We propose a scalable preconditioned primal-dual gradient algorithm for solving partial differential equations (PDEs). We multiply the PDE with a dual function to obtain an inf-sup problem whose loss function involves lower-order differential operators. The Primal-Dual hybrid gradient (PDHG) algorithm is leveraged for this saddle point problem. By introducing suitable precondition operators to the proximal steps in the PDHG algorithm, we obtain an alternative natural gradient ascent-descent optimization scheme for updating the primal and the adversarial neural network parameters. We apply the Krylov subspace method to evaluate the natural gradients efficiently. A posterior convergence analysis is established for the time-continuous version of the proposed method.

The algorithm is tested on various types of linear and nonlinear PDEs with dimensions ranging from 1 to 50. We compare the performance of the proposed method with several commonly used deep learning algorithms, such as physics-informed neural networks (PINNs), the DeepRitz method, and weak adversarial networks (WANs) using the Adam or L-BFGS optimizers. The numerical results suggest that the proposed method performs efficiently and robustly and converges more stably.

Consider a countable discrete group Γ and its subgroup space-Sub(Γ), the collection of all subgroups of Γ. Sub(Γ) is a compact metrizable space with respect to the Chabauty topology (the topology induced from the product topology on {0, 1} Γ). The normal subgroups of Γ are the fixed points of (Sub(Γ), Γ). Furthermore, the Γ-invariant probability measures of this dynamical system are known as invariant random subgroups (IRSs). Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. They generalize the notion of normal subgroups. Among many other remarkable results, they strengthen the well-known Margulis’s normal subgroup Theorem. More recently, in a joint work with Hartman and Oppelmayer, we introduced the notion of Invariant Random Algebra (IRA), an invariant probability measure on the collection of sub algebras of L(Γ). Motivated by the works of Glasner and Lederle, in ongoing joint work with Yair Glasner, Yair Hartman, and Yongle Jiang, we introduce the notion of Boomerang subalgebras in the context of L(Γ). In this talk, we shall show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group comes from a Boomerang subgroup. We shall also discuss its connection to understanding IRAs in such groups.

One definition of key polynomials is as the weight generating functions of key tableaux. Assaf and Schilling introduced a crystal structure on key tableaux and related it to Morse–Schilling crystals on reduced factorizations for permutations via weak Edelman–Greene insertion. In this talk, we consider generalizations of both crystals depending on a flag. We extend weak EG insertion to a bijection between our flagged objects and show that the recording tableau gives a crystal isomorphism. As an application, we show that flagged key tableaux have a natural Demazure crystal structure, whose characters recover Reiner and Shimozono’s flagged key polynomials.

The main objects of study in algebraic geometry are varieties, or solution sets to polynomial equations.  A fundamental goal of the subject is the classification of algebraic varieties.  I will survey some of the steps of classification, including the minimal model program and construction of moduli spaces.  This talk is intended to be accessible to all graduate students, regardless of background in algebraic geometry. 

On the symmetric space for $G_2$, there exist various submanifolds $G_D$ corresponding to the stabilizer of a norm $D$ vector. We show that when a suitable automorphic form is integrated against the $G_D$, the resulting numbers assemble to give a half-integral weight classical modular form. Although this is already implied by a result of Kudla-Millson, we give a simpler proof that avoids the complications in their paper.

Classical results in asymptotic statistics show that the Fisher information matrix controls the difficulty of estimating a statistical model from observed data. In this work, we introduce a companion measure of robustness of an estimation problem: the radius of statistical efficiency (RSE) is the size of the smallest perturbation to the problem data that renders the Fisher information matrix singular. We compute RSE up to numerical constants for a variety of test bed problems, including principal component analysis, generalized linear models, phase retrieval, bilinear sensing, and matrix completion. In all cases, the RSE quantifies the compatibility between the covariance of the population data and the latent model parameter. Interestingly, we observe a precise reciprocal relationship between RSE and the intrinsic complexity/sensitivity of the problem instance, paralleling the classical Eckart–Young theorem in numerical analysis.

Let $G$ be a discrete group. A $G$-space $X$ is called a $G$-boundary if the action $G \curvearrowright X$ is minimal and strongly proximal. In this talk, we shall prove a continuous version of the well-studied Intermediate Factor Theorem in the context of measurable dynamics. When a product group $G = \Gamma_1 \times \Gamma_2$ acts (by a product action) on the product of corresponding $\Gamma_i$-boundaries $\partial \Gamma_i$, we show that every intermediate factor $$X \times (\partial \Gamma_1 \times \partial \Gamma_2) \rightarrow Y \rightarrow X$$ is a product (under some additional assumptions on $X$). We shall also compare it to its measurable analog proved by Bader-Shalom. This is a recent joint work with Yongle Jiang.

Asymptotically conical shrinking Ricci solitons are an important class of potential singularity models for the Ricci flow in dimension four and higher, and give rise to solutions which, near infinity, flow smoothly into a cone at the singular time. We will present some uniqueness and nonexistence results which can be inferred from this latter fact using a unique continuation result valid at the singular time, and discuss their applications to the classification problem for asymptotically conical shrinkers.

Inverse Reinforcement Learning (IRL) is a compelling technique for revealing the rationale underlying the behavior of autonomous agents.  IRL seeks to estimate the unknown reward function of a Markov decision process (MDP) from observed agent trajectories. However, IRL needs a transition function, and most algorithms assume it is known or can be estimated in advance from data. It therefore becomes even more challenging when such transition dynamics is not known a-priori, since it enters the estimation of the policy in addition to determining the system's evolution. When the dynamics of these agents in the state-action space is described by stochastic differential equations (SDE) in It\^{o} calculus, these transitions can be inferred from the mean-field theory described by the Fokker-Planck (FP) equation. We conjecture there exists an isomorphism between the time-discrete FP and MDP that extends beyond the minimization of free energy (in FP) and maximization of the reward (in MDP). We identify specific manifestations of this isomorphism and use them to create a novel physics-aware IRL algorithm, FP-IRL, which can simultaneously infer the transition and reward functions using only observed trajectories.  We employ variational system identification to infer the potential function in FP, which consequently allows the evaluation of reward, transition, and policy by leveraging the conjecture. We demonstrate the effectiveness of FP-IRL by applying it to a synthetic benchmark and a biological problem of cancer cell dynamics, where the transition function is inaccessible. This is joint work with Chengyang Huang, Sid Srivastava, Kenneth Ho, Kathy Luker, Gary Luker and Xun Huan.

We consider properly immersed two-sided stable minimal hypersurfaces of dimension n. We illustrate the validity of curvature estimates for n \leq 6 (and associated Bernstein-type properties with an extrinsic area growth assumption). For n \geq 7 we illustrate sheeting results around "flat points". The proof relies on PDE analysis. The results extend respectively the Schoen-Simon-Yau estimates (obtained for n \leq 5) and the Schoen-Simon sheeting theorem (valid for embeddings).

It is a well-known theorem in combinatorics that everything is counted by the Catalan numbers. These numbers are ubiquitous in mathematics, often appearing where you least expect them — such as in the HOMFLYPT polynomial of torus knots. In this talk, we’ll take a journey into algebraic combinatorics by pursuing the question: why do the torus knots know about Catalan numbers? Along the way, we’ll encounter braid groups, Hecke algebras, complex reflection groups, and more! This talk is based on a paper by Pavel Galashin, Thomas Lam, Minh-Tam Quang Trinh, and Nathan Williams.