Weekly Bulletin

Floer homology and the related invariants of 4-manifolds has given us deep insight in smooth differential topology in dimensions 3 and particularly 4. The theory has yielded insights like existence of exotic differentiable structures on 4 dimensional euclidean space, complex curves minimize genus in complex projective space, killing the Hauptvermuntung, there even appear to be connection to the 4 color map theorem. This course will build up Floer homology of three manifolds from scratch. The focus will be on Instanton Floer homology but we will mention other versions and develop applications as the course goes on.

In this course we will investigate questions of weak turbulence theory by using as explicit example of wave interactions the solutions to periodic and nonlinear Schrödinger equations. We will start with Strichartz estimates on periodic setting, then we will move to well-posedness. We will then present two different ways of introducing the evolution of the energy spectrum. We will first work on a method proposed by Bourgain and involving the growth of high Sobolev norms. Then, we will give some ideas of how to derive rigorously the effective dynamics of the energy spectrum itself (wave kinetic equation), when one considers weakly nonlinear dispersive equations.

In the groundbreaking works [2,3], Vishik proved non-uniqueness for the 2D Euler equation with forcing, below the Yudovich well-posedness class. A nice exposition of this result can be found in [1]. In this talk, we present a simpler proof and show how to extend the result to the Surface Quasi-Geostrophic (SQG) equation. Specifically, we prove non-uniqueness for the forced $\alpha$-SQG equation in $H^s$ for any $s<1+\alpha$ and $0\leq\alpha\leq 1$. This family of active scalar equations interpolates between the 2D Euler equation ($\alpha=0$) and the SQG equation ($\alpha=1$). This is a joint work with Castro, Faraco and Solera. [1] D. Albritton, E. Brué, M. Colombo, C. De Lellis, V. Giri, M. Janisch, and H. Kwon. Instability and non-uniqueness for the 2D Euler equations, after M. Vishik. Annals of Mathematics Studies. Princeton University Press, Princeton, 2024. [2] M. Vishik. Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I. arXiv:1805.09426, 2018. [3] M. Vishik. Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II. arXiv:1805.09440, 2018.

As firstly conjectured by Erdös and Turán in 1936, in 1972 Szemerédi proved that any positive density subset of \(N\) contains arbitrary long arithmetic progressions. Determinant contributions came from very different fields: harmonic analysis, graph theory and ergodic theory. This theorem uncovered deep connections between these fields and sits at the foundation of the celebrated Green-Tao theorem about arithmetic progressions of prime numbers. During the talk we will introduce the notion of density of a subset of \(N\) and we will motivate the statement of the theorem. We will then turn our attention to Roth's theorem (Szemerédi's theorem for arithmetic progressions of length 3): we will sketch the harmonic analysis proof by Roth (1953) and we will mention Szemerédi's alternative one, that exploits the 'subgraph removal lemma' and opened the way to a proof of Szemerédi's theorem. Finally we will discuss Furstenberg alternative proof (1977) of Szemerédi's theorem, based on his 'Correspondence principle' between subsets of \(N\) and measure preserving dynamical systems.

I will discuss the problem of affinely embedding self-similar sets in the line into other such sets. Conjecturally, embedding is precluded when the contraction ratios of the defining maps are incommensurable. This is closely related to conjectures on intersections of fractals, but in the open cases even the embedding problem is challenging. I will describe recent joint work with Amir Algom and Meng Wu in which we confirm the conjecture whenever the contraction ratios are algebraic numbers, and also for a.e. choice of parameters. I will discuss the proof. If time permits, I will explain how this is related to the dimension of alpha-beta sets, and describe recent examples that show that the problem is more delicate than anticipated.

I will discuss examples of birational models of M_{g,n} that on the one hand are given as GIT-quotients for torus actions on explicit affine schemes, and on the other hand admit modular descriptions. I’ll focus mostly on cases g=0 and g=1.

There has been a lot of interest in understanding which knots are characterized by which Dehn surgeries. In this talk, I'll propose studying a 4-dimensional version of this question: which knots are determined by the orientation-preserving diffeomorphism types of which traces? I'll discuss several results that are in stark contrast with what is known about characterizing slopes; for example, that every algebraic knot is determined by its 0-trace. Moreover, every positive torus knot is determined by its n-trace for any n <= 0, whereas no non-positive integer is known to be a characterizing slope for any positive torus knot besides the right-handed trefoil. Our proofs use tools from Heegaard Floer homology and results about surface homeomorphisms.

Interacting particle systems provide flexible and powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. However, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computationally. Therefore, a common strategy is to derive effective equations that describe the time evolution of the empirical particle density. A prototypical example that we will consider is the formal identification of a finite system of particles with the singular Dean-Kawasaki equation. Our aim is to introduce a well-behaved nonlinear SPDE that approximates the Dean-Kawasaki equation for a particle system with mean-field interaction both in the drift and the noise term. We want to study the well-posedness of these nonlinear SPDE models and to control the weak error of the SPDE approximation with respect to the particle system using the technique of transport equations on the space of probability measures. This is the joint work with H. Kremp, N. Perkowski and J. Xiaohao.

When applying multivariate extreme values statistics to analyze tail risk in compound events defined by a multivariate random vector, one often assumes that all dimensions share the same extreme value index. While such an assumption can be tested using a Wald-type test, the performance of such a test deteriorates as the dimensionality increases. This paper introduces a novel test for testing extreme value indices in a high dimensional setting. We show the asymptotic behavior of the test statistic and conduct simulation studies to evaluate its finite sample performance. The proposed test significantly outperforms existing methods in high dimensional settings. We apply this test to examine two datasets previously assumed to have identical extreme value indices across all dimensions. This is a joint work with Liujun Chen (USTC)

In this talk, we study the causal distributionally robust optimization (DRO) in both discrete and continuous-time settings. The framework captures model uncertainty, with potential models penalized in function of their adapted Wasserstein distance to a given reference model. The strength of model uncertainty is parameterized via a penalization parameter, and we compute the first-order sensitivity of the value of causal DRO with respect to this parameter. Moreover, we investigate the case where a martingale constraint is imposed on the underlying model, as is the case for pricing measures in mathematical finance. We introduce different scaling regimes, which allow us to obtain the continuous-time sensitivities as nontrivial limits of their discrete-time counterparts. Our proofs rely on novel methods. In particular, we introduce a pathwise Malliavin derivative, which agrees with its classical counterpart under the Wiener measure, and we extend the adjoint operator, the Skorokhod integral, to regular martingale integrators and show it satisfies a stochastic Fubini theorem.

Modern deep learning in combination with satellite data offers great opportunities to protect nature at global scale. I will present ongoing research to map crops at country-scale, for species distribution modeling, to estimate vegetation parameters such as biomass and vegetation height, and how conflicts can be monitored remotely. Traditional approaches usually must be adapted for specific ecosystems and regions. It is therefore very difficult to carry out homogeneous, large-scale modeling with high spatial and temporal resolution and, at the same time, good accuracy. Data-driven approaches, especially modern deep learning methods, promise great potential here to achieve globally consistent, transparent assessments of our environment. Bio: Jan Dirk Wegner leads the EcoVision Lab at the DM3L at University of Zurich as an Associate Professor. Jan was PostDoc (2012-2016) and senior scientist (2017-2020) in the Photogrammetry and Remote Sensing group at ETH Zurich after completing his PhD (with distinction) at Leibniz Universität Hannover in 2011. His main research interests are at the frontier of machine learning, computer vision, and remote sensing to solve scientific questions in the environmental sciences and geosciences. Jan was granted multiple awards, among others an ETH Postdoctoral fellowship and the science award of the German Geodetic Commission. He was selected for the WEF Young Scientist Class 2020 as one of the 25 best researchers world-wide under the age of 40 committed to integrating scientific knowledge into society for the public good. Jan is vice-president of ISPRS Technical Commission II, associated faculty of the ETH AI Center, director of the PhD graduate school "Data Science" at University of Zurich, and his professorship is part of the Digital Society Initiative at University of Zurich. Together with colleagues, Jan is chairing the CVPR EarthVision workshops.

The adjoint hypersurface of a simple convex polytope is uniquely defined by interpolation conditions. Its defining equation is the numerator of a meromorphic volume form associated to the polytope, called its canonical form. That form has applications in optimization and particle physics. We extend these assertions to amplituhedra in the Grassmannian of lines in three-space. Joint work with Kristian Ranestad and Rainer Sinn.