Events

Associated to any loop of Lagrangian submanifolds is a cohomology class of the basepoint: its flux. However, contrary to the symplectic setting, this does not define a morphism on the fundamental group of the space of Lagrangian submanifolds. I will explain how this can be rectified by considering the monodromy of the loop and how the algebraic properties of the ensuing flux-monodromy morphism are related to topological properties of the Hamiltonian orbit of the basepoint. Finally, I will provide explicit computations for certain Lagrangian tori, revealing novel structure in the associated fundamental groups. This is joint work with Joé Brendel and Rémi Leclercq.

This talk is based on joint work with Emanuel Milman (Technion) and Hiroshi Tsuji (Institution of Science Tokyo). The Brascamp–Lieb inequality (for multilinear functionals), originally introduced in 1976 as a generalization of Young’s convolution inequality, has since been found to be useful across a wide range of fields. For example, in 1991, within convex geometry, Ball showed that the Brascamp–Lieb inequality implies the reverse isoperimetric inequality. More recently, it has played a critical role in harmonic analysis, particularly in the context of the Fourier restriction conjecture and the Kakeya conjecture (e.g. Bennett—Carbery—Tao). In this talk, we focus on inequalities for symmetric convex bodies within convex geometry. Specifically, we consider the Blaschke–Santaló inequality and its multilinear version, a Talagrand-type inequality for the Wasserstein barycenter, the Gaussian correlation inequality and its strengthened version. On the analytic side, we also include a Laplace transform bound and an improvement of Borell’s reverse hypercontractivity. We will report on new developments obtained by reinterpreting this collection of inequalities from the perspective of the Brascamp–Lieb inequality.

Linear programming (LP) is a central problem in optimization with broad applications across science, engineering, and business. Interior point methods (IPMs) are among the most widely used algorithms for solving LPs and are known for their fast convergence both in theory and in practice. This talk surveys several recent theoretical developments in IPMs for LP. I will show how dynamic data structures can be used to implement IPM iterations more efficiently, leading to faster and nearly-optimal solvers for both general and structured LPs. I will also outline an ongoing project on improving the iterative complexity of IPMs for random LPs, providing insight into why these methods perform so well empirically.

<div style="font-size: 12pt; font-family: Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif;" data-olk-copy-source="MessageBody">The classical Kakeya problem in Euclidean space asks how "small" a set can be if it contains a unit line segment in every direction. More generally, packing sets contain all images of a given set under a collection of transformations. Finite field analogues offer a combinatorial and algebraic perspective and motivate our work.     </div> <div style="font-size: 12pt; font-family: Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif;"> </div> <div style="font-size: 12pt; font-family: Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif;">We study sets E in the affine plane over a finite field that are invariant under a large subgroup R of SL_2. We prove that if |R|>c|E|^{3/2}, then E must be contained in a line. The exponent 3/2 is sharp. We also state a conjecture in the Euclidean setting. This is joint work with Thang Pham and Le Quang-Hung.</div>

<p style="text-align: justify;"><span style="color: #000000;"><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px;">Consider the random walk on a nilmanifold \(X\) induced by random automorphisms on some fixed starting point \(x_0\).<br>By a theorem of Bourgain-Furman-Lindenstrauss-Mozes, for \(X\) a torus and \(x_0\) irrational, the random walk approaches a uniform distribution at an exponential rate. <br>Benoist-Quint partially extends this to a nilmanifold, without giving any rate. Bekka-Guivarc'h shows a spectral gap exists in this setting.<br>I will report on an upcoming work with Weikun He and Elon Lindenstrauss that generalizes these results and our previous result on Heisenberg nilmanifolds, giving an effective rate of equidistribution from random walks by automorphisms on 2-step nilmanifolds.</span></span></p>

We show how tropical geometry provides an effective tool for cohomological computations on (partial) compactifications of the moduli space of principally polarized abelian varieties. As an application, we extend the Fourier-Mukai transform to the torus rank 1 partial compactification and use it to define a weight filtration on the Chow ring of the partial compactification of the universal family. We also discuss extensions to more general toroidal compactifications, which allow us to study the moduli space of log abelian varieties and reinterpret results by Faltings-Chai.

Every knot leaves a trace in the 4-dimensional world. The trace of a knot is the smooth 4-manifold obtained by attaching a 2-handle to the 4-ball along a knot in the 3-sphere. We will introduce the relevant notions and present a strategy to disprove the smooth 4-dimensional Poincaré conjecture by finding knot traces with certain exotic properties. In the second part of the talk, we will discuss different methods to search for such exotic knot traces. This talk will mainly be based on joint work with Jonathan Spreer.

Every monotone Lipschitz map from the circle into the Euclidean plane admits a monotone Lipschitz extension to the disc. This property generally fails for non-smooth metric surface targets. However, by passing to the slightly larger class of Sobolev mappings, one can prove an analogous extension result. In this talk, we show how such extensions can be constructed using a suitable collar construction combined with energy minimization techniques.

Classical hypothesis testing requires the significance level alpha be fixed before any statistical analysis takes place - violating this requirement is a mortal (yet frequently and often unwittingly committed!) sin in classical testing. The requirement is stringent: it prohibits updating alpha during (or after) an experiment due to changing concern about the cost of false positives, or to reflect unexpectedly strong evidence against the null. Perhaps most disturbingly, witnessing a p-value p << \alpha vs p-just-slightly-smaller-than alpha has no statistical relevance for any downstream decision-making, making the evidence conveyed by a p-value (or a classical confidence interval) exceedingly hard to interpret. Here we show that tests based on e-values (wikipedia), a modern alternative to the. p-value, do allow for risk control with a data-dependent alpha. While e-values have been mostly associated with their suitability for anytime-valid tests and confidence, we consider their post-hoc validity at least as important. We show that, under a suitable (weak) definition of 'admissibility', every admissible post-hoc valid test must be based on e-values. If the significance level is fixed in advance and the notion of 'admissibility' is strengthened, this new result - really a complete class theorem - reduces to the familiar Neyman-Pearson Lemma. Throughout the talk we illustrate our results using various types of e-values, including the newly defined "conditional" ones. Literature: G. Beyond Neyman-Pearson: e-values enable hypothesis testing with a data-driven alpha. Proceedings National Academy of Sciences of the USA (PNAS), 2024. B. Chugg, T. Lardy, A. Ramdas, G. On Admissibility in Post-Hoc Hypothesis Testing. Intern. Journal of Approx. Reasoning, 2026.

<p>We shall present a recent work on  the semiclassical limit of NLS and Hartree type equations for infinitely many particles under suitable assumptions on  the Wigner transform of the initial  datum. The formal limit is a singular  kinetic equation of Vlasov type  where the force field is given by the gradient of the density.<br>We will recall previous results on the well-posedness of this type of kinetic equations<br>and explain how to get uniform estimates suitable to justify the semiclassical limit.<br>Joint work with Daniel Han-Kwan (Nantes)</p>

Neural generative models optimize global error measures but offer no guarantees on structural properties. This talk presents three settings where structure matters and standard methods fail, along with theory-guided fixes. (1) Extreme values: replacing Gaussian latent noise with Fréchet inputs in GANs enables universal approximation of tail dependence functions. (2) Order statistics: classical representations (Sukhatme, Schucany) decompose ranked data generation into neural-approximable blocks, with complexity independent of sample size. (3) Diffusion fine-tuning: Fisher's identity yields a gradient-free iterative tilting algorithm that steers generation via black-box rewards, with controllable error. Each contribution comes with approximation guarantees and experimental validation.

The equidistribution of Frobenius conjugacy classes acting on Galois representations is a corner stone of number theory going back to at least Dirichlets theorem on primes in arithmetic progressions. The goal of this talk is to give an introduction into such equidistribution theorems and to present recent equidistribution theorems for Tannakian monodromy groups. I will also discuss the cohomological vanishing theorems, which are a crucial input for the equidistribution theorems, in this talk.

On the universal family of abelian varieties, two key formulas have been known for a long time: [e] = \theta^g/g! and \theta^{g+1}=0, where $e$ is the class of the zero section and $\theta$ is the symmetric theta divisor. The moduli space of abelian varieties can be partially compactified by torus rank 1 degenerations. These are semistable families that contain a semiabelian variety. Two basic questions arise: how can we express the class of the zero section in terms of simpler "tautological" classes? What are the relations between tautological classes? We use Fourier transforms and weight decomposition to address both questions. This is a continuation of the talk on Wednesday by J. Feusi, and it is based on joint work with Y. Bae, J. Feusi and S. Molcho.