Weekly Bulletin

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

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).

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.

Together in recent work with G. Freixas, we study a relative intersection theory, with values in line bundles, originally introduced by Deligne and developed by e.g. Elkik. Whereas one usually intersects n divisors on an n-dimensional variety and gets a number, here intersects n+1 divisors and gets a line. <BR> It amounts to a categorical refinement of direct images in Chow-theory. We introduce a formalism and categorical framework that allows us to give life to many of the usual intersection theoretic notions to categorified levels. <BR> A great motivation for developing this was to study categorified versions of the Grothendieck-Riemann-Roch theorem, conjectured by Deligne. While being of general interest, it has also turned out to be important in Arakelov theory, algebraic geometry, mirror symmetry, and other fields. I will also report on some partial results in this direction.

''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.

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.

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 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.