Bo\'az Klartag (The Weizmann Institute of Science)
Isoperimetric inequalities in high-dimensional convex sets
10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Roberto Colombo (EPFL)
Abstract:
Gradient flows in the Wasserstein space of probability measures have proven very useful for providing new interpretations of many parabolic partial differential equations in relation to Optimal Transport theory. Recently, it has been observed that this formalism can also be used to describe the learning dynamics of certain (continuous) two-layer neural networks. In a model case, this evolution corresponds to the Wasserstein gradient flow of the energy given by a negative Sobolev distance to a fixed target measure. The resulting active-scalar continuity equation shares several analogies with the vorticity formulation of the Euler equations for 2D fluids, while exhibiting markedly different qualitative long time behavior. A natural question is to identify conditions under which the system converges to the target, possibly with an explicit rate of convergence. In joint work with Lénaïc Chizat, Maria Colombo, and Xavier Fernández-Real, we address this question from a PDE perspective, obtaining precise exponential or polynomial convergence rates under suitable smoothness assumptions. In this seminar, we will introduce Wasserstein gradient flows of negative Sobolev discrepancies and describe the main ideas behind the aforementioned quantitative convergence results. pdf_version
12:15 • Universität Basel, Seminarraum 05.001
Christopher Kuo (MPIM Bonn)
Perverse microsheaves on contact manifolds abstract
Abstract:
Kashiwara showed in 1996 that the categories of microlocalized D-modules can becanonically glued to give a sheaf of categories over a complex contactmanifold. In this talk, we explain how the corresponding sheaf of categoriescan be defined on the perverse sheaf side. One of the main points is themicrolocal definition of perversity due to Kashiwara and Schapira, togetherwith its contact-geometric interpretation via a result of Nadler and Shende. Iftime permits, we will comment on the comparison between these twoconstructions, namely, the microlocal Riemann–Hilbert correspondence. This isjoint work with Laurent Côté, David Nadler, and Vivek Shende.
13:00 • EPF Lausanne, CM 1 517
Prof. Dr. Boaz Klartag (Weizmann Institute of Science, IL)
Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid abstract
Abstract:
We prove that in any dimension n there exists an origin-symmetric ellipsoid {\\mathcal{E}} \\subset {\\mathbb{R}}^n of volume c n^2 that contains no points of {\\mathbb{Z}}^n other than the origin, where c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in {\\mathbb{R}}^n whose density is at least cn^2 \\cdot 2^{-n}. Previously known constructions of sphere packings in {\\mathbb{R}}^n had densities of the order of magnitude of n \\cdot 2^{-n}, up to logarithmic factors. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least c n^2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Christof Seiler (Klinik für Rheumatologie, USZ)
Uncertainty Quantification of Prediction Models for Differential Expression Analysis abstract
Abstract:
Differential expression analyses for single-cell RNA sequencing typically use empirical Bayes methods such as DESeq2, edgeR, limma, and MAST. These approaches perform univariate statistical testing by modeling gene expression with generalized linear models and borrow strength across genes to stabilize variance estimates. In this talk, I will introduce a framework that borrows strength also for the estimation of the gene expression itself by predicting a gene of interest from the other genes. Our R package, conformeR, combines counterfactual prediction with conformal prediction to leverage the multivariate structure of the data and increase statistical power. This is joint work with Justine Leclerc.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Rostislav Grigorchuk (Texas A & M University)
A hunt for spectral gaps abstract
Abstract:
In my talk I will describe an approach to studying spectra of groups and graphs which originated thirty years ago during my first visit to Geneva and was developed in collaboration with Laurent Bartholdi, Pierre de la Harpe, Tatiana Nagnibeda, Christophe Pittet and other colleagues. We will discuss such topics as graphs whose spectrum is a Cantor set, pure point spectrum vs continuous spectrum, the question of Alain Valette "Can one hear the shape of a group? ", Ramanujan graphs coming from automata groups.
16:15 • Université de Genève, Conseil Général 7-9, Room 1-15
Philippe Anjolras
Path-connectedness for the incompressible Euler equation abstract
Abstract:
<p style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; caret-color: #000000; color: #000000;">In 2009, Camillo De Lellis and László Székelyhidi Jr. proved a non-uniqueness result for weak solutions to the incompressible Euler equation, through a convex integration method coming from geometry (Nash 54', Gromov 86'). A consequence of their proof is the following surprising result : the set of weak solutions of the equation is dense in $L^{\\infty}_t L^2_x$ for the weak topology of this space. By generalizing the geometric framework of their approach, I will explain how to show that the set of weak solutions is path-connected for the strong topology, and I will give the main ideas of the proof of De Lellis and Székelyhidi. </p>
16:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Martin Lieberherr (MNG Rämibühl)
Computational Physics – Beispiele aus Schülerprojekten und Unterrichtsvorbereitung abstract
Abstract:
Computational Physics ist in den 1950er Jahren entstanden und hat sich als mächtige Methode für Theorie und Experiment etabliert. Sie verbindet Physik, Mathematik und Informatik. Die Gymnasien sollten da nicht zurückstehen, sondern die Schülerinnen und Schüler mit entsprechenden, stufengerechten Fähigkeiten ausstatten. Der Zeitpunkt ist ideal, denn die neuen Rahmenlehrpläne fordern Interdisziplinarität.Im Vortrag werden Beispiele aus dem Projektunterricht "Fourierreihen" und dem "computational physics lab" vorgestellt sowie einige Beispiele, die ich für die Unterrichtsvorbereitung und persönliche Weiterbildung verwendet habe.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Prof. Dr. Lukasz Delong (University of Warsaw)
Universal inference for testing calibration of mean estimates within the Exponential Dispersion Family abstract
Abstract:
The calibration of mean estimates, which requires that predictions are, on average, equal to the observed responses, is a critical property for reliable decision-making, particularly in actuarial and financial applications. In this presentation, first, we review classic approaches for validating the mean-calibration and introduce the Likelihood Ratio Test (LRT) within the Exponential Dispersion Family (EDF). Next, we investigate the framework of universal inference to test the mean-calibration. We develop a sub-sampled split Likelihood Ratio Test (sub-sampled split LRT) within the EDF that provides finite-sample guarantees with universally valid critical values. We investigate type I error, power and e-power of our sub-sampled split LRT, compare the sub-sampled split LRT to the classic LRT, study the effect of sub-sampling of training and test sets on the split LRT, and propose novel test statistics based on the sub-sampled split LRT to enhance performance the test. In our numerical experiments, we demonstrate that the sub-sampled split LRT and our modifications are appealing alternatives to the classic LRT and achieve high power in detecting miscalibration, offering a practical and powerful toolkit for validating the calibration of mean estimates.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43