Abstract: In this talk, I will first describe how classical Dieudonne module of finite flat group schemes and p-divisible groups can be recovered from crystalline cohomology of classifying stacks. Then, I will explain how in mixed characteristics, using classifying stacks, one can define Dieudonné module of a finite locally free group scheme as a prismatic F-gauge (prismatic F-gauges have been recently introduced by Drinfeld and Bhatt--Lurie), which gives a fully faithful functor from finite locally free group schemes over a quasi-syntomic algebra to the category of prismatic F-gauges. This can be seen as a generalisation of the work of Anschütz--Le Bras on "prismatic Dieudonne theory" to torsion situations.
Abstract: We introduce a multi-modal model for scientific problems, named PROSE-PDE. Our model, designed for bi-modality to bi-modality learning, is a multi-operator learning approach which can predict future states of spatiotemporal systems while simultaneously recovering the underlying governing equations of the observed physical system. We focus on training distinct one-dimensional time-dependent nonlinear constant coefficient partial differential equations. In addition, we will discuss some extrapolation studies related to generalizing physical features and predicting PDE solutions whose models or data were unseen during the training. We show through numerical experiments that the utilization of the symbolic modality in our model effectively resolves the well-posedness problems with training multiple operators and thus enhances our model's predictive capabilities.
Abstract: Satellite operations are a valuable method of constructing complicated knots from simpler ones, and much work has gone into understanding how various knot invariants change under these operations. We describe a new way of computing the (UV=0 quotient of the) knot Floer complex using an immersed Heegaard diagram obtained from a Heegaard diagram for the pattern and the immersed curve representing the UV=0 knot Floer complex of the companion. This is particularly useful for (1,1)-patterns, since in this case the resulting immersed diagram is genus one and the computation is combinatorial. In the case of one-bridge braid satellites the immersed curve invariant for the satellite can be obtained directly from that of the companion by deforming the diagram, generalizing earlier work with Watson on cables. This is joint work with Wenzhao Chen.
Abstract: The goal of this talk is to give an overview of the advantages and disadvantages of having geometric locality in quantum error-correcting codes. Starting with an introduction to the surface code, I will highlight the nice features of a geometrically local 2D stabilizer code. However, we will also examine the limitations that arise from imposing geometric locality, and how these limitations come about, particularly with regard to the code parameters and the allowable set of logical gates. Finally, we will explore some interesting techniques such as magic state distillation and code-switching that can be used to overcome these limitations.
Abstract: The spread of epidemics is structured by time distributions, including the now-famous “serial interval” between when an individual experiences symptoms, and when the person that they infect experiences symptoms. This is often used to represent the “generation interval” between when the same two individuals were infected, but these can be importantly different. Defining these time distributions clearly, and describing how they relate to each other, and to key parameters of disease spread, poses interesting theoretical and practical questions.
I will discuss how transmission intervals link the “speed” and “strength” of epidemics, issues in their estimation, and their role in helping monitor changes in the parameters underlying disease, with examples from COVID-19, rabies and HIV.
Abstract: Coercivity thresholds are a central theme in geometry. They appear classically in the Yamabe problem (constant scalar curvature in a conformal class), in the Nirenberg problem (prescribed curvature on the 2-sphere), and in numerous problems on determining best constants in Sobolev embeddings and related functionals inequalities. In 1980's Aubin and Tian introduced the first such thresholds in the Kahler-Einstein problem and their study has been a central and still very active field. In 1988 Tian observed that these thresholds have quantum versions and he posed the so-called Stabilization Problem: do the equivariant quantum thresholds become constant (and hence equal to the classical thresholds)? Cheltsov conjectured that these invariants coincide with the algbero-geometric log canonical thresholds (lct), and this was verified by Demailly (2008). The best result so far has been Birkar's theorem (2019) that shows that the quantum lcts are constant along a subsequence in the absence of group actions. Over the past 20 years, previous works have claimed a solution to Tian's problem in the toric case, but it turns out that they assume without justification monotonicity of these invariants in the quantization parameter. In joint work with C. Jin we offer a new approach and solve Tian's problem in the toric case. Surprisingly, the equivariant lcts are constant already from the first quantum level. For more general Grassmannian lcts we offer counterexamples to stabilization and determine when it holds. The key new ideas are understanding the effect of finite group actions on these invariants, and relating these thresholds to support and gauge functions from convex geometry. Time permitting I will discuss extensions and generalizations to other invariants.
Abstract: Local-global limits let one transfer results from measurable combinatorics to asymptotic results about sequences of graphs. Folklore has it that local-global limits of graphs capture the same information as weak equivalence classes of pmp actions and (continuous) existential theories of probability algebras with automorphisms. I'll try to explain some of these connections and report on recent work generalizing local-global limits to hypergraphs and other structures.
Abstract: Basis Pursuit Denoising (BPDN) is a cornerstone of compressive sensing, statistics and machine learning. Its applicability to high-dimensional signal reconstruction, feature selection, and regression problems has motivated much research and effort to develop algorithms for performing BPDN effectively, yielding state-of-the-art algorithms via first-order optimization, coordinate descent, or homotopy methods. Recent work, however, has questioned the efficiency, robustness and accuracy of these state-of-the-art algorithms for BPDN. For example, the glmnet package for BPDN, which is state-of-the-art due to its claimed efficiency, lacks robustness and can yield inaccurate solutions that lead to many so-called false discoveries. Another example is existing homotopy methods for BPDN; most require technical assumptions that may not hold in practice to compute exact solution paths. Without an exact robust and efficient algorithm, these shortcomings will continue to hinder BPDN for high-dimensional applications. In this talk, I will present a novel homotopy algorithm based on differential inclusions that efficiently and robustly computes a solution to BPDN exactly up to machine precision. I will present some numerical experiments to illustrate the efficiency of our algorithm and discuss various theoretical implications of our algorithm.
Abstract: The classical question of determining which varieties are rational has led to a huge amount of interest and activity. On the other hand, one can consider a complementary perspective - given a smooth projective variety whose nonrationality is known, how "irrational" is it? I will survey recent developments, with an emphasis on surfaces and open problems.
Abstract: The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing mean field limits of controlled multi-agent systems, which arise in applications coming from macroeconomics, social behavior, and telecommunications. Justifying this continuum description is often nontrivial and is sensitive to the type of stochastic noise influencing the population. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of the well-posedness for a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces, and how this well-posedness allows for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type.
Abstract: In this talk, I will introduce the spectral network for Landau-Ginzburg Model. The good thing about LG model is that the supersymmetric ground states of this model are given by critical points of the potential functions. When connecting two ground states one may explicitly write down the BPS solitons, and the BPS indices for these solitons will correspond to the intersection numbers of vanishing cycles. In this talk, I will try to explain these correspondences and as a result, how the use of spectral network explains the Picard-Lefschetz for these vanishing cycles.
Abstract: The Inverse-Wishart (IW) distribution is a standard and popular choice of priors for covariance matrices and has attractive properties such as conditional conjugacy. However, the IW family of priors has crucial drawbacks, including the lack of effective choices for non-informative priors. Several classes of priors for covariance matrices that alleviate these drawbacks, while preserving computational tractability, have been proposed in the literature. These priors can be obtained through appropriate scale mixtures of IW priors. However, the high-dimensional posterior consistency of models which incorporate such priors has not been investigated. We address this issue for the multi-response regression setting ( q responses, n samples) under a wide variety of IW scale mixture priors for the error covariance matrix. Posterior consistency and contraction rates for both the regression coefficient matrix and the error covariance matrix are established in the "large q , large n " setting under mild assumptions on the true data-generating covariance matrix and relevant hyperparameters. In particular, the number of responses q=q_n is allowed to grow with n , but with q_n = o(n). Also, some results related to the inconsistency of the posterior are provided.
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