### Stably irrational hypersurfaces of small slopes

Stefan Schreieder
We show that over any uncountable field of characteristic different from two, a very general hypersurface of dimension $n > 2$ and degree at least $log_2 (n) + 2$ is not stably rational. This significantly improves earlier results of Kollár and Totaro. As a byproduct of our proof, we obtain new counterexamples to the integral Hodge conjecture, answering a question of Voisin and Colliot-Thélène – Voisin.

### Counting curves of given type, revisited

Juan Souto
Mirzakhani wrote two papers studying the asymptotic behaviour of the number of curves of a given type (simple or not) and with length at most $L$. In this talk I will explain a new independent proof of Mirzakhani’s results. This is joint work with Viveka Erlandsson.

### Torsion groups do not act on 2-dimensional CAT(0) complexes

Piotr Przytycki
We show, under mild hypotheses, that if each element of a finitely generated group acting on a 2-dimensional CAT(0) complex has a fixed point, then the action is trivial. In particular, all actions of finitely generated torsion groups on such complexes are trivial. As an ingredient, we prove that the image of an immersed loop in a graph of girth 2π with length not commensurable to π has diameter > π. This is related to...

### Computing Hecke operators for cohomology of arithmetic subgroups of $SL_n(Z)$

Mark W. McConnell
We will describe two projects. The first which is joint with Avner Ash and Paul Gunnells, concerns arithmetic subgroups $\Gamma$ of $G = SL_4(Z)$. We compute the cohomology of $\Gamma \setminus G/K$, focusing on the cuspidal degree $H^5$. We compute a range of Hecke operators on this cohomology. We fi Galois representations that appear to be attached to the Hecke eigenclasses, based on the operators we have computed. We have done this for both non-torsion...

### Bayesian inference and mathematical imaging - Part 3: probability and convex optimisation

Marcelo Pereyra
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation...

### Big mapping class groups - lecture 3

Danny Calegari
Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe...

### Willmore stability and conformal rigidity of minimal surfaces in $\mathbb{S}^{n}$

Rob Kusner
A minimal surface $M$ in the round sphere $\mathbb{S}^{n}$ is critical for area, as well as for the Willmore bending energy $W=\int\int(1+H^{2})da$. Willmore stability of $M$ is equivalent to a gap between −2 and 0 in its area-Jacobi operator spectrum. We show the $W$-stability of $M$ persists in all higher dimensional spheres if and only if the Laplacian of $M$ has first eigenvalue 2. The square Clifford 2-torus in $\mathbb{S}^{3}$ and the equilateral minimal 2-torus...

### Schubert calculus and self-dual puzzles

Iva Halacheva
Puzzles are combinatorial objects developed by Knutson and Tao for computing the expansion of the product of two Grassmannian Schubert classes. I will describe how selfdual puzzles give the restriction of a Grassmannian Schubert class to the symplectic Grassmannian in equivariant cohomology. The proof uses the machinery of quantum integrable systems. Time permitting, I will also discuss some ideas about how to interpret and generalize this result using Lagrangian correspondences and Maulik-Okounkov stable classes. This...

### Understanding quadratic forms on lattices through generalised theta series

Lynne Walling
Siegel introduced generalised theta series to study representation numbers of quadratic forms. Given an integral lattice $L$ with quadratic form $q$, Siegel’s degree $n$ theta series attached to $L$ has a Fourier expansion supported on $n$-dimensional lattices, with Fourier coefficients that tells us how many times $L$ represents any given $n$-dimensional lattice. Siegel proved that this theta series is a type of automorphic form. In this talk we explore how the theory of automorphic forms,...

### Identités de $q$-séries et de partitions

Jehanne Dousse
Les $q$-séries (parfois appelées séries basiques hypergéométriques) sont des séries construites en utilisant les $q$-factorielles $(a;q)_n := (1-a)(1-aq)...(1-aq^{n-1}).$ On les retrouve dans de nombreux domaines des mathématiques tels que la combinatoire, la théorie des nombres, la théorie des groupes et la physique mathématique. Sous l'influence de Ramanujan, les $q$-séries ont souvent été étudiées en relation avec les partitions d'entiers. Nous commencerons par une introduction générale aux $q$-séries et étudierons quelques identités classiques, puis nous verrons...

### Understanding the growth of Laplace eigenfunctions (part 1 of 2)

Yaiza Canzani
In this talk we will discuss a new geodesic beam approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of $L^{2}$ mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we obtain...

### ​Construction of lattices defining fake projective planes - Lecture 6

Donald I. Cartwright

### Intermittent weak solutions of the 3D Navier-Stokes equations

​I will discuss recent developments concerning the non-uniqueness of distributional solutions to the Navier-Stokes equation.

### Topics on $K3$ surfaces - Lecture 1: $K3$ surfaces in the Enriques Kodaira classification and examples

Alessandra Sarti
Aim of the lecture is to give an introduction to $K3$ surfaces, that are special algebraic surfaces with an extremely rich geometry. The most easy example of such a surface is the Fermat quartic in complex three-dimensional space. The name $K3$ was given by André Weil in 1958 in honour of the three remarkable mathematicians: Kummer, Kähler and Kodaira and of the beautiful K2 mountain at Cachemire. The topics of the lecture are the following:...

### Big mapping class groups - lecture 1

Danny Calegari
Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe...

Tim Steger

### On the general linear group over arithmetic orders and corresponding cohomology groups

Joachim Schwermer
Orders in finite-dimensional algebras over number fi give rise to interesting locally symmetric spaces and algebraic varieties. Hilbert modular varieties or arithmetically defined hyperbolic 3-manifolds, compact ones as well as noncompact ones, are familiar examples. In this talk we discuss various cases related to the general linear group $GL(2)$ over orders in division algebras defined over some number field. Geometry, arithmetic, and the theory of automorphic forms are interwoven in a most fruitful way in...

### The challenge of linear-time Boltzmann sampling

Andrea Sportiello
Let $X_{n}$ be an ensemble of combinatorial structures of size $N$, equipped with a measure. Consider the algorithmic problem of exactly sampling from this measure. When this ensemble has a ‘combinatorial specification, the celebrated Boltzmann sampling algorithm allows to solve this problem with a complexity which is, typically, of order $N(3/2)$. Here, a factor $N$ is inherent to the problem, and implied by the Shannon bound on the average number of required random bits, while...

### Transfer operators for Sinai billiards - lecture 1

We will discuss an approach to the statistical properties of two-dimensional dispersive billiards (mostly discrete-time) using transfer operators acting on anisotropic Banach spaces of distributions. The focus of this part will be our recent work with Mark Demers on the measure of maximal entropy but we will also survey previous results by Demers, Zhang, Liverani, etc on the SRB measure.

### Bayesian inference and mathematical imaging - Part 2: Markov chain Monte Carlo

Marcelo Pereyra
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation...

### Quantum character varieties at roots of unity

Pavel Safronov
Character varieties of closed surfaces have a natural Poisson structure whose quantization may be constructed in terms of the corresponding quantum group. When the quantum parameter is a root of unity, this quantization carries a central subalgebra isomorphic to the algebra of functions on the classical character variety. In this talk I will describe a procedure which allows one to obtain Azumaya algebras via quantum Hamiltonian reduction. As an application, I will show that quantizations...

### Some new dynamical applications of smooth parametrizations for C∞ systems - lecture 3

David Burguet
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$...

### $L^2$ curvature for surfaces in Riemannian manifolds

Ernst Kuwert
For surfaces immersed into a compact Riemannian manifold, we consider the curvature functional given by the $L^{2}$ integral of the second fundamental form. We discuss an area bound in terms of the energy, with application to the existence of minimizers. This is joint work with V. Bangert.

### On the B-Semiampleness Conjecture

Enrica Floris
An lc-trivial fibration $f : (X, B) \to Y$ is a fibration such that the log-canonical divisor of the pair $(X, B)$ is trivial along the fibres of $f$. As in the case of the canonical bundle formula for elliptic fibrations, the log-canonical divisor can be written as the sum of the pullback of three divisors: the canonical divisor of $Y$; a divisor, called discriminant, which contains informations on the singular fibres; a divisor, called...

### Hodge-GUE Correspondence

Di Yang
An explicit relationship between certain cubic Hodge integrals on the Deligne–Mumford moduli space of stable algebraic curves and connected GUE correlators of even valencies, called the Hodge–GUE correspondence, was recently discovered. In this talk, we prove this correspondence by using the Virasoro constraints and by deriving the Dubrovin–Zhang loop equation. The talk is based on a series of joint work with Boris Dubrovin, Si-Qi Liu and Youjin Zhang.

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• Audiovisual
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