Abstract: Any finite-dimensional p-adic representation of the absolute Galois group of a p-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their construction to the fundamental group of a p-adic affine variety with a semi-stable chart, and give a canonical formation of Sen's theory independently of the choice of the chart, which is even new in the case of local fields. Our construction relies on a descent theorem in the p-adic Simpson correspondence developed by Tsuji. When the representation comes from a Q_p-representation of a p-adic analytic group quotient of the fundamental group, we describe the Lie algebra action of its inertia subgroups in terms of the Sen operators, which is a generalization of a result of Sen and Ohkubo.
Abstract:Â Machine learning in non-Euclidean spaces have been rapidly attracting attention in recent years, and this talk will give some examples of progress on its mathematical and algorithmic foundations. A sequence of developments that eventually leads to non-Euclidean generative modeling will be reported. More precisely, I will begin with variational optimization, which, together with delicate interplays between continuous- and discrete-time dynamics, enables the construction of momentum-accelerated algorithms that optimize functions defined on manifolds. Selected applications, namely a generic improvement of Transformer, and a low-dim. approximation of high-dim. optimal transport distance, will be described. Then I will turn the optimization dynamics into an algorithm that samples from probability distributions on Lie groups. If time permits, the performance of this sampler will also be quantified, without log-concavity condition or its common relaxations. Finally, I will describe how this sampler can lead to a structurally-pleasant diffusion generative model that allows users to, given training data that follow any latent statistical distribution on a Lie group, generate more data exactly on the same manifold that follow the same distribution. If time permits, applications such as to quantum data will be briefly mentioned.
Abstract: In our paper "geomorphology of Lagrangian ridges" we showed how one may make a Lagrangian submanifold transverse to any Lagrangian distribution at the expense of introducing a certain combinatorial Lagrangian singularity which we called a "Lagrangian ridge". This result was essential to our existence theorem for arboreal skeleta of polarized Weinstein manifolds. I will explain a 1-parametric version of this story, which will be essential to our uniqueness theorem for arboreal skeleta of polarized Weinstein manifolds (up to positive Reidemeister moves). This is joint with Y. Eliashberg and D. Nadler.
Abstract: How can we reliably test whether a quantum computer has achieved an advantage over existing classical computers? A promising approach is to base these tests ("proofs of quantumness") on cryptographic hardness assumptions. Such assumptions are the foundation for encryption and authentication protocols, and as such they are well-studied. Brakerski et al. (arXiv:1804.00640) introduced an interactive proof quantumness based on a standard lattice-based assumption (learning with errors). What would it take to make cryptographic proofs of quantumness realizable on near-term devices? I will explore this question and exhibit some of the mathematics involved in this topic, with a focus on the paper "Depth-efficient proofs of quantumness" by Z. Liu and A. Gheorghiu (arXiv:2107.02163).
Abstract: Mathematical models have provided a general framework for understanding the dynamics and control of infectious disease. Many compartmental models are limited in that they do not account for the range of behavioral feedbacks that have been observed in the response to emerging infections. Here we expand on the SIR compartmental model framework by introducing a general class of behavioral feedbacks that encompasses both individual responses and non-pharmaceutical interventions. By linking transmission dynamics and behavior, this new class of models can capture the interplay of disease incidence, behavioral response, and controls such as vaccination. Within this wide class of behavioral models, we consider a minimal set of assumptions which we call: Reactivity, Resilience, and Boundedness. For example, Reactivity assumes that the immediate change in activity level at any given time depends only on the current activity level and the current disease incidence, but we do not assume any specific form for this reactivity function. Using these minimal assumptions on the response, we prove mathematically the existence of two new endemic equilibria depending on the vaccination rate: one in the presence of low vaccination but with reduced societal activity (the ``new normal"), and one with return to normal activity which requires a vaccination rate that is lower than would be required for disease elimination. We show how the stability of the various equilibria (disease free, old normal, and new normal) depends on Resilience and Boundedness of the response as well as the vaccination rate.
Abstract: It is well known that for a formula $\varphi$ one of the following is true: either there are arbitrarily large finite sets $A$ such that the number of consistent $\varphi$-types over $A$ is $2^{|A|}$ or the number of consistent $\varphi$-types over any finite set $A$ is bounded by a polynomial in the size of $A$. Bhaskar showed that a similar dichotomy is true when we count the number of consistent types over trees of parameters: the number of consistent $\varphi$-types is either exponential or bounded by a polynomial in the depth of the tree. Chase and Freitag found a unified way to explain the upper bounds on the number of consistent types in both cases by counting what they called banned sequences. In this talk, I will present a simplified argument for counting banned sequences and will discuss possible extensions. This is joint work with Vince Guingona.
Abstract: We target large-scale computational methods and parallel algorithms centered around a challenging application: global-scale high-resolution mantle convection. Estimating parameters in mantle convection and plate tectonics models from surface observations results in an optimization problem. The forward problem for mantle flow is governed by highly nonlinear, heterogeneous, and incompressible Stokes equations. Solving these governing equations is already a major challenge at extreme computing scales. Adding an outer loop for parameter estimation adds another dimension of solver challenges.
The computational methods for the forward problem rely heavily on adaptive meshes for local 1-km-resolution of the globe. The methods further include inexact Newton-Krylov with a combination of "BFBT" and multigrid preconditioning for the saddle point linear systems in each Newton step. Scalable parameter estimation is enabled by analytically derived adjoint Stokes equations (i.e., optimize-then-discretize) within a Newton's method for optimizing in the parameter space. Uncertainties of parameters are revealed by local approximations (of the Bayesian posterior) at the MAP point by computing a Gauss-Newton approximation of the Hessian. We show inference on cross-sectional models of the Pacific and the first global inference results for 1-km-resolving models.
Abstract: Problem setting is a critical precursor to problem solving. It involves the art of formulating the right problem statement. The importance of this phase is underscored by the fact that without a well-defined problem, finding the right tools and techniques for problem solution becomes a cumbersome and often futile endeavor. This transition from problem setting to problem solving is integral to the larger paradigm of knowledge development. While AI tools have made tremendous strides in recent years, they remain dependent on the foundation laid by human intelligence. Mathematicians, with their ability to discern patterns and relationships, data, and variables, play a vital role in this stage. In this course, I will introduce basic mathematical concepts from both traditional machine learning and scientific machine learning. Scientific machine learning, which integrates data-driven machine learning algorithms with physics-based digital models, provides an ideal platform for the virtuous merging of problem setting and problem solving, facilitated by a profound domain knowledge. During the course, the reference application will focus on the development of a mathematical simulator for the cardiac function.
Abstract: For any cubic surface over a field k, the number of its lines that are defined over k must be one of 0, 1, 2, 3, 5, 7, 9, 15, or 27. But which of these possibilities is actually realized by some cubic surface? I will talk about joint work-in-progress with Enis Kaya, Sam Streeter, and Happy Uppal on this question. The answer depends on the arithmetic of k and requires a thorough understanding of minimal del Pezzo surfaces and their coincidences under blowups.
Abstract: I'll discuss spectral gaps in the context of graphs, hyperbolic surfaces, and unitary representations of discrete groups. The main focus will be on spectral gaps that are (asymptotically) optimal. One famous example of this phenomenon is a theorem of Friedman stating that random d-regular graphs on a large number of vertices are almost Ramanujan. Now, analogs of this result are known for hyperbolic surfaces. What underpins some of the recent progress in this area are notions of strong spectral gaps arising from operator algebras that I'll also explain. My talk will also contain some very basic open questions that I hope will be of broad appeal.(Talk runs 2:00-4:00PM)
Abstract: BPS states appear very frequently in geometric applications of quantum field theory.This talk aims to explain BPS states, especially the case of BPS states in N = (2, 2) supersymmetric field theories in two dimensions, and how spectral networks count the BPS indices.
Abstract: Predictive inference under a general regression setting is gaining more interest in the big-data era. In terms of going beyond point prediction to develop prediction intervals, two main threads of development are Conformal Prediction and Model-free Prediction. Recently, Chernozhukov et al. [2021] proposed a new conformal prediction approach exploiting the same uniformization procedure as in the Model-free Bootstrap of Politis [2015]. Hence, it is of interest to compare and further investigate the performance of the two methods. In the paper at hand, we contrast the two approaches via theoretical analysis and numerical experiments with a focus on conditional coverage of prediction intervals. We discuss suitable scenarios for applying each algorithm, underscore the importance of conditional vs. unconditional coverage, and show that, under mild conditions, the Model-free bootstrap yields prediction intervals with guaranteed better conditional coverage compared to quantile estimation. We also extend the concept of ‘pertinence’ of prediction intervals in Politis [2015] to the nonparametric regression setting, and give concrete examples where its importance emerges under finite sample scenarios. Finally, we define the new notion of ‘conjecture testing’ that is the analog of hypothesis testing as applied to the prediction problem.
Abstract: Donaldson and Uhlenbeck-Yau established the classical result that on a compact Kahler manifold, an irreducible holomorphic vector bundle admits a Hermitian metric solving the Hermitian-Yang-Mills equation if and only if the vector bundle is Mumford-Takemoto stable. Motivated by the characterization of supersymmetric B-branes in string theory and mirror symmetry, Collins-Yau asked if a line bundle admits a solution of the deformed Hermitian-Yang-Mills (dHYM) equation is equivalent to it is stable for certain Bridgeland stability conditions. In this talk, we will discuss a partial answer to this question for a set of line bundles on a Weierstrass elliptic K3 surface. This is joint work with Tristan Collins, Jason Lo, and Shing-Tung Yau.
Abstract: Given an analytic family of smooth projective varieties over a complex manifold B, one can construct a holomorphic vector bundle over B whose fibers carry a polarized Hodge structure of weight k. This family of Hodge structures can be abstracted to the notion of a variation of Hodge structure (VHS) over B. In this talk, I will first describe the geometric VHS in the above setting and then define the abstract notion of VHS following E. Cattani's notes on VHS: https://webusers.imj-prg.fr/~fouad.elzein/Hodge.pdf
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