Abstract: This workshop is devoted to the mathematical legacy of Todd Drumm. The primary topics are affine manifolds, constant-curvature Lorentzian geometry, and links with hyperbolic geometry, as well as the geometry of the bidisk.
Abstract: In this talk, I'll introduce a family of algebraic varieties that has recently appeared in several distinct areas of mathematics. This family goes by the name of "braid varieties" and associates a variety to any positive braid. I'll discuss how braid varieties arise in symplectic topology (my specialty); in algebraic geometry through flags and constructible sheaves; and in algebraic combinatorics through cluster theory. I'll use braid varieties to explore the symplectic problem of classifying Lagrangian fillings of Legendrian links, and I'll talk about how cluster theory provides some new insight into this old problem.
Abstract: This seminar will provide an overview of how continuous stochastic processes have been applied to the study of animal movement ecology using data from GPS tracking devices. I will present the mathematical foundations of these applications and discuss how we statistically fit the stochastic process models to diverse biological datasets. I will then give an overview of the wide range of applications that my colleagues and I have found for these approaches, including such biological topics as: 1) animal home ranges, migration, and space use 2) behavioral evidence for learning and disease states 3) route-based movement by carnivores 4) consumer-resource interactions
Movement data from GPS tracking devices typically feature a high degree of temporal autocorrelation, often at multiple scales. Over the years, our work has dealt with such data in a variety of statistical contexts, including: 1) timeseries analysis 2) kernel density estimation 3) path estimation via kriging 4) estimation of probability ridges 5) comparative (i.e., phylogenetically controlled) analyses The talk will present results from joint work with mathematicians Leonid Koralov and Mark Lewis; past-postdocs Christen Fleming, Eliezer Gurarie, and Michael Noonan; past-PhD students Justin Calabrese and Nicole Barbour; current PhD students Frank McBride, Marron McConnell, Gayatri Anand, Stephanie Chia, Qianru Liao, and Phillip Koshute; current undergraduate Zachary Tomares; and hundreds of biologists. Open questions abound and span a wide range of difficulty. I have access to mountains of animal movement data and am eager for collaborators.
Abstract: The human population presents a richly documented natural laboratories for understanding polymicrobial dynamics of infectious disease pathogens, providing unique opportunities to answer broader questions about diversity and stability of ecological communities. First, inspired by Modern Coexistence Theory, I lay out Pathogen Invasion Theory (PIT) for predicting the outcome of pathogen competition. PIT reveals that mutual invasion of competing strains is near-universal across major human pathogens. Instead, what determines strain co-circulation is the subsequent persistence of competing strains, which depend on the dynamics of the susceptible host populations. Then, using COVID-19 intervention as an example, I examine how pathogen communities respond to perturbations and quantify ecological resilience across major human respiratory pathogens. Resulting estimates provide insights into the susceptible host dynamics and persistence. Finally, I present a case study, illustrating how subtle changes in population-level susceptibility, driven by an expansion of childcare facilities in Japan, translates to complex outbreak dynamics.
Abstract: We study the local linear convergence behavior of the Alternating Direction Method of Multipliers (ADMM) when applied to Semidefinite Programming (SDP). While ADMM is widely perceived as slow and only capable of achieving medium-accuracy solutions—due to both its sublinear worst-case complexity and empirical evidence of slow convergence—we challenge this conventional view. Specifically, we establish a new sufficient condition for local linear convergence: as long as the converged primal-dual solution satisfies strict complementarity, ADMM achieves local linear convergence, regardless of nondegeneracy conditions. Our proof relies on a direct local linearization of the ADMM update operator and a refined error bound for projection onto the positive semidefinite cone. This new bound improves upon prior results and highlights the anisotropic nature of projection residuals. We support our theoretical findings with extensive numerical experiments, demonstrating that ADMM exhibits local linear convergence across a broad class of SDP instances, including those where nondegeneracy fails. Additionally, we identify problem instances where ADMM performs poorly and trace these difficulties to near-violations of strict complementarity—a phenomenon that mirrors recent observations in linear programming. Finally, our experiments reveal intriguing connections between local linear convergence and rank identification. Joint work with Shucheng Kang (Harvard) and Xin Jiang (Cornell).
Abstract: Curve counting theory on Calabi-Yau and Fano threefolds has been a central topic in enumerative geometry. For Calabi-Yau fourfolds, DT4 virtual cycles can be defined on the Hilbert scheme of two-dimensional subschemes (or other stable pair type moduli) using the Borisov-Joyce/Oh-Thomas theory. These classes, however, become trivial when the Hodge locus of the surface class has positive codimension. We reduce the theory and prove that the resulting invariants remain deformation invariant along the Hodge locus. In pursuit of understanding the structure of invariants counting surfaces on Calabi-Yau fourfolds, we turn our attention to the moduli space of stable two-dimensional sheaves. For surfaces with mild singularities, we propose a conjecture that the pushforward of the (reduced) virtual cycle to the Chow variety has modular properties. This is joint work with M. Kool and H. Park.
Abstract: We introduce a family of intrinsic Gaussian noises on compact manifolds that we call “colored noise” on manifolds. With this noise, we study the parabolic Anderson model (PAM) on compact manifolds. Under some curvature conditions, we show the well-posedness of the PAM and provide some preliminary (but sharp) bounds on the second moment of the solution. It is interesting to see that global geometry enters the play in the well-posedness of the equation.
Abstract: We introduce an open class of discrete dynamical systems generated by differentiable self-covering maps on closed manifolds, which we call virtually expanding. We show that, for such systems, the Perron–Frobenius operator is quasi-compact on a Sobolev space of positive order. We also derive a few consequences of this quasi-compactness. We conjecture that most volume-expanding self-maps on closed manifolds are virtually expanding and present a partial result in support of this conjecture.
Abstract: The Wasserstein barycenter plays a fundamental role in averaging measure-valued data under the framework of optimal transport (OT). However, there are tremendous challenges in computing and estimating the Wasserstein barycenter for high-dimensional distributions. In this talk, we will discuss some recent progress in advancing the statistical and computational frontiers of optimal transport barycenters. We first introduce a multimarginal Schrödinger barycenter (MSB) based on the entropy regularized multimarginal optimal transport problem that admits general-purpose fast algorithms for computation. By recognizing a proper dual geometry, we derive sharp non-asymptotic rates of convergence for estimating several key MSB quantities (cost functional, Schrödinger coupling and barycenter) from point clouds randomly sampled from the input marginal distributions. We will also consider the computation exact (i.e., unregularized) Wasserstein barycenter, which can be recast into a nonconvex-concave minimax optimization. By alternating between the primal Wasserstein and dual potential Sobolev optimization geometries, we introduce a linear-time and linear-space Wasserstein-Descent H-Ascent (WDHA) algorithm and prove its algorithmic convergence to a stationary point.
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