On a two dimensional Stein space with isolated, normal singularities, finite type boundary and locally algebraic Bergman kernel, we find an estimate of the local algebraic degree of the Bergman kernel in terms of the type of the boundary. As an application, we characterize two dimensional ball quotients as finite type Stein spaces with a rational Bergman kernel.

In this talk, I will explore the interesting geometries that emerge in high-dimensional attraction basins, which are important in applications such as protein folding, cell differentiation, and neural networks. As a paradigmatic model, I will consider networks of coupled Kuramoto oscillators and show that high-dimensional basins generally cannot be approximated by simple convex shapes. Instead, they have tentacle-like structures where most of the basin volume is concentrated. Next, I will show that introducing non-pairwise couplings among Kuramoto oscillators can make basins deeper but smaller—the attractors become linearly more stable but much harder to find due to basins shrinking dramatically. Time allowing, I will also briefly mention a few related projects, including learning basins with reservoir computing and modeling circadian clocks with Kuramoto oscillators.

The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. We introduce the notion of biexactness for von Neumann algebras, which allows us to place many previous solidity results in a more systematic context, and naturally leads to extensions of these results. We will also discuss examples of solid factors that are not biexact. This is a joint work with Jesse Peterson.

The class of SQP methods solve nonlinear constrained optimization problems by solving a related sequence of simpler problems. These SQP subproblems involve minimization of a quadratic model of the Lagrangian function subject to linearized constraints. In contrast to the quasi-Newton approach, which maintains a positive definite Hessian approximation, Second-derivative SQP methods use the exact Hessian of the Lagrangian. In this context, we will discuss an adaptive convexification strategy that makes minimal matrix modifications while ensuring the subproblem iterates are bounded and the solution defines a descent direction for the relevant Lagrangian. This talk will focus on adaptive convexification of stabilized SQP methods, as well as their connection with primal-dual interior methods.

We prove a conjecture of K. Marton, widely known as the polynomial Freiman– Ruzsa conjecture, in characteristic 2. The argument extends to odd characteristic, with details to follow in a subsequent paper. This is a joint work with Timothy Gowers, Ben Green and Terence Tao.  

The singularity formation problem is a central question in fluid dynamics, and it is still widely open for fundamental models such as the 3d incompressible Euler equations and the Navier-Stokes equations. In this talk, I will first review the singularity formation problem, and I will describe how particle transport poses the main challenge in constructing blow-up solutions — an effect known as “regularization by transport.” I will then outline a new mechanism, arising from the classical Taylor-Couette instability, allowing us to overcome regularization by transport in the 3d Euler equations, thereby constructing the first swirl-driven singularity in R^3. This is joint work with Tarek Elgindi (Duke University).

Topology plays a fundamental role in controlling the dynamics of adaptive biological and physical systems, from chromosomal DNA and biofilms to cilia carpets and worm collectives. Despite their long history, the subtle interplay between topology, geometry and mechanics in tangled elastic filaments remains poorly understood. To uncover the topological principles underlying the dynamics of knotted and tangled matter, we first develop a mapping between human-designed elastic knots and long-range ferromagnetic spin systems. This mapping gives rise to topological counting rules that predict the relative mechanical stability of commonly used climbing and sailing knots. Building upon this framework, we then examine the adaptive topological dynamics exhibited by California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using blackworm locomotion datasets, we construct stochastic trajectory equations that explain how the dynamics of individual active filaments control their emergent topological state. By identifying the principles behind stability and adaptivity in living tangled matter, our results have applications in understanding broad classes of adaptive, self-optimizing biological systems.

Let $H < G$ both be noncompact connected semisimple real algebraic groups and $\Gamma < G$ be a lattice. In the work of Gorodnik--Weiss, they showed that the distribution of dense $\Gamma$-orbit on homogeneous space $G/H$ is asymptotically the same as $G$-orbit on $G/H$. One key ingredient in their proof is Shah's theorem derived from the famous Ratner's theorem. In this talk, we report an effective version of this result in the case $(G, H, \Gamma) = (\mathrm{SL}_3(\mathbb{R}), \mathrm{SO}(2, 1), \mathrm{SL}_3(\mathbb{Z}))$. The proof uses recent progress by Lindenstrauss--Mohammadi--Wang--Yang towards an effective version of Ratner's theorem. We also prove the general case if an effective version of Ratner's theorem is provided. The talk is based on an ongoing work with Pratyush Sarkar.

In this talk, I will describe the regularity theory for surfaces minimizing the prescribed mean curvature functional over isotopies in a closed Riemannian 3-manifold, which is a prescribed mean curvature counterpart of the celebrated regularity result of Meeks, Simon and Yau about minimizers of the area functional over isotopies. 

Whereas for the area functional minimizers over isotopies are smooth embedded minimal surfaces, minimizers of the prescribed mean curvature functional turn out to be C^{1,1} immersions which can have a large self-touching set where the mean curvature vanishes. 

Even though the proof broadly follows the same general strategy as in the case of the area functional, several new ideas are needed to deal with the lower regularity setting. This regularity theory plays an important role in Z. Wang-X. Zhou’s recent proof of the existence of 4 embedded minimal spheres in a generic metric on the 3-sphere.

The results in this talk are joint work with Douglas Stryker (Princeton). 

Sharp geometric and functional inequalities play an important role in analysis,  PDEs and differential geometry. In this talk, we will describe our works in recent years on sharp higher order Poincare-Sobolev and Hardy-Sobolev-Maz'ya inequalities on real and complex hyperbolic spaces and noncompact symmetric spaces of rank one. The approach we have developed crucially relies on the Helgason-Fourier analysis on hyperbolic spaces and establishing such inequalities for the GJMS operators.  Best constants for such inequalities will be compared with the classical higher order Sobolev inequalities in Euclidean spaces. The borderline case of such inequalities, such as the Moser-Trudinger and Adams inequalities will be also considered. 

The concept of p-adic families of automorphic forms has far reaching applications in number theory. In this talk, we will discuss one of the first examples of such a family, built from the Eisenstein series, before allowing this to inform a construction of a family on an exceptional group of type G_2.

[pre-talk at 1:20PM in APM 6402]

Modern statistical learning is featured by the high-dimensional nature of data and over-parameterization of models. In this regime, analyzing the dynamics of the used algorithms is challenging but crucial for understanding the performance of learned models. This talk will present recent results on the dynamics of two pivotal algorithms: Approximate Message Passing (AMP) and Stochastic Gradient Descent (SGD). Specifically, AMP refers to a class of iterative algorithms for solving large-scale statistical problems, whose dynamics admit asymptotically a simple but exact description known as state evolution. We will demonstrate the universality of AMP's state evolution over large classes of random matrices, and provide illustrative examples of applications of our universality results. Secondly, for SGD, a workhorse for training deep neural networks, we will introduce a novel mathematical framework for analyzing its implicit regularization. This is essential for SGD's ability to find solutions with strong generalization performance, particularly in the case of over-parameterization. Our framework offers a general method to characterize the implicit regularization induced by gradient noise. Finally, in the context of underdetermined linear regression, we will show that both AMP and SGD can provably achieve sparse recovery, yet they do so from markedly different perspectives.

Integrable lattice models play a pivotal role in the investigation of microscopic multi-particle systems, with their continuum limits forming the foundation of the macroscopic effective theory. These models have found broad applications in condensed matter physics and numerical analysis. In this talk, I will discuss our recent work on the continuum limit of some differential-difference equations. Using the Ablowitz--Ladik system (AL) as our prototypical example, we establish that solutions to this discrete model converge to solutions of the cubic nonlinear Schr\"odinger equations (NLS). Notably, we consider merely $L^2$ initial data which combines both slowly varying and rapidly oscillating components, and demonstrate convergence to a decoupled system of NLS. This surprising result highlights that a sole NLS does not suffice to encapsulate the AL evolution in such a low-regularity setting reminiscent of the thermal equilibrium state. I will also explain the framework of our proof and how it has been successfully extended to address more general lattice approximations to NLS and mKdV.

F-bundle is a formal version of variation of non-commutative Hodge structures. I will explain basic ideas and properties of F-bundles, present an explicit construction of F-bundles associated to blowups, and discuss its relation with Iritani’s work on quantum cohomology. Joint work with Katzarkov, Kontsevich and Pantev.