Weekly Bulletin

The structure of a topological space can be studied by the existence of simplices and their relations. An important topic in Symplectic Geometry is understanding intersections of Lagrangian submanifolds. The Fukaya category of a symplectic manifold categorifies this intersection theory. Biran and Cornea proved that Lagrangian cobordisms induce exact simplices in the Fukaya category of certain compact Lagrangian submanifolds in certain symplectic manifolds. In this talk, we transfer this result to include certain non-compact Lagrangian submanifolds. The second part discusses how many exact simplices can be recovered by Lagrangian cobordisms. One way to measure this is by comparing the Lagrangian cobordism group to the Grothendieck group of the Fukaya category. We compute the Lagrangian cobordism groups of Weinstein manifolds and get a first example where the two groups do not agree.

Verlinde series are generating functions of Euler characteristics of line bundles on the Hilbert schemes of points on a surface. Formulas for Verlinde series were determined for surfaces with K=0 by Ellingsrud, Göttsche, and Lehn. More recently, Göttsche and Mellit determined Verlinde series for surfaces with K^2=0, and gave a conjectural formula in the general case. In this talk, I will give a formula for the Euler characteristics of line bundles on the Hilbert schemes of points on CP1 x CP1, and a combinatorial (but less explicit) formula for ample line bundles on the Hilbert schemes of points on Hirzebruch surfaces. By structural results of Ellingsrud, Göttsche, and Lehn, this determines the Verlinde series for all surfaces. The proof is based on a new combinatorial description of the equivariant Verlinde series for the affine plane.

Which functions preserve positive semidefiniteness (psd) when applied entrywise to psd matrices? This question has a long history beginning with Schur, Schoenberg, and Rudin, and has also recently received renewed attention due to applications in high-dimensional statistics. I will explain some of these developments, and connections to other areas. These include analysis results on Fourier sequence preservers (by Rudin) and moment sequence preservers (in recent work). Next come contributions of Loewner, Karlin, and their students: FitzGerald, Horn, Micchelli, and Pinkus, on entrywise maps in all dimensions and in a fixed dimension. I will end with some recent results that describe hitherto undiscovered connections to combinatorics, via a novel graph invariant. (Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, Bala Rajaratnam, and Terence Tao.)

In 1962, Yudovich established the well-posedness of the two-dimensional incompressible Euler equations within the class of solutions with bounded vorticity. Since then, a central unresolved problem has been the question of uniqueness within the broader class of solutions with L^p-vorticities. Recent years have witnessed significant progress in this investigation. In my talk, I aim to provide an overview of these developments and highlight recent results obtained thanks to the convex integration method.

''An algebraic integer is a complex number which is a root of a monic irreducible polynomial with integer coefficients. The complete set of zeros of such a polynomial is called a conjugate set of algebraic numbers. Bounding the maximum absolute value of elements in these sets from below has been studied intensively over the years by number theorists. We will call this maximum the house of an algebraic integer. In 1965, Schinzel and Zassenhaus proposed the following conjecture. There exists an absolute positive constant C such that the house of every non-zero algebraic integer which is not a root of unity is at least 1 + C/d. The above conjecture was proved in 2019 by Dimitrov. In this talk we will introduce the relevant notions and go over Dimitrov's proof.

This talk concerns critical points $u$ of polyconvex energies of the form $f(X) = g(det(X))$, where $g$ is (uniformly) convex. It is not hard to see that, if $u$ is smooth, then $\det(Du)$ is constant. I will show that the same result holds for Lipschitz critical points $u$ in the plane. I will also discuss how to obtain rigidity for approximate solutions. This is a joint work with A. Guerra.

The advent of generative AI has turbocharged the development of a myriad of commercial applications, and it has slowly started to permeate to scientific computing. In this talk we discussed how recasting the formulation of old and new problems within a probabilistic approach opens the door to leverage and tailor state-of-the-art generative AI tools. As such, we review recent advancements in Probabilistic SciML – including computational fluid dynamics, inverse problems, and particularly climate sciences, with an emphasis on statistical downscaling. Statistical downscaling is a crucial tool for analyzing the regional effects of climate change under different climate models: it seeks to transform low-resolution data from a (potentially biased) coarse-grained numerical scheme (which is computationally inexpensive) into high-resolution data consistent with high-fidelity models. We recast this problem in a two-stage probabilistic framework using unpaired data by combining two transformations: a debiasing step performed by an optimal transport map, followed by an upsampling step achieved through a probabilistic conditional diffusion model. Our approach characterizes conditional distribution without requiring paired data and faithfully recovers relevant physical statistics, even from biased samples. We will show that our method generates statistically correct high-resolution outputs from low-resolution ones, for different chaotic systems, including well known climate models and weather data. We show that the framework is able to upsample resolutions by 8x and 16x while accurately matching the statistics of physical quantities – even when the low-frequency content of the inputs and outputs differs. This is a crucial yet challenging requirement that existing state-of-the-art methods usually struggle with.

Time-change equations are a generalization of ordinary differential equations which are driven by the random, irregular, and possibly densely discontinuous sample paths of the typical stochastic process. They can be thought of as a multiparameter version of the method of time-change and can be given a pathwise theory. Time-change equations can lead to deep results on weak existence and uniqueness of stochastic differential equations and posses a robust strong approximation theory. However, time-change equations are not restricted to Markovian or semimartingale settings. In this talk, we will go through some examples of time-change equations which can be succesfully analyzed (such as (multidimensional) affine processes, sticky Lévy processes or Doeblin´s mostly unknown proposal for diffusion processes) as well as some open problems they suggest.

In studies ranging from clinical medicine to policy research, complete data are usually available from a population P, but the quantity of interest is often sought for a related but different population Q. In this talk, we consider the unsupervised domain adaptation setting under the label shift assumption. In the first part, we estimate a parameter of interest in population Q by leveraging information from P, where three ingredients are essential: (a) the common conditional distribution of X given Y, (b) the regression model of Y given X in P, and (c) the density ratio of the outcome Y between the two populations. We propose an estimation procedure that only needs some standard nonparametric technique to approximate the conditional expectations with respect to (a), while by no means needs an estimate or model for (b) or (c); i.e., doubly flexible to the model misspecifications of both (b) and (c). In the second part, we pay special attention to the case that the outcome Y is categorical. In this scenario, traditional label shift adaptation methods either suffer from large estimation errors or require cumbersome post-prediction calibrations. To address these issues, we propose a moment-matching framework for adapting the label shift, and an efficient label shift adaptation method where the adaptation weights can be estimated by solving linear systems. We rigorously study the theoretical properties of our proposed methods. Empirically, we illustrate our proposed methods in the MIMIC-III database as well as in some benchmark datasets including MNIST, CIFAR-10, and CIFAR-100.

Although practised as an art and science for ages, cryptography had to wait until the mid-​twentieth century before Claude Shannon gave it a strong mathematical foundation. However, Shannon's approach was rooted in his own information theory, itself inspired by the classical physics of Newton and Einstein. When quantum theory is taken into account, new vistas open up both for codemakers and codebreakers. Is this a blessing or a curse for the protection of privacy? As we shall see, it can go both ways, with an emphasis on quantum cryptography, from its humble origins more than a half-​century ago to its glorious future.

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