Dr. Vikramaditya Giri (Universität Zürich)
A Spectral Gap for Spinors on Hyperbolic Surfaces abstract
Abstract:
<p>We'll recall the spectral theory of the Laplacian on functions on a hyperbolic surface and show that as the genus goes to infinity, one can't obtain a uniform spectral gap above 1/4 - the bottom of the \\(L^2\\) spectrum of the hyperbolic plane. We'll show that this obstruction goes away when one instead considers the spectrum of the Dirac operator on spinors on a hyperbolic surface with a choice of spin structure. We'll sketch a construction of these surfaces with such a uniform spectral gap and note that these surfaces can be taken to be arithmetic.</p> <p>
Based on joint work with Anshul Adve.</p>
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Marius Zeinhofer
Geometric Optimization in Scientific Machine Learning abstract
Abstract:
We discusses an “optimize-then-project” approach for applications in scientific machine learning. The key idea is to design algorithms at the infinite-dimensional level and subsequently discretize them in the tangent space of the neural network ansatz. We illustrate this approach in the context of the variational Monte Carlo method for quantum many-body problems, where neural quantum states have recently emerged as powerful representations of high-dimensional wavefunctions. In this setting, we recover the celebrated stochastic reconfiguration algorithm, interpreting it as a projected Riemannian L2 gradient descent method. We further explore extensions to Riemannian Newton methods, and conclude with considerations related to the scalability of these schemes.
13:30 • EPF Lausanne, CM 1 517
Olivier Schiffmann
Hecke operators, COHAs and chi-independence for symplectic surfaces abstract
Abstract:
In his study of Gopakumar-Vafa curve counting invariants on CY 3folds, Toda conjectured that the (BPS) cohomology of moduli stacks of semistable one-dimensional sheaves only depends on the curve class (and not on the Euler characteristics) of sheaves considered. This allows one to extend the computation of these invariants to the singular setup. We prove this conjecture (and in fact a slightly stronger statement, relative over the Chow variety) for CY3 folds of the form $X=S\\times \\mathbb{A}^1$, where $S$ is a symplectic surface. For this we use the theory of (2d and 3d) cohomological Hall algebras, as well as Hecke operators of punctual modifications on surfaces. As a corollary, we obtain a computation of both the intersection homology of the (coarse) moduli space and the homology of the moduli stack of semistable coherent sheaves on a projective K3 or abelian surface for any Mukai vector and (generic) polarization. We also obtain an extension of Markman \'tautological generation of the cohomology\' theorem to the case of nonprimitive Mukai vectors. This is joint work with Ben Davison, Lucien Hennecart, Tasuki Kinjo and Eric Vasserot.
14:15 • EPF Lausanne, CM 1 517
Dr. Katerina Papagiannouli (University of Pisa)
Abstract:
Bayesian neural networks, in the overparameterized and infinite-width regime, are now well understood. Under mild assumptions, their prior converges to a Gaussian process (NNGP), and both Bayesian inference and training dynamics can be described by kernel methods. Although, these infinite-width limits provide tractable models and sharp theoretical insights, they also exhibit a fundamental rigidity: the induced feature representation becomes fixed and independent of data. As a result, feature learning disappears in the infinite-width limit, and Bayesian inference reduces to kernel regression with a predetermined kernel.In this talk, I present a complementary large-deviation perspective on wide Bayesian neural networks. Rather than studying typical Gaussian fluctuations, we analyse exponentially rare, but statistically dominant, configurations that govern posterior concentration as width grows. At this scale, Bayesian inference becomes variational: posterior mass concentrates near minimizers of an explicit functional rate function defined directly on predictors.Our main result shows that, in contrast to the Gaussian-process limit, the posterior large-deviation rate function involves a joint optimization over predictors and internal covariance kernels. This nested variational structure leads to data-dependent kernel selection and provides a mechanism for feature learning that persists even in the infinite-width regime. In particular, we prove that the posterior-optimal kernel generically differs from the NNGP kernel.
16:15 • Universität Bern, Hörsaal B78, ExWi, Sidlerstrasse 5, 3012 Bern
Dr. David Wiedemann (TU Dortmund)
Interface conditions for Maxwell’s equations by homogenization of thin inclusions: transmission, reflection or polarization abstract
Abstract:
We investigate the time-harmonic Maxwell equations in complex geometries, with particular emphasis on configurations that model polarization filters or Faraday cages. The domains under consideration contain perfectly conducting inclusions that are periodically distributed along a surface. We analyze the asymptotic regime in which both the size of the inclusions and the distance between them tend to zero, and we derive the resulting effectiveinterface conditions. Depending on the geometric properties of the inclusions, the limiting system exhibits markedly different behaviors: perfect transmission, perfect reflection orpolarization effects
16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Dr. Sahar Rabin Diskin (ETH Zürich, Switzerland)
Mixing time and diameter of the percolated hypercube abstract
Abstract:
We study bond percolation on the d-dimensional hypercube Q^d with edge retention probability p=c/d. It is well known that when c>1 is fixed, a unique giant component emerges. In this regime, we resolve conjectures of Bollobás, Kohayakawa, and Łuczak (1994) and of Benjamini and Mossel (2003), showing that the typical diameter of the giant component is Θ(d), and that the mixing time of the lazy random walk on it is Θ(d^2). In the talk, we will introduce the notion of mixing time and its connection to expansion properties of subsets of the giant. We will then discuss some of the key obstacles in obtaining this result, and in particular why classical sprinkling techniques are insufficient for this problem. Finally, we will explain how our new approach - based on analysing the effect of small perturbations and establishing stability under thinning - overcomes these obstacles. This method also yields tight large-deviation estimates for the size of the giant. Based on joint work with Michael Anastos, Lyuben Lichev, and Maksim Zhukovskii.
17:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H12