Events

Abstract: character variety is the space parametrizing flat local systems on a Riemann surface. I will give an overview of some open problems and recent progress in understanding the cohomology of these varieties, and how they connect to combinatorics and questions in low-dimensional topology.

One fundamental aim of algebraic geometry is to classify algebraic varieties up to isomorphisms. This is, however, much too difficult, already for surfaces, and it is much more reasonable to study them up isomorphisms between dense open sets, so-called birational maps. The group of birational self-maps is finite for some varieties and very large for others. The latter is the case for the simplest variety, the affine space, and we call its group of birational self-maps Cremona group. It is not finite dimensional and has been well-studied in dimension two. However, much less is known about these groups in higher dimensions. In this talk, I will present some properties of Cremona groups in higher dimension.

https://www.math.uzh.ch/mat074

In an influential article from the 1970s, Albert Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the n-ball is simple for n ≥ 3, asked if the same statement holds in dimension 2. In a joint work with Cristofaro-Gardiner and Humiliére, we proved that the group of compactly supported area-preserving homeomorphisms of the 2-disc is not simple. This answers Fathi's question and settles what is known as the simplicity conjecture in the affirmative.

The two models mentioned in the title are natural examples of dynamical systems preserving an infinite measure. Because of their periodicity, they can be represented by a Z^d-extension over a chaotic probability preserving dynamical system (resp. Sinai billiard, geodesic flow on a compact surface). Thus, their ergodic properties are closely related to those of the underlying probability preserving chaotic system (studied namely by Sinai, Bunimovich, Chernov, Young, Ratner, Pesin, etc.) and in particular with the local limit theorem established by resp. Domokos Szász and Tamás Varjú and Yves Guivarc'h and J. Hardy. When the horizon is finite, the free flight is bounded, and powerful tools can be used to establish many strong results, such as quantitative recurrence results, expansions in mixing, limit theorems for Birkhoff sums, for pin-ball, for non-stationary Birkhoff sums and for solutions of perturbed differential equations (results in collaboration with Benoît Saussol, with Dima Dolgopyat and Péter Nándori, with Damien Thomine, results by Nasab Yassine and Maxence Phalempin). Finally we will also state results in the more difficult case of the Lorentz gas in infinite horizon (results in collaboration with Dalia Terhesiu, and also with Ian Melbourne).

Discs are among the simplest manifolds, but their groups of diffeomorphisms can be very complicated. I will describe the geometric techniques that were used to understand these groups in low dimensions, their relationship to stable homotopy theory and number theory in high dimensions, and recent breakthroughs in understanding their homotopy type. This talk will be aimed at a broad audience.

In this talk, we focus on the Schrödinger operator with inverse-square potential L_a=−\Delta+a/|x|^2, a\geq−(d−2)^2/4, d\geq2. We will discuss the boundedness of wave operators in certain Sobolev spaces, which lead to a series of interesting inequalities, such as dispersive estimates, Strichartz estimates and uniform Sobolev inequalities. We will explain how to construct the wave operators using Mellin transform and spherical harmonic decomposition, and prove that they are W^{s,p}-bounded for certain p and s which depend on a. This talk is based on joint work with Changxing Miao and Jiqiang Zheng.

A typical problem in extremal graph theory asks to determine how many edges a (hyper)graph can have if it has a given number of vertices and no copies of a given (hyper)graph H. While this problem is well understood for most graphs, much less is known for hypergraphs. In this talk I will give an overview of the status of this problem for graphs, and present a new result in this direction about hypergraphs.

Algebraic geometry studies solution sets of polynomial equations. For instance, over the complex numbers, one may examine the topology of the solution set, whereas over a finite field, one may count its points. For polynomials with integer coefficients, these two fundamental invariants are intimately related via cohomological comparison theorems and trace formulas for the action of Frobenius. I will present recent results regarding point counts over finite fields and the cohomology of moduli spaces of curves that resolve longstanding questions in algebraic geometry and confirm more recent predictions from the Langlands program.

One of the main objectives of number theory is solving Diophantine equations, that is polynomial equations with integer coefficients for integer solutions. A fruitful theme has been studying local solutions, in other words, solutions over real numbers and solutions modulo powers of prime numbers. Even if there are local solutions everywhere, it does not mean there are integer (global) solutions. The first and the most fundamental object measuring such a local-global obstruction is the ideal class group. The ideal class group of a finite field extension of the field of rational numbers is a finite abelian group. The order of this group is an important arithmetic invariant. Dirichlet's class number formula relates this order to the leading term at s = 0 of a complex analytic function, namely the Dedekind zeta function. L-functions are generalisations of zeta functions and play a central role in modern number theory. In the 1970s, Stark, in an attempt to generalise Dirichlet's class number formula, made a series of conjectures relating leading terms of Artin L-functions at s = 0 to arithmetic invariants. In the 1980s, Gross and Tate refined these conjectures and formulated their p-adic analogues. In this talk, I will present a formulation of some instances of Stark's conjectures. In my recent joint work with Samit Dasgupta we resolved the Brumer-Stark conjecture, the Gross-Stark conjecture and the tower of felds conjecture of Gross. I will present some ideas that go into the proof of these conjectures.

''In cryptography, we are always looking for hard mathematical problems on which to build secure protocols for exchanging messages. Current cryptography is based on the difficulty of integer factorisation and the Discrete Logarithm Problem. Unfortunately, both of these problems can be solved on (sufficiently powerful) quantum computers in an acceptable time thanks to Shor's algorithm (1994). Hence the need to look for new problems that are also hard on quantum computers. A good proposal in this direction is the Isogeny Path Problem, which gave rise to Isogeny-Based Cryptography. We will take a friendly look at the problem and the cryptosystems based on it.

Knots in contact manifolds are interesting objects to study. Contact structures come in two flavors- tight and overtwisted. In this talk, I'll focus on knots in overtwisted manifolds. The knots that we really care about in this setting are known as non-loose or exceptional knots. I'll define what these knots are and then mention some of their existence and classification results. If time permits, I'll talk about how one can contruct a family of non-loose knots via cabling. This is based on joint work with Etnyre, Min and Mukherjee. No background knowledge of contact topology will be assumed.

Onsager proposed a closed-form expression of the free energy of the Ising model in 1944. The method of Kac and Ward is particularly elegant and it has recently be made rigorous by Lis and Aizenman-Warzel. I will show how to extend it to the triangular lattice, with coupling constants of arbitrary signs. This is ongoing work with Georgios Athanasopoulos.

We will give an overview of the history and main results in the theory of harmonic maps in dimension two. We will in particular emphasise their relation to minimal surfaces.

Euclidean field theories have been extensively studied in the mathematical literature since the sixties, motivated by high-energy physics and statistical mechanics. Formally, such a theory is given by a Gibbs measure associated with a Euclidean action functional over a space of distributions. In this talk I explain how some such theories arise as high-density limits of interacting Bose gases at positive temperature. This provides a rigorous derivation of them starting from a realistic microscopic model of statistical mechanics. I focus on field theories with a quartic, local or nonlocal, interaction in dimensions <= 3. Owing to the singularity of the Gaussian free field in dimensions higher than one, the interaction is ill-defined and has to be renormalized by infinite mass and energy counterterms. The proof is based on a new functional integral representation of the interacting Bose gas. Based on joint work with Jürg Fröhlich, Benjamin Schlein, and Vedran Sohinger.

In diesem Vortrag werden wir die mathematische und rechnerische Modellierung von Spielen, Videospielen und Puzzles untersuchen, die im Rahmen von Forschungsprojekten in Mathematik und Informatik entwickelt wurden. Dazu gehören Schiebepuzzles, farbige Würfel, Tessellierungsprobleme und Kunstgalerieprobleme mit Türmen und Königinnen. Hierbei werden wir Werkzeuge aus der Kombinatorik, diskreten Konfigurationsräumen, diskreter Geometrie, algorithmischem Denken, diskreter Wahrscheinlichkeitstheorie, kombinatorischer Spieltheorie, künstlicher Intelligenz und mehr kennenlernen. Zum Abschluss gibt es eine Ausstellung, bei der jeder die Spiele, Videospiele und Puzzles spielen kann, die im Vortrag vorgestellt wurden.

The theory of affine processes has been recently extended to continuous stochastic Volterra equations. These so-called affine Volterra processes overcome modeling shortcomings of affine processes by incorporating path-dependent features and trajectories with regularity different from the paths of Brownian motion. More specifically, singular kernels yield rough affine processes. This paper extends the theory by considering affine stochastic Volterra equations with jumps. This extension is not straightforward because the jump structure and possible singularities of the kernel may induce explosions of the trajectories. This study also provides exponential affine formulas for the conditional Fourier-Laplace transform of marked Hawkes processes. This is joint work with Alessandro Bondi and Giulia Livieri.

In this talk, we will consider some special sets of reals, which on one side originate in real analysis, general topology and algebra and which on the other can be defined in terms of elementary set theoretic operations on the reals. These sets carry a surprisingly complex infinite-combinatorial structure and their study easily brings us to the boundaries of our axiomatic systems. Typical examples of such special families are maximal almost disjoint families, maximal independent families, bounded and unbounded families, as well as maximal cofinitary groups. We will discuss some recent advances and trends in their study, and outline interesting remaining open questions.

In this talk we review results on several types of harmonic weak Maass forms that are related to integral even weight newforms. We start with a brief introduction to the theory of harmonic weak Maass forms. These can be related to classical modular forms via a certain differential operator, the so-called \xi-operator. Starting with an integral weight newform, we will review different constructions of integral weight harmonic weak Maass forms via (generalized) Weierstrass zeta functions that map to the newform under the \xi-operator. A second construction via theta liftings gives a half-integral weight harmonic weak Maass form whose coefficients are given by periods of certain meromorphic modular forms with algebraic coefficients and periods of the integer even weight newform. This is joint work with Jens Funke, Michael Mertens, and Eugenia Rosu resp. Jan Bruinier and Markus Schwagenscheidt.

There are several different notions of "complexity" for a topological space M. For instance, when M is a manifold one can study: the Lebesgue covering dimension; sum of Betti numbers; minimal number of Morse critical values; or the Lusternik–Schnirelmann category. Similarly, given a triangulated category C, one can measure its complexity using invariants such as the Rouquier dimension; diagonal dimension; or minimal length of presentation as a homotopy colimit. In this talk, I will discuss some of the relations between these categorical invariants and topological invariants when the category C is the Fukaya category of the cotangent bundle of M (equivalently, the category of modules over chains on the based loop space on M). I will mostly focus on - introducing the above invariants of topological spaces and categories and - discussing how Lagrangian cobordisms play a role in bounding the diagonal dimension of C in terms of the minimal number of critical values of a Morse function on M.