Rational curves are intricately linked to the birational geometry of varieties containing them.  Certain curves, called free curves, have the nicest deformation properties.  However, it is unknown whether mildly singular Fano varieties contain free rational curves in their smooth locus.  In this talk, we discuss free curves of higher genus. Using recent results about tangent bundles, we prove that any klt Fano variety has higher genus free curves.  We then use the existence of such free curves to get some applications: we prove the existence of free rational curves in terminal Fano threefolds; obtain an optimal upper bound on the length of extremal rays in the Kleiman-Mori cone of any klt pair; and study the fundamental group of the smooth locus of a Fano variety. This is joint work with Brian Lehmann and Eric Riedl.

A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure complexity of Borel equivalence relations. Recently, a lot of research has been done on hyperfiniteness of the orbit equivalence relation on the Gromov boundary induced by various group actions on hyperbolic spaces. In this talk, I will explain my attempt to explore this connection of Borel complexity and geometric group theory for another intensively studied geometric object, which is CAT(0) cube complexes. More precisely, we prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite.

In this talk, I will discuss recent work on the problem of singularity formation in the incompressible porous medium (IPM) equation. We construct Lipschitz continuous solutions of the IPM equation which vanish on the boundary of the domain and blow-up in finite time. At the blow-up point, the flow is hyperbolic with points approaching the boundary from the interior and escaping tangent to the boundary.

We discuss a variety of proofs that $\sqrt{2}$ is irrational. In doing so we’ll discuss what makes two proofs distinct, morals in math and methods of finding rational solutions to polynomials.

Abstract: Expander graphs are sparse graphs with strong connectivity properties. Chapman, Linial and Peled asked whether there exist families of expander graphs with high levels of regularity, that is not only the number of edges containing a given vertex needs to be constant but also the number of triangles containing a given edge etcetera. We answer this question positively constructing families of expander graphs as quotient graphs of 1-skeleta of infinite polytopes (1-skeleton means only retain the vertex-edge information of the polytope). The latter are Wythoffian polytopes, which are obtained from Coxeter groups by decorating the associated Coxeter diagram. The specific higher regularity properties depend on this diagram. Expansion stems from superapproximation of the Cayley graphs associated to the Coxeter group, which is a number-theoretic way to study the rate of convergence of random walks on these graphs. The Cayley graphs and the 1-skeleta are quasi-isometric (that is equal on a large scale) which implies that one forms an expanding family if and only if the other does. 

Based on joint work with Marston Conder, Alexander Lubotzky and Francois Thilmany.

Size (or length) generalization is a key challenge in designing neural modules to perform algorithmic tasks. Specifically, when can a neural model with bounded complexity generalize to problem instances of arbitrary size? In this talk, I will focus on approaches to achieve size generalization by "aligning" the neural models with certain algorithmic structures, so as to facilitate a neural model learning "procedures" instead of merely fitting data. I will first present a theoretical result to show the benefit of algorithmic alignment in extrapolating for the graph shortest path distance estimation. We will then present examples of designing practical and efficient neural models for various geometric optimization problems via algorithmic alignments. 

Biane’s free multiplicative Brownian motion b_t is the large-N limit of the Brownian motion in the general linear group GL(N;C) and can be viewed as the solution to a free stochastic differential equation driven by a circular Brownian motion. I will consider random walk approximations to b_t, which are discrete approximations to the solution of the SDE. These approximations have the form of a product of steps, each of which is the identity plus a multiple of a circular element. We are able to compute the Brown measure of the model with a fixed number of steps using the linearization method. We are then able to let the number of steps tend to infinity and recover the previously computed Brown measure of b_t itself. 

A key step in the argument is a new freeness result for block elements. In general, matrices with freely independent entries are not freely independent in the ordinary sense but only in the “operator valued” sense . But we show that in some interesting examples, we do obtain freeness in the ordinary sense. We also show that for a fixed number of steps, the empirical eigenvalue distribution of the corresponding matrix model converges to the Brown measure of the free model.

This is joint work with Bruce Driver, Ching Wei Ho, Todd Kemp, Yuriy Nemish, Evangelos Nikitopolous, and Félix Parraud. The talk will be self-contained and have lots of pictures.

The investigation of the dimension of Bergman spaces has long been a central topic in several complex variables, uncovering profound connections with potential theory and function theory since the pioneering work of Carleson, Wiegerinck, and others in the 1960s. We investigate the dimension of $p$-Bergman spaces associated with pseudoconvex domains in $\mathbb{C}^n$. By constructing $L^p$-versions of the extension theorems of Ohsawa and Ohsawa-Takegoshi, we establish several geometric and potential-theoretic criteria that ensure the spaces are infinite-dimensional. Sufficient conditions for the infinite dimensionality of $p$-Bergman spaces of complete N-circled fibered Hartogs domains, balanced domains, and weighted $p$-Fock spaces are obtained by applying the mentioned $L^p$-analogs of extension theorems and generalizing a sufficient condition of Jucha.

Dipping a wire of metal or plastic in soapy water and taking it out is a favorite classroom experiment: typically the soapy water will form a thin film which is attached to the wire. The classical Plateau laws, stated by the Belgian physicist Joseph Plateau in the nineteenth century, assert that, away from the wire, the local geometry of a soap film is described locally by the following list of shapes: a 2-dimensional plane, three halfplanes meeting at a common line with equal angles, and the cone over the 1-dimensional skeleton of a regular tetrahedron. 

Is there a similar list of possible shapes for the points where the film touches its ``boundary'', namely the wire of the classroom experiment? The classical Plateau laws were translated into a mathematical theorem by Jean Taylor in the seventies: in a nutshell Taylor's theorem rigorously classifies 2-dimensional conical shapes which minimize the area. In this talk I will illustrate a recent joint work with Federico Glaudo, classifying conical shapes which minimize the area and include a boundary line: the corresponding list suggests an analog of Plateau's laws at the boundary of the soap film, which are very much in agreement with both real-life and numerical experiments.

Adaptive optimizers such as Adam and Shampoo are workhorses of modern machine learning, enabling efficient training of large-scale models across architectures and domains. In this talk, we will present a unified framework for adaptive optimizers with structured preconditioners, encompassing a variety of existing methods and introducing new ones. Our analysis reveals the fundamental interplay between preconditioner structures and loss geometries, highlighting in particular that more adaptivity is not always helpful. Furthermore, the dominance of adaptive methods has recently been challenged by the surprising effectiveness of simpler normalized steepest descent (NSD)–type methods such as Muon, while a consensus has emerged that both families of methods succeed by exploiting the non-Euclidean geometry of the loss landscape. Building on the proposed framework, we show that the convergence of adaptive optimizers is governed by a notion of adaptive smoothness, which contrasts with the standard smoothness assumption leveraged by NSD. In addition, although adaptive smoothness is a stronger condition, it enables acceleration via Nesterov momentum, which cannot be achieved under the standard smoothness assumption in non-Euclidean settings. Finally, we develop a notion of adaptive gradient variance that parallels adaptive smoothness and yields qualitatively improved guarantees compared to those based on standard gradient variance.