Abstract: This talk is based on a series of papers with Yang Li (Cambridge). I will introduce the problem of counting special Lagrangians in Calabi-Yau 3-folds and Fueter sections to define new numerical and Floer-theoretic invariants. The key challenges are the non-compactness problems and the wall-crossing phenomenon. Donaldson-Segal proposed a weighted count of special Lagrangians to remedy this problem, where the weight is determined by counting Fueter sections. However, the compactness of the space of Fueter sections is an open problem, first raised by Taubes in 1998, motivated by defining new invariants of 3-manifolds. In this work, we prove a compactness theorem for Fueter sections and also, if time permits, a local version of the Donaldson-Scaduto conjecture. This conjecture is expected to play a major role in proving compactness results for special Lagrangians in Lefschetz-fibered Calabi-Yau 3-folds.
Abstract: Abstract. In these two lectures, we will introduce a class of non- linear transforms called the FBI transforms (after Fourier, Bros, and Iagolitzner) that characterize local and microlocal regularity of functions (and distributions) in C∞, real analytic, and Gevrey spaces. If time permits, we will present applications to linear and nonlinear pdes with complex-valued coefficients.
Abstract: We give a geometric characterization of the quantitative joint non-integrability, introduced by Asaf Katz, of strong stable and unstable bundles of partially hyperbolic measures and sets in dimension 3. This is done via the use of higher order templates for the invariant bundles. Using the recent work of Katz, we derive some consequences, including the measure rigidity of uu-states and the existence of physical measures. This is a joint work with Alex Eskin and Rafael Potrie. We also discuss a work-in-progress with Artur Avila, Sylvain Crovisier, Alex Eskin, Rafael Potrie and Amie Wilkinson on the unstable foliation and u-states of Anosov diffeomorphism.
Abstract: We briefly define and motivate the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discuss the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group. We give some intuition for the proof. No prior knowledge on fixed price or higher rank will be assumed.
Abstract: In this talk, we will present a highly parallel and derivative-free martingale neural network method, based on the probability theory of Varadhan’s martingale formulation of PDEs,
to solve Hamilton-Jacobi-Bellman (HJB) equations arising from stochastic optimal control problems (SOCPs), as well as general quasilinear parabolic partial differential equations (PDEs).
In both cases, the PDEs are reformulated into a martingale problem such that loss functions will not require the computation of the gradient or Hessian matrix of the PDE solution, and can be computed in parallel in both time and spatial domains. Moreover, the martingale conditions for the PDEs are enforced using a Galerkin method realized with adversarial learning techniques, eliminating the need for direct computation of the conditional expectations associated with the martingale property. For SOCPs, a derivative-free implementation of the maximum principle for optimal controls is also introduced. The numerical results demonstrate the effectiveness and efficiency of the proposed method, which is capable of solving HJB and quasilinear parabolic PDEs accurately and fast in dimensions as high as 10,000.
Abstract: The theory of local models has been a very successful tool for the study of Shimura varieties with parahoric level structure, and the theory is now very developed in that setting. For level structure which is deeper than Iwahori level, many complications arise, and the subject is in its infancy. I will first review the basic theory of local models for Iwahori level, concentrating on the general linear and general symplectic group cases. The main goal will be to explain what can be said about local models when the level structure is $\Gamma_1(p)$, which is slightly deeper than Iwahori level. For PEL Shimura varieties of Siegel type, I will define the local models using a linear algebra incarnation of Oort-Tate generators of finite flat group schemes of order $p$, and then I will explain how one uses a variant of Beilinson-Drinfeld Grassmannians and Gaitsgory's central functor adapted to pro-p Iwahori level, to study the nearby cycles on the special fibers. This is based on joint work in progress with Qihang Li and Benoit Stroh.
Abstract: Generalized linear mixed models (GLMM) with crossed random effects are well known not only for the computational challenges involved in numerically evaluating the maximum likelihood estimator (MLE) but also for the theoretical challenges in studying asymptotic behavior of the MLE under these models. In fact, not until 2012 has consistency of the MLE been established for GLMM with crossed random effects (Jiang 2013). Now, another part of the asymptotic behavior, that is, asymptotic normality of the MLE for GLMM with crossed random effects has also been established (Jiang 2025). This talk provides an overview of this “amazing journey”, focusing on the methodology developments for overcoming the theoretical challenges.
References: Jiang, J. (2013), The subset argument and consistency of MLE in GLMM: Answer to an open problem and beyond, Ann. Statist. 41, 177-195. Jiang, J. (2025), Asymptotic distribution of maximum likelihood estimator in generalized linear mixed models with crossed random effects, Ann. Statist., in press.