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Abstract: In this talk, we will focus on operator learning — a framework for approximating mappings between function spaces that has broad applications in PDE-related problems. We will begin by discussing the mathematical foundations of operator approximation, which inform the design of neural network architectures and provide a basis for analyzing the performance of trained models on test samples. Specifically, we will introduce the neural scaling law, which characterizes error convergence in relation to network size and generalization error in relation to training dataset size. Building on these theoretical insights, we will present a distributed learning algorithm based on the theoretical architectures to address two key computational challenges: (1) efficiently handling heterogeneous problems where input functions exhibit vastly different properties, and (2) multi-operator learning, where use one network model to approximate multiple operators simultaneously such that the model can be extrapolated and rapidly adapt to the new problems.

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Abstract: We will review some models of chemotaxis featuring some mechanisms that prevent overcrowding. These are variants of classical Patlak-Keller-Segel systems that model the population density of some organisms who follow a chemical signal that they themselves produce. Then, we will describe some recent results by Antoine Mellet, Inwon Kim, Yijing Wu, Jeremy Sheung-Him Wu, and myself that show, in a singular limit formally corresponding to that of a large population, that the organisms arrange themselves into a region of constant positive density. Finally, we derive mean-curvature-driven free boundary problems describing the evolution of the boundary of this region, which allows to understand the migration of the population in the limit.

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Abstract: On the cusp of the 100 year anniversary, Khinchin's theorem implies a strong 0-1 law for the real line; namely, the set of real numbers within distance q^{-2-\epsilon} of infinitely many rationals p/q is Lebesgue measure 0 for \epsilon>0, and full measure for \epsilon=0. In these lectures, I will present an analogous result for horoball packings in Gromov hyperbolic metric spaces. As an application, we prove a logarithm law; that is, we prove asymptotics for the depth in the packing of a typical geodesic.  This is joint work with Giulio Tiozzo.

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Abstract: Since its spillover to humans some 12,000 years ago, malaria, a deadly parasitic disease transmitted between humans via the bite of an infected adult female Anopheles mosquitoes, remains one of the deadliest infectious diseases of mankind. Much progress has been recorded in the battle against malaria over the last decade or two, prompting a renewed quest to significantly reduce its burden (by 90% by 2030) or eradicate it by 2040. Unfortunately, these laudable concerted global efforts are threatened by several challenges, such as widespread resistance to all the currently-available insecticides used in vector control, evolution of drug resistance, climate change, land-use changes, emergence of invasive species, human mobility (rural-urban migration), and quality of public health infrastructure and care. I will discuss some of these advances and challenges, in addition to presenting a mathematical framework for gaining deeper qualitative insight into the transmission dynamics and control of the disease in endemic areas. Some pertinent questions, such as those related to whether climate change will lead to a shift or range expansion of the malaria vector and whether or not the eradication goals can be achieved using existing control resources, will be discussed. Relevant bifurcations, and their implications vis a vis the persistence and/or extinction of the malaria vector and, consequently, the disease, will also be discussed. If time permits, I will discuss the potential utility of some of the DNA-based biocontrol methods (such as the release of transgenic mosquitoes) advanced by entomologists against the malaria vector.

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Abstract: In joint work with Gianluca Basso, we explore the class of Polish groups whose universal minimal flows admit a comeager orbit. By work of Ben Yaacov, Melleray, and Tsankov, this class contains all Polish groups with metrizable universal minimal flow, and by an example of Kwiatkowska, this inclusion is strict. We isolate the correct generalization of this class of Polish groups to the class of all topological groups. We call these the topological groups with "tractable minimal dynamics (TMD)." One way of phrasing what makes this class "tractable" is an "abstract Kechris-Pestov-Todorcevic correspondence" which characterizes membership in TMD using a Ramsey-theoretic property of the group. In particular, this implies that TMD is absolute between models of set theory. We also state some conjectures to the effect that any topological group not in TMD has "wild" minimal dynamics.

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Abstract: TBA

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Abstract: A $\textit{cluster}$ of length $l$ in a permutation from $S_n$ is a set of $l$ consecutive numbers that appear in any order in $l$ consecutive positions in the permutation.For $\eta\in S_3$, let $S_n^{\text{av}(\eta)}$ denote the set of permutations in $S_n$ that avoid the pattern $\eta$, and let $E_n^{\text{av}(\eta)}$ denote the expectationwith respect to the uniform probability measure on $S_n^{\text{av}(\eta)}$.For $n\ge l\ge2$ and $\tau\in S_l^{\text{av}(\eta)}$,let $N^{(n)}_{l}(\sigma)$ denote the number ofclusters of length $l$ in $\sigma$ and let $N^{(n)}_{l;\tau}(\sigma)$ denote the number of clusters of length $l$ whose order is the pattern $\tau$.We obtain explicit formulas for$E_n^{\text{av}(\eta)} N^{(n)}_{l;\tau}$ and $E_n^{\text{av}(\eta)}N^{(n)}_{l}$.These exact formulas yield asymptotic formulasas $n\to\infty$ with $l$ fixed, and as $n\to\infty$ with $l=l_n\to\infty$.In particular, for fixed $l$, depending on $\eta$ and $\tau$, the expectation $E_n^{\text{av}(\eta)} N^{(n)}_{l;\tau}$ either grows linearly in $n$ or is bounded and bounded away from zero.The expectation $E_n^{\text{av}(\eta)}N^{(n)}_{l}$ always has linear growth in $n$, and the maximum threshold size of $l_n$ for which $E_n^{\text{av}(\eta)}N^{(n)}_{l_n}$ remains bounded away from zero depends on $\eta$. We also obtain a weak law of large numbers.Analogous results are obtained for $S_n^{\text{av}(\eta_1,\cdots,\eta_r)}$, the permutations in $S_n$ avoiding the patterns $\{\eta_i\}_{i=1}^r$, where $\eta_i\in S_{m_i}$, in the case that $\{\eta_i\}_{i=1}^n$ are all \it simple\rm\ permutations. A particular case of this, where we can calculate the asymptotic behavior explicitly, is the set of \it separable\rm\ permutations, which corresponds to $r=2$, $\eta_1=2413,\eta_2=3142$.

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Abstract: Given a compact Riemann surface, nonabelian Hodge theory relates topological and algebro-geometric objects associated to it. Namely, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field. This gives a transcendental matching between two very different moduli spaces associated with the Riemann surface: the character variety (parameterizing representations of the fundamental group) and the Hitchin moduli space (parameterizing Higgs bundles). In 2010, de Cataldo, Hausel, and Migliorini proposed the P=W conjecture, which predicted that the Hodge theory of the character variety is determined by the topology of the Hitchin space, imposing surprising constraints on each side. In this talk, I will introduce the conjecture and review its recent proofs; time permitting, I will try to explain how this phenomenon relates to other geometric questions.

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Abstract: In the first lecture, given a proper map $f: X \rightarrow Y$, I introduced the perverse filtration on the cohomology of $X$, which measures the singularities of the fibers of $f$. In this talk, when $X$ is an abelian fibration over $Y$, I will explain a technique for studying this filtration via the Fourier-Mukai transform on DCoh(X), the derived category of coherent sheaves of X. This approach is a natural extension of ideas of Beauville and Deninger-Murre for studying Chow groups of abelian schemes. As an application, we get a proof of the P=W conjecture, introduced in the last lecture, but also other conjectures lifting these filtrations to Chow groups. Joint work with Junliang Shen and Qizheng Yin.

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Abstract: Chemotactic singularity formation in the context of the Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong -- this effect is conjectured to hold for more general classes of nonlinear PDEs. In this talk, I will introduce Keller-Segel equation coupled with several incompressible flows forced by buoyancy, including a flow characterized by Darcy's law for incompressible porous media and a Navier-Stokes equation. We prove that, in contrast to previous results driven by passive mixing flows, such active fluid coupling is capable of suppressing singularity formation via an interesting dynamical, nonlinear mixing mechanism which is manifested in a delicate kinetic-potential energy balance. This talk is based on a recent work by the author and an earlier joint work with Alexander Kiselev and Yao Yao.

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Abstract: Graphical models have become an important tool for summarizing conditional relations in a multivariate time series. Typically, the partial covariance is used as a measure of conditional dependence and forms the basis for construction of the interaction graph. However, for many real time series the outcomes may not be Gaussian and/or could be a mixture of different outcomes. For such time series using the partial covariance as a measure of conditional dependence may lead to misleading results. The aim of this talk is to develop graphical models for non-Gaussian time series. We propose a broad class of time series models which are specifically designed to succinctly encode the graphical model in its coefficients. For each univariate component in the time series, we model its conditional distribution with a distribution from the exponential family. We derive conditions under which the conditional specification leads to a well-defined strictly stationary time series. Further, we show that the time series is geometrically mixing and obtain an approximate Gibbs sampler to simulate sample paths.

(Based on joint work with Dr. Suhasini Subba Rao, Texas A&M University)

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Abstract: The D-equivalence conjecture of Bondal and Orlov predicts that birational Calabi-Yau varieties have equivalent derived categories of coherent sheaves. I will explain how to prove this conjecture for hyperkahler varieties of K3^[n] type (i.e. those that are deformation equivalent to Hilbert schemes of K3 surfaces). This is joint work with Junliang Shen, Qizheng Yin, and Ruxuan Zhang.