moreover, these results extend to the PageRank scores of the model, thus confirming a version of the so-called power-law hypothesis. Joint work with Xing Shi Cai, Pietro Caputo and Matteo Quattropani.

The integral of mean curvature squared is a conformal invariant that measures the distance from a given immersion to the standard embedding of a round sphere. Following work of Robert Bryant who showed that all Willmore spheres in the 3-sphere are conformally minimal, Robert Kusner proposed in the early 1980s to use the Willmore energy to obtain an “optimal” sphere eversion, called the min-max sphere eversion.

We will present a method due to Tristan Rivière that permits to tackle a wide variety of min-max problems, including ones about the Willmore energy. An important step to solve Kusner’s conjecture is to determine the Morse index of branched Willmore spheres, and we show that the Morse index of conformally minimal branched Willmore spheres is equal to the index of a canonically associated matrix whose dimension is equal to the number of ends of the dual minimal surface.

Elastic strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (tensile force produces elongation) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of tensile force is applied the stretch of the string is maximized (becomes inextensible). In this talk, we discuss a model for these stretch-limited elastic strings, in what way they model elastic behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.

In this talk, we discuss the feasibility of algorithms based on deep artificial neural networks (DNN) for the solution of high-dimensional PDEs, such as those arising from stochastic control and games. In the first part, we show that in certain cases, DNNs can break the curse of dimensionality in representing high-dimensional value functions of stochastic control problems. We then exploit policy iteration to reduce the associated nonlinear PDEs into a sequence of linear PDEs, which are then further approximated via a multilayer feedforward neural network ansatz. We establish that in suitable settings the numerical solutions and their derivatives converge globally, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration. Numerical experiments on Zermelo's navigation problem and on consensus control of interacting particle systems are presented to demonstrate the effectiveness of the method. This is joint work with Kazufumi Ito, Christoph Reisinger and Wolfgang Stockinger.

The periods of a Calabi-Yau manifold are of interest both to number theorists and to physicists. To a number theorist the primary object of interest is the zeta function. I will explain what this is, and why this is of interest also to physicists. For applications it is important to be able to calculate the local zeta function for many primes p. I will set out a method, adapted from a procedure proposed by Alan Lauder that makes the computation of the zeta function practical, in this sense, and comment on the form of the results. This talk is based largely on the recent paper hepth 2104.07816 and presents joint work with Xenia de la Ossa and Duco van Straten.

The elliptic Hall algebra has appeared in many different contexts in representation theory and geometry under different names.  We will explain how this algebra is categorified by the quantum Heisenberg category.  This diagrammatic category is modelled on affine Hecke algebras and can be viewed as a deformation of the framed HOMFLYPT skein category underpinning the HOMFLYPT link invariant.  Using the categorification of the elliptic Hall algebra, one can construct large families of representations for this algebra.