Weekly Bulletin

Mirror symmetry is a subject at the crossroad of symplectic geometry, complex analytic geometry and mathematical physics. Informally, it is a conjectural duality between Calabi-Yau varieties: given such a variety X (or, rather, a family of such objects), there should be a second variety (or family) Y, called the "mirror" of X. The pair (X,Y) should be characterised by the fact that symplectic invariants of X should correspond to holomorphic invariants of Y. Nevertheless, the construction of Y out of X stays an open major problem of the theory: it has been solved in countless many examples, but not in general. In 2006 Kontsevich and Soibelman proposed an attack strategy based on non-archimedean geometry. In this exposé I will survey the history of this program, focusing on the key moments and the recent advances dues to S. Keel, J. Nicaise, C. Xu, T. Y. Yu and myself.

Gibbs u-states are a distinguished class of measures invariant by a partially hyperbolic diffeomorphism of a manifold. These measures and the related SRB measures have been studied extensively in various situations. There are some preliminary indications that homogeneous dynamics methods are relevant to this circle of questions. I will try to sketch this connection.

We discuss the relation between hypersurface singularities (e.g. ADE, $\widetilde{E}_{6},\widetilde{E}_{7},\widetilde{E}_{8}$, etc) and spectral invariants, which are symplectic invariants coming from Floer theory.

Consider the sparse approximation or best subset selection problem: Given a vector y and a matrix A, find a k-sparse vector x that minimizes the residual ||Ax-y||. This sparse linear regression problem, and related variants, plays a key role in high dimensional statistics, compressed sensing, and more. In this talk we focus on the trimmed lasso penalty, defined as the L_1 norm of x minus the L_1 norm of its top k entries in absolute value. We advocate using this penalty by deriving sparse recovery guarantees for it, and by presenting a practical approach to optimize it. Our computational approach is based on the generalized soft-min penalty, a smooth surrogate that takes into account all possible k-sparse patterns. We derive a polynomial time algorithm to compute it, which in turn yields a novel method for the best subset selection problem. Numerical simulations illustrate its competitive performance compared to current state of the art.

Random matrices frequently appear in many different fields — physics, computer science, applied and pure mathematics. Oftentimes the random matrix of interest will have non-trivial structure — entries that are dependent and have potentially different means and variances (e.g. sparse Wigner matrices, matrices corresponding to adjacencies of random graphs, sample covariance matrices). However, current understanding of such complex random matrices remains lacking. In this talk, I will discuss recent results concerning the spectrum of sums of independent random matrices with a.s. bounded operator norms. In particular, I will demonstrate that under some fairly general conditions, such sums will exhibit the following universality phenomenon — their spectrum will lie close to that of a Gaussian random matrix with the same mean and covariance. No prior background in random matrix theory is required — basic knowledge of probability and linear algebra are sufficient. (joint with Ramon van Handel) Pre-print link: https://web.math.princeton.edu/~rvan/tuniv220113.pdf

We showcase tropical geometry as a tool for geometric counting problems. A nice feature of tropical geometry is that many techniques can be applied simultaneously over various ground fields, e.g. for complex or real counting problems. Our prime example will be the count of bitangent liens to a smooth plane quartic. Already Plücker knew that a smooth complex plane quartic curve has exactly 28 bitangents. Bitangents of quartic curves are related to a variety of mathematical problems. They appear in one of Arnold's trinities, together with lines in a cubic surface and 120 tritangent planes of a sextic space curve. In this talk, we review known results about counts of bitangents under variation of the ground field. Special focus will be on counting in the tropical world, and its relations to real and arithmetic counts. We end with new results concerning the arithmetic multiplicity of tropical bitangent classes, based on joint work with Sam Payne and Kris Shaw.

This cycle of talks wants to highlight how ideas from tropical geometry have contributed not only to the solution, but also to the development of enumerative geometric problems regarding moduli spaces of curves, and maps from curves to curves. We will spend a little of time reviewing the origins of this story, i.e. the development of tropical Hurwitz numbers as combinatorial analogues for the classical Hurwitz numbers. We will discuss a more recent interpretation that views tropical Hurwitz numbers as the natural computation for the intersection number of the double ramification cycle with an element of the log Chow ring of the moduli space of curves (called in this case the branch polynomial, as it is presented as a piecewise polynomial function on the moduli spaces of tropical curves) which is determined by the tropical moduli space of covers of the projective line. We will see that from the tropical perspective analogous piecewise polynomial functions may be associated to $k$-DR cycles (cycles arising from spaces of twisted pluri-differentials), thus giving rise to $k$-analogues of Hurwitz numbers (called leaky Hurwitz numbers) that enjoy many of the algebro-combinatorial properties of Hurwitz numbers - such as piecewise polynomiality and wall crossings. We will present some work in progress which intends to incorporate descendants into these pictures. Tropical algorithms are developed that give rise to some intruiguingly simple formulas in the case when one point is fully ramified. The material presented is based on many years of joint work with several people, including Paul Johnson, Hannah Markwig, Dhruv Ranganathan and Johannes Schmitt.

Strong typicality <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

We suggest a new definition of higher-dimensional automata motivated by cocompact quotients of buildings. We construct infinite series of such automata and produce very explicit constructions of Ramanujan higher-dimensional graphs.

Shanonn's lossless source coding theorem; Stein's Lemma for hypothesis testing <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

The scattering of electromagnetic waves from obstacles with wave- material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken here is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. The fields are then evaluated in the exterior domain by a known representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge-Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. The well-posedness analysis of the boundary integral equation as well as the error analysis of the numerical methods relies on frequency-explicit bounds in the Laplace domain. These are then transferred to the time domain and combined with known approximation estimates of the numerical methods. The talk is based on joint work with Balázs Kovács and Jörg Nick.

Billiards in polygons can exhibit bizarre behavior, some of which can be explained by deep connections to several seemingly unrelated branches of mathematics. These include algebraic geometry, Teichmuller theory and ergodic theory on homogeneous spaces. The talk will be an introduction to these ideas, aimed at a general mathematical audience.

More information: https://eth-its.ethz.ch/activities/its-science-colloquium.htmlcall_made

We will present a new quantitative approach to the problem of proving hydrodynamic limits from microscopic stochastic particle systems, namely the zero-range and the Ginzburg-Landau process with Kawasaki dynamics, to macroscopic partial differential equations. Our method combines a modulated Wasserstein-distance estimate comparing the law of the stochastic process to the local Gibbs measure, together with stability estimates a la Kruzhkov in weak distance and consistency estimates exploiting the regularity of the limit solution. It is simplified as it avoids the use of the block estimates. This is a joint work with Daniel Marahrens and Clément Mouhot (University of Cambridge).

In this talk, we will study Nash equilibria in two player continuous time stochastic differential games with diffusion control, and where the Brownian motions driving the state processes are correlated. We consider zero-sum ranking games, in the sense that the criteria to optimize only depends on the difference of the two players' state processes. We explicitly compute the players' equilibrium strategies, depending on the correlation of the Brownian motions driving the two state equations: in particular, if the correlation coefficient is smaller than some explicit threshold, then the equilibrium strategies consist of strong controls, whereas if the correlation exceeds the threshold, then the equilibrium controls are mixed strategies. To characterize these equilibria, we rely on a relaxed formulation of the game based on solutions to martingale problems, allowing the players to randomize their actions. The talk is based on a joint work with Stefan Ankirchner (University of Jena) and Julian Wendt (University of Jena).