Veranstaltungen

The Rauzy gasket is a fractal subset of the two dimensional simplex which is an important subset of parameter space in numerous dynamical and topological problems. Arnoux conjectured that the Hausdorff dimension of the Rauzy gasket is strictly less than 2, and since then there has been considerable interest in computing its Hausdorff dimension. In this talk we will also discuss a natural class of measures supported on the Rauzy gasket (the stationary measures for the projective action of the generators of the Rauzy gasket). We will show how recent developments from the theory of self-affine sets can be adapted to compute the dimension of these measures, which will allow us to establish an exact value for the Hausdorff dimension of the Rauzy gasket.

Q-Gorenstein toric contact manifolds provide an interesting class of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams. The main goal of this talk is to describe how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to the Ehrhart (quasi-)polynomial of its toric diagram. This is part of joint work with L. Macarini and M. Moreira, arXiv:2202.00442. If time permits, I will also briefly discuss an application to the study of periodic Reeb orbits on lens spaces (part of joint work with L. Macarini and H. Liu, arXiv:2211.16470).

The developments of topological insulators have provided a new avenue of creating interface modes (or edge modes) in photonic/phononic structures. Such created modes have a distinct property of being topologically protected and are stable with respect to perturbations in certain classes. In this talk, we will report recent results on the existence of in-gap interface modes that are bifurcated from Dirac points in photonic/phononic structures. Both one-dimensional and two-dimensional structures will be discussed.

The Doeblin Harris method is a well established tool in the study of long time asymptotics for Markov processes. In recent years, it has gained popularity in the PDE community, while adapting to problems for which more traditional tools such as e.g. entropy methods did not seem directly applicable. In the talk we present a simple PDE model involving partial diffusion for which such a strategy turned up fruitful. Extensions to a wider class of models coming from neurosciences raise some interesting questions related to pointwise lower bounds for Green's functions in situations where Hörmander's iterated bracket condition is not satisfied, at least not everywhere. This is joint and ongoing work with Delphine Salort.

I will discuss various aspects of the enumerative geometry of curves in an algebraic torus, formulated as the logarithmic enumerative geometry of a toric pair. I will explain how logarithmic double ramification cycles can be used to give a complete, albeit complex, solution to the logarithmic GW theory of all toric varieties, relative to their full toric boundary. I will then explain what happens in various special geometries when this solution can be made explicit. One simplification leads very quickly to traditional tropical correspondence theorems, recovering work of Mikhalkin, Nishinou-Siebert, and others. Another simplification leads to tropical refined curve counting, recovering work of Bousseau via integrable systems techniques. I will then explain how the logarithmic GW/DT conjectures (aka the LMNOP conjectures) come into the story via “triple double” ramification cycles. The talk is based on joint work with A. Urundolil Kumaran and D. Maulik, and touches upon forthcoming work of P. Kennedy-Hunt, Q. Shafi, and A. Urundolil-Kumaran.

Simplicial volume is a homotopy invariant for compact manifolds introduced by Gromov that measures the complexity of a manifold in terms of singular simplices. A celebrated question by Gromov (~’90) asks whether all oriented closed connected aspherical manifolds with zero simplicial volume also have vanishing Euler characteristic. In this talk, we will describe the problem and we will show counterexamples to some variations of the previous question. Moreover, we will describe some new strategies to approach the problem as well as the relation between Gromov’s question and other classical problems in topology. This is part of a joint work with Clara Löh and George Raptis.

The Chow groups of a blowup of a smooth variety along a smooth subvariety are described in Fulton's book using Grothendieck's "key formula", involving the Chow groups of the blown up variety, the center of blowup, and the Chern classes of its normal bundle. If interested in weighted blowups, one expects everything to generalize directly. This is in hindsight correct, except that at every turn there is an interesting and delightful surprise, shedding light on the original formulas for usual blowups, especially when one wants to pin down the integral Chow ring of a stack theoretic weighted blowup. As an application, one obtains a quick derivation of a formula, due to Di Lorenzo-Pernice-Vistoli and Inchiostro, of the Chow ring of the moduli space \bar{M}_{1,2}.

Variational formulations are a popular tool to analyse linear PDEs (eg. neutron diffusion, Maxwell equations, Stokes equations ...), and it also provides a convenient basis to design numerical methods to solve them. Of paramount importance is the inf-sup condition, designed by Ladyzhenskaya, Necas, Babuska and Brezzi in the 1960s and 1970s. As is well-known, it provides sharp conditions to prove well-posedness of the problem, namely existence and uniqueness of the solution, and continuous dependence with respect to the data. Then, to solve the approximate, or discrete, problems, there is the (uniform) discrete inf-sup condition, to ensure existence of the approximate solutions, and convergence of those solutions to the exact solution. Often, the two sides of this problem (exact and approximate) are handled separately, or at least no explicit connection is made between the two. In this talk, I will focus on an approach that is completely equivalent to the inf-sup condition for problems set in Hilbert spaces, the T-coercivity approach. This approach relies on the design of an explicit operator to realize the inf-sup condition. If the operator is carefully chosen, it can provide useful insight for a straightforward definition of the approximation of the exact problem. As a matter of fact, the derivation of the discrete inf-sup condition often becomes elementary, at least when one considers conforming methods, that is when the discrete spaces are subspaces of the exact Hilbert spaces. In this way, both the exact and the approximate problems are considered, analysed and solved at once. In itself, T-coercivity is not a new theory, however it seems that some of its strengths have been overlooked, and that, if used properly, it can be a simple, yet powerful tool to analyse and solve linear PDEs. In particular, it provides guidelines such as, which abstract tools and which numerical methods are the most “natural” to analyse and solve the problem at hand. In other words, it allows one to select simply appropriate tools in the mathematical, or numerical, toolboxes. This claim will be illustrated on classical linear PDEs, and for some generalizations of those models.

Due to the recent development in machine learning, data science and signal processing more and more data is generated, but only part of it might be necessary in order to already make predictions in a sufficiently good manner. Therefore, the question arises to best approximate a signal b by linear combinations of no more than \sigma vectors from a suitable dictionary A. Additionally, many areas of application - as for example portfolio selection theory, sparse linear discriminant analysis, general linear complementarity problems or pattern recognition - require the solution x to satisfy certain polyhedral constraints. This talk presents an analysis of the NP-hard minimization problem min{||b-Ax||: x \in [0,1]^n, |supp(x)| \leq \sigma} and its natural relaxation min{||b-Ax||: x \in [0,1]^n, \sum x_i \leq \sigma}. Our analysis includes a probabilistic view on when the relaxation is exact. We will also consider the problem from a deterministic point of view and provide a bound on the distance between the images of optimal solutions of the original problem and its relaxation under A. This leads to an algorithm for generic integer matrices A \in \mathbb{Z}^{m \times n} and achieves a polynomial running time provided that m and ||A||_{\infty} are fixed.

Toric contact manifolds provide an interesting class of contact manifolds. In this mini-​course we will introduce them, show how the ones with zero first Chern class can be determined by certain integral convex polytopes, called toric diagrams, and how to directly read relevant contact invariants from these toric diagrams. Plenty of hands-​on examples and some applications will be provided.

More information: https://math.ethz.ch/fim/activities/minicourses.htmlcall_made

Let C be a Riemann surface of genus g and let p₁, ..., p_n be n distinct points on C. For an n-tuple of integers (a₁, ..., a_n) which sum up to zero, one can ask when the holomorphic line bundle O_C(a₁.p₁ + ... + a_n.p_n) is trivial. The complete answer to this question was given by Abel in 19th century. Now one can ask a similar question over the moduli space of marked Riemann surfaces or its Deligne-Mumford compactification. How can we compactify this locus and can we compute the homology class of this locus systematically? We will see a quantitative answer to this question and its generalisation.

General relativity makes spectacular predictions about our world, predictions which have captured the popular imagination more than any other part of physics: gravitational waves, black holes, spacetime singularities. For the mathematician, however, perhaps the most spectacular thing about these predictions is not their exoticness, but, on the contrary, the fact that they all correspond to well-defined mathematical concepts: Indeed, it was precisely through mathematics that these predictions of general relativity were first discovered—originally to much controversy and objection!—and the qualitative mathematical analysis of the Einstein equations remains one of the most powerful ways to understand the great conceptual questions of the theory. This talk will describe some past contributions of mathematics to general relativity and some of the big open conjectures which mathematics hopes to answer in the future.

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

We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems from an iid sample. This procedure is the first one that exhibits the optimal statistical performance in heavy tailed situations and also applies in highdimensional settings. We discuss the portfolio optimization problem as a special instance. Joint work with Shahar Mendelson.

I will lightly discuss a few problems closely connected to Vinogradov's Mean value theorems going beyond the optimal conjecture thereof (proved by Wooley and Bourgain--Demeter--Guth in 2015). Depending on time and the audience's interest I will focus on one of the problems.

The start of 2020 saw a then-novel coronavirus, SARS-CoV-2, spread rapidly across the globe. A stand-out characteristic of the resulting pandemic has been the incredible level of genomic surveillance applied, which has resulted in over 15 million SARS-CoV-2 publicly-available genomes to date. In this talk I will provide a brief introduction to a statistical inference framework known as Bayesian phylodynamics. This framework combines ideas from computational phylogenetics and population genetics to produce model-based inferences of key epidemiological parameters using pathogen sequences and other epidemiological data. I will then go on to discuss some recent applications of this framework to the inference of basic SARS-CoV-2 reproductive numbers, case count dynamics, as well as the quantification of the genetic evidence for the effectiveness of Swiss contact tracing efforts.

Given a log Calabi-Yau pair (Y,D), the intrinsic mirror symmetry construction of Gross-Siebert builds the mirror family X of (Y,D) as a geometric generating function of the genus 0 punctured Gromov-Witten invariants of (Y,D). String theory loosely predicts that period integrals on X equal generating functions of genus 0 log Gromov-Witten invariants of (Y,D). In this joint work with Helge Ruddat and Bernd Siebert, we verify this prediction for pairs (Y,D) with Y a smooth Fano variety and D a smooth anticanonical divisor, and for insertions curve classes on D. A key to this is that insertions of (Y,D) determine Lagrangians of X via tropical geometry.