Weekly Bulletin

Deep neural networks (DNNs) are singular statistical models whose loss landscapes exhibit complex degeneracies - features that defy explanation through classical regular statistical theory. Drawing on tools from singular learning theory, we introduce the Local Learning Coefficient (LLC), a quantity that rigorously quantifies the degree of degeneracy in DNNs. While the LLC is rooted in the singular framework, it recovers familiar notions of model complexity in regular or "minimally singular" regimes. We develop a scalable estimator for the LLC and apply it across a range of architectures, including deep linear networks with up to 100M parameters, ResNet image classifiers, and transformers. Empirical results demonstrate that the LLC sheds light on a range of deep learning phenomena, including in-context learning in transformers and the competition between energy and entropy during training dynamics, as well as how standard training heuristics influence effective model complexity. Ultimately, the LLC provides a practical instantiation of singular learning theory in modern deep learning, offering new perspectives on the interplay between overparameterization, generalization, and parsimony.

The natural question of how much smoother integral currents are with respect to their initial definition goes back to the late 1950s and to the origin of the theory with the seminal article of Federer and Fleming. In this seminar I will explain how closely one can approximate an integral current representing a given homology class by means of a smooth submanifold. This is a joint study with William Browder and Camillo De Lellis, based on some previous preliminary work of the former author together with Frederick Almgren.

Models of non-Newtonian fluids play an important role in science and engineering and their mathematical analysis and numerical approximation have been active fields of research over the past decade. This lecture is concerned with the analysis of numerical methods for the approximate solution of a system of nonlinear partial differential equations that arise in models of chemically-reacting viscous incompressible non-Newtonian fluids, such as the synovial fluid found in the cavities of synovial joints. The synovial fluid consists of an ultra-filtrate of blood plasma that contains hyaluronic acid, whose function is to reduce friction during movement. The shear-stress appearing in the model involves a power-law type nonlinearity, where the power-law exponent depends on a spatially varying nonnegative concentration function, expressing the concentration of hyaluronic acid, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove convergence of the sequence of numerical approximations to a solution of this coupled system of nonlinear partial differential equations one has to derive a uniform Hölder norm bound on the sequence of approximations to the concentration in a setting where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely a bounded function with no additional regularity. This necessitates the development of a discrete counterpart of the De Giorgi--Nash--Moser theory, which is then used, in conjunction with various compactness techniques, to prove the convergence of the sequence of numerical approximations to a weak solution of the coupled system of nonlinear partial differential equations under consideration

Recent advances in the understanding of higher-rank diagonalizable actions on homogeneous spaces by Einsiedler and Lindenstrauss have opened the door to natural couplings of distributions of objects like periodic geodesics or complex multiplication points on the modular surface, or the distribution of integer points on large spheres. In this talk we will discuss such a natural coupling by considering an indefinite quaternary quadratic vector space (V, Q). To each rational plane in V one can naturally attach three arithmetic objects which are associated to quadratic forms. The first two objects arise from the plane and its orthogonal complements with respect to Q, the third (accidental) object is constructed via the Clifford algebra of (V, Q). These arithmetic objects lead to tuples consisting of three geodesics and a point. We discuss their constructions and simultaneous equidistribution using the joining classification result of Einsiedler and Lindenstrauss under a splitting condition. This is joint work with Menny Aka and Andreas Wieser.

Intrinsic Mirror Symmetry (Gross, Siebert) associates to a log Calabi-Yau variety (Y,D) a geometric generating function with support the tropicalisation of (Y,D), and with invariants the corresponding punctured Gromov-Witten invariants. This construction defines a ring and a mirror family. The enumerative mirror conjecture then states that various period integrals on the mirror family compute various Gromov-Witten invariants of (Y,D), not necessarily punctured. I will describe some examples, where this is realized, and where one obtains some new relations for Gromov-Witten invariants. Joint work with Siebert and Ruddat.

A metric space <i>X</i> satisfies a Euclidean isoperimetric inequality for <i>n</i>-spheres, if every <i>n</i>-sphere <i>S ⊂ X</i> bounds a ball <i>B ⊂ X</i> with vol_<i>n</i>+1(<i>B</i>)≤ <i>C</i> · vol_<i>n</i>(<i>S</i>)^{(<i>n</i>+1)/<i>n</i>}. Every CAT(0) space <i>X</i> satisfies Euclidean isoperimetric inequalities for 1-spheres with the sharp constant <i>C</i>=1/4π. Moreover, if such inequalities hold with a constant strictly smaller than 1/4π, then <i>X</i> has to be Gromov hyperbolic. In particular, a sharp isoperimetric gap appears. In the talk I will focus on the case <i>n</i>=2, namely fillings of 2-spheres by 3-balls. This is based on joint work with Drutu, Lang and Papasoglu.

Eigenvarieties are geometric objects that parametrise finite-slope p-adic automorphic forms. It is discovered by mathematicians that the geometry of eigenvarieties encodes interesting arithmetics of finite-slope automorphic forms. It is natural to ask what happens if one is interested in more general automorphic forms (not necessarily the finite-slope ones). In this talk, I will report on an ongoing joint work with Andrew Graham, where we construct eigenstacks, which see beyond finite-slope families of automorphic forms.

Fourier analysis is a powerful tool in analysis. In the context of abelian schemes, Fourier-Mukai transformation and the weight decomposition play a similar role. For degenerate abelian fibrations, the relative group structure disappears and understanding the intersection theory leads to many interesting questions. In this talk, I will connect Fourier transform between compactified Jacobians over the moduli space of stable curves and logarithmic Abel-Jacobi theory. As an application, I will compute the pushforward of monomials of divisor classes on compactified Jacobians via the top degree part of the twisted double ramification formula of all codimensions. This is a joint work with Samouil Molcho and Aaron Pixton.