Weekly Bulletin

In this talk I will introduce a new notion of approximability for metric spaces that can be seen as a categorification of a concept introduced by Turing for metric groups and as a generalization of total-boundedness. I will explain how recent technological advances in symplectic topology and persistence category theory allow us to talk about approximablity of spaces of Lagrangian submanifolds and discuss applications to rigidity and complexity of Lagrangians, as well as potential relations to open problems in Lagrangian topology. This talk is based on joint work with Paul Biran and Octav Cornea.

<p class="p1">Feynman integrals are ubiquitous in quantum field theory computations. I will review some of their properties and discuss modern methods for computing them, based on differential equations. The structure of these differential equations is linked to the underlying geometry of the integral. While Feynman integrals associated with the Riemann sphere are generally well understood, more complicated geometries appear in cutting-edge calculations. Despite significant progress in recent years, these functions defy available methods and remain the object of active study.</p> <p class="p1">Finally, I will present an example of a cutting-edge calculation, of relevance for the analysis of experimental data from CERN's Large Hadron Collider, showing the interplay between analytic and algebraic complexity that challenges the available methods.</p>

Fluids are often thought of as continua, with their evolution governed by PDEs. In computational models, the PDEs are approximated by finite-dimensional truncations, leading to systems whose evolution for prescribed initial data are typically unique. What about the original PDEs? For smooth initial data and smooth solutions, the evolution can often be proved to be unique without much difficulty. The situation becomes more subtle when non-smooth objects are involved. In this talk, I will review both positive and negative results in this direction.

More information: https://math.ethz.ch/de/news-und-veranstaltungen/veranstaltungen/vortragsreihen/heinz-hopf-preis.htmlcall_made

We discuss the construction of a class of global, dynamical solutions to the 3d incompressible Euler equations near the stationary state given by uniform "rigid body" rotation. At the heart of this result is a dispersive effect due to rotation, which we present with some context. This is based on joint work with Y. Guo and B. Pausader (Brown University).

As a model of the viscous interaction of planar vortices, we consider the solution of the two-dimensional Navier-Stokes equations with singular initial data corresponding to a finite collection of point vortices. In the high Reynolds number regime, the evolution of the vortex centers is described, at least for small times, by the classical Helmholtz-Kirchhoff system, and the vortex cores are slightly deformed due to the mutual interactions. Longer timescales can be reached in simple situations corresponding to relative equilibria of the point vortex system. For example, in the case of a vortex dipole with opposite circulations, an acccurate approximation of the solution can be constructed, which takes into account the leading order correction to the translation speed due to finite size effects. The evolution is similar for ill-prepared initial data, corresponding to smooth and radially symmetric vortices, except that the solution first undergoes a short transient period during which each vortex adapts its shape to the exterior strain. This talk relies on joint works with Martin Donati, Michele Dolce, and Vladimir Sverak.

<p>Given a compact hyperbolic surface of fixed topology, we consider its Laplace eigenvalues together with the structure constants for multiplication with respect to a suitable orthonormal basis of automorphic forms. These numbers obey algebraic constraints analogous to the conformal bootstrap equations in physics. In this talk I will present two results. The first is a converse theorem for these constraints: any collection of numbers satisfying the constraints must come from a hyperbolic surface. The second is an application of the constraints to subconvexity for triple product L-functions. This second result is joint with James Bonifacio, Petr Kravchuk, Dalimil Mazac, Sridip Pal, Alex Radcliffe, and Gordon Rogelberg. No knowledge of physics or L-functions will be assumed.</p>

The incompressible Navier–Stokes and Euler equations are central to mathematical fluid dynamics, yet their well-posedness theory remains one of the great open problems in PDEs. In recent years, remarkable progress has been made in constructing and understanding non-unique solutions to these equations, often revealing unexpected instability mechanisms. In this talk I will survey some of these developments and highlight connections with the pioneering contributions of Vladimir Sverák.

<p>Minimal codewords of a linear code reveal its important structural properties and are required, for instance, in secret-sharing schemes and certain decoding algorithms. A nonzero codeword is said to be minimal if its support does not properly contain the support of any other nonzero codeword. Determining minimal codewords of a general linear code is NP-hard, so one typically exploits the specific structure of a given code. Here, we consider this problem for projective Reed Muller (PRM) codes of order 2.</p> <p>PRM codes of order 2 are evaluation codes obtained by evaluating quadratic forms over a finite field \(F\) at the \(F\)-rational points of the corresponding projective space. To characterize their minimal codewords, we reduce the problem to the following geometric question: given two quadrics such that the \(F\)-rational points of one are contained in the other, can they differ? Our main result is that for absolutely irreducible quadrics, this almost never happens. In this talk, we present a complete answer to this question, thereby classifying the minimal codewords of PRM codes of order 2.</p> <p>This is joint work with Alain Couvreur.</p>

It it well-known that the dynamics of three point-vortices in the plane is integrable. The results go back to W. Groebli (1877), H. Poincare (1893), and others. Over the years, many perspectives on the topic have been developed. In this talk I will discuss a recent work with Thierry Gallay that approaches the computations from a geometric viewpoint based on the Hopf fibration. Our main motivation has been to understand the possibilities for regularizing three-vortex collisions.

We introduce a new technique to prove bounds for the spectral radius of a random matrix, based on using Jensen's formula to establish the zerofreeness of the associated characteristic polynomial in a region of the complex plane. Our techniques are entirely non-asymptotic, and we instantiate it in three settings: (i) The spectral radius of non-asymptotic Girko matrices---these are asymmetric matrices M ∈ ℂ^{n × n} whose entries are independent and satisfy 𝔼 Mᵢⱼ = 0 and 𝔼 |Mᵢⱼ|² ≤ 1/n. (ii) The spectral radius of non-asymptotic Wigner matrices---these are symmetric matrices M ∈ ℂ^{n × n} whose entries above the diagonal are independent and satisfy 𝔼 Mᵢⱼ = 0, 𝔼 |Mᵢⱼ|² ≤ 1/n, and 𝔼 |Mᵢⱼ|⁴ ≤ 1/n. (iii) The second eigenvalue of the adjacency matrix of a random d-regular graph on n vertices, as drawn from the configuration model. In all three settings, we obtain constant-probability eigenvalue bounds that are tight up to a constant. Applied to specific random matrix ensembles, we recover classic bounds for Wigner matrices, as well as results of Bordenave--Chafaï--García-Zelada, Bordenave--Lelarge--Massoulié, and Friedman, up to constants.

<div dir="auto">Vladuts characterized in 1999 the set of finite fields k such that all elliptic curves defined over k have a cyclic group of rational points. Under the conjecture of infinitely many Mersenne primes, this set is infinite. </div> <div dir="auto">In this talk, I study the question in higher dimension. Precisely, I prove that there is no such fields. This is related with the existence of some totally real algebraic integers having some arithmetic properties.</div> <div dir="auto">I am going to present a result about cyclicity of maximal abelian varieties as well, and how this is related to some totally positive algebraic integers.</div> <div dir="auto">In both problems, open questions about totally real algebraic integers arise, some of which are addressed from an algorithm point of view.</div>

The circle can be seen as the boundary at infinity of the hyperbolic plane. We give a 1-to-2 dimensional holographic interpretation of the Schwarzian action, by showing that the Schwarzian action (which is a function of a diffeomorphism of the circle) is equal to the hyperbolic area enclosed by an "Epstein curve" in the disk. A dimension higher, the Epstein construction was used to relate the Loewner energy (a function of a Jordan curve related to SLE and Brownian loop measures) to renormalized volume in hyperbolic 3-space. In this talk I will explain how to construct the Epstein curve, how the bi-local observables of Schwarzian field theory can be interpreted as a renormalized hyperbolic length using the same Epstein construction, and (time permitting) discuss a bit what we know so far about the relationship between the Schwarzian action and the Loewner energy. This is based on joint work with Franco Vargas Pallete and Yilin Wang.

We introduce kernel density machines (KDM), a nonparametric estimator of a Radon--Nikodym derivative, based on reproducing kernel Hilbert spaces. KDM applies to general probability measures on countably generated measurable spaces under minimal assumptions. For computational efficiency, we incorporate a low-rank approximation with precisely controlled error that grants scalability to large-sample settings. We provide rigorous theoretical guarantees, including asymptotic consistency, a functional central limit theorem, and finite-sample error bounds, establishing a strong foundation for practical use. Empirical results based on simulated and real data demonstrate the efficacy and precision of KDM.

The Manin constant c of an elliptic curve E over Q is the nonzero integer that scales the differential ω determined by the normalized newform f associated to E into the pullback of a Néron differential under a minimal parametrization \phi : X_0(N) ---> E. Manin conjectured that c = ±1 for optimal parametrizations. We show that in general c | deg(\phi), under a minor assumption at 2 and 3, which improves the status of the Manin conjecture for many E. Our core result that gives this divisibility is the containment ω ∈ H^0(X_0(N), Ω), which we establish by combining automorphic methods with techniques from arithmetic geometry (here the modular curve X_0(N) is considered over Z and Ω is its relative dualizing sheaf over Z). To overcome obstacles at 2 and 3, we analyze nondihedral supercuspidal representations of GL_2(Q_2) and exhibit new cases in which X_0(N) has rational singularities over Z. The talk is based on joint work with Abhishek Saha and Michalis Neururer.

Many notions in log geometry are invariant under log modification. To formalize this, we introduce the so-called m-type morphisms and their associated topologies and discuss their properties. In particular, this lead us to discover a mistake in the definition of the full log étale site introduced by Kazuya Kato and Fujiwara 30 years ago. Moreover, I will also present a fix for this problem. Time permitting, I will talk about applications of our topologies. All the above results are joint with Xianyu Hu.

We study the universal compactified Jacobians of degree d, as constructed by Kass and Pagani over the moduli space of stable curves. These spaces compactify the universal degree-d Jacobian over the moduli of smooth curves, and depend on a choice of universal stability condition. I will discuss the proof of how the Hodge numbers of these compactified Jacobians are independent of both the degree d and the choice of stability condition. This is joint work with Rahul Pandharipande, Dan Petersen, and Johannes Schmitt.