Speaker: Neza Korenjak (UT Austin) - Abstract: Free groups are negatively curved, but they still admit properly discontinuous actions on affine space, which is flat. This allows us to utilize hyperbolic geometry to study affine actions. In this talk, we will generalize an approach of Danciger-Gueritaud-Kassel for constructing proper affine actions in three dimensions. We use a hyperbolic surface to construct higher strip deformations, which can be used to define proper actions of Fuchsian free groups on affine (4n-1)-space for any n.
When: Mon, December 5, 2022 - 2:00pm Where: Kirwan Hall 1308
Abstract: Free groups are negatively curved, but they still admit properly discontinuous actions on affine space, which is flat. This allows us to utilize hyperbolic geometry to study affine actions. In this talk, we will generalize an approach of Danciger-Gueritaud-Kassel for constructing proper affine actions in three dimensions. We use a hyperbolic surface to construct higher strip deformations, which can be used to define proper actions of Fuchsian free groups on affine (4n-1)-space for any n.
Abstract: I will speak on joint work with Linus Hamann and Kieu Hieu Nguyen. For odd unramified unitary groups over a p-adic field, there is a Langlands correspondence using trace formula techniques due to Mok and Kaletha--Minguez--Shin--White. Another correspondence was constructed recently by Fargues--Scholze using p-adic geometry. We show these correspondences are compatible by first proving an analogous result for unitary similitude groups by studying the cohomology of local Shimura varieties. If time permits, we will discuss applications to the Kottwitz conjecture and eigensheaf conjecture of Fargues.
Speaker: Jeff Danciger (UT Austin) -abstract: A convex real projective surface is one obtained as the quotient of a properly convex open set in the projective plane by a discrete subgroup of SL(3,R), called the holonomy group, that preserves this convex set. The most basic examples are hyperbolic surfaces, for which the convex set is an ellipse, and the holonomy group is conjugate into SO(2,1). In this case, the eigenvalues of elements of the holonomy group are symmetric. More generally, the asymmetry of the eigenvalues of the holonomy group is a natural measure of how far a convex real projective surface is from being hyperbolic. We study the problem of determining which elements (and more generally geodesic currents) may have maximal eigenvalue asymmetry. We will present some limited initial results that we hope may be suggestive of a bigger picture. Joint work with Florian Stecker.
When: Mon, December 5, 2022 - 3:00pm Where: Kirwan Hall 3206
Abstract: We will discuss the existence and stability of traveling surface waves in porous media. These waves are solutions to the one-phase Muskat problem, which concerns the dynamics of the free boundary of a fluid flow modeled by Darcy’s law.
Abstract: Data-driven epidemiological modeling has allowed us to exploit a wealth of data related to the Covid-19 pandemic to provide real-time forecasts of the impact of the pandemic; providing key insights to the public and policy makers both in the US and abroad. These models have been used to assess the impact of public health interventions and estimate critical resource needs under great uncertainty. During the course of the COVID-19 pandemic, JHU-APL has developed the “Bucky Model”; an infectious disease model framework that models the spread of the COVID-19 pandemic at a sub-national level. At its base, the Bucky model is a geographically distributed SEIR model. Here, we will describe the JHUAPL-Bucky model as well as discuss how the model has been operationalized to support forecasts provided to both the CDC and the United Nations Office for the Coordination of Humanitarian Aid.
Abstract: In a series of lectures at Princeton University between 1935 and 1937, John von Neumann developed a continuous version of projective geometry: the central objects of this study, continuous geometries, are complete, complemented, modular lattices whose algebraic operations possess certain natural continuity properties. Beyond classical finite-dimensional projective geometries, the class of continuous geometries contains, for instance, every orthocomplemented complete modular lattice (e.g., the projection lattice of any finite von Neumann algebra).
In the course of his analysis, von Neumann established the following remarkable coordinatization theorem: every complemented modular lattice (with an order at least four) is isomorphic to the lattice of principal left ideals of some (up to isomorphism unique) regular ring. Furthermore, he proved that every irreducible continuous geometry possesses a unique dimension function (with values in the closed real unit interval), which then induces a compatible complete metric on the corresponding coordinatizing ring and thus furnishes the latter with a natural topology. The topological groups of units of such ”continuous rings” exhibit very peculiar dynamical behavior.
In the talk, I will give a brief overview of von Neumann’s continuous geometry and report on some recent progress in understanding the structure and dynamics of topological groups of units of continuous rings.
Abstract: In many scientific areas, deterministic models (e.g., differential equations) use numerical parameters. Often, such parameters might be uncertain or noisy. A more honest model should therefore provide a statistical description of the quantity of interest. Underlying this numerical analysis problem is a fundamental question - if two "similar" functions push-forward the same measure, would the new resulting measures be close, and if so, in what sense? We will first show how the probability density function (PDF) of the quantity of interest can be approximated. We will then discuss how, through the lense of the Wasserstein-distance, our problem yields a simpler and more robust theoretical framework.
Finally, we will take a steep turn to a seemingly unrelated topic: the computational sampling problem. In particular, we will discuss the emerging class of sampling-by-transport algorithms, which to-date lacks rigorous theoretical guarantees. As it turns out, the mathematical machinery developed in the first half of the talk provides a clear avenue to understand this latter class of algorithms.
Abstract: When the line bundle is not positive, another useful tool to study the cohomologies of line bundles is Demailly's holomorphic Morse inequalities. We will extend Bergman kernel to (0,q)-forms valued in a holomorphic line bundle and discuss local holomorphic Morse inequalities due to Berman in his Ph.D thesis. Finally, if time permits, I will mention some works in my master thesis regarding this direction.
Abstract: The smallest number of parameters needed to define an algebraic covering space is called its essential dimension. Questions about this invariant go back to Klein, Kronecker and Hilbert and are related to Hilbert's 13th problem. In this talk, I will give a little history, and then explain a new approach which relies on recent developments in p-adic Hodge theory.
Abstract: High dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves, results in empirically observed "stickiness" and poor theoretical mixing properties -- lack geometric ergodicity. In this talk, we introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of the Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy-tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the ``blessings of dimensionality'' that the mixing time decreases with dimension.
(join work with Krzysztof Łatuszyński and Gareth O. Roberts)
Abstract: Abstract:A basic fact of geometry is that there are no length-preserving smooth maps from a spherical cap into a plane. But what happens if you try to press a curved elastic shell into a plane anyways? It wrinkles along a remarkable pattern of peaks and troughs, the arrangement of which is sometimes random, sometimes not depending on the shell. After a brief introduction to the mathematics of thin elastic sheets and shells, this talk will focus on a new set of simple, geometric rules we have discovered for predicting wrinkles driven by confinement. These rules are the latest output from an ongoing study of elastic confinement using the tools of Gamma-convergence and convex analysis. The asymptotic expansions they encode reveal a beautiful and unexpected connection between opposite curvatures — apparently, surfaces with positive or negative Gaussian curvatures can be paired according to the way that they wrinkle when confined. Our predictions match the results of numerous experiments and simulations done with Eleni Katifori (U. Penn) and Joey Paulsen (Syracuse). Underlying their analysis is a certain class of interpolation inequalities of Gagliardo-Nirenberg type, whose best prefactors encode the optimal patterns.
Abstract: In the study of K-stability, Fujita and Li proposed the valuative criterion of K-stability on Fano varieties, which has played an essential role of the algebraic theory of K-stability. Recently, Dervan-Legendre considered the valuative criterion of polarized varieties, which is a generalization of Fujita-Li criterion on Fano varieties. We will show that valuative stability is an open condition. We would like to study the valuative criterion for the Donaldson's J-equation. Motivated by the beta-invariant of Dervan-Legendre, we introduce a notion, the so-called valuative J-stability and prove that J-stability implies valuative J-stability. If time permits, we show the upper bound of the volume of K-semistable polarized toric varieties as an application of valuative stability.
Abstract: The Mumford-Tate group of an abelian variety A over the complex numbers is an algebraic group G, defined in terms of the complex geometry of A, more specifically its Hodge structure. If A is defined over a number field K, then a remarkable result of Deligne asserts that the ℓ -adic cohomology of A gives rise to a G -valued Galois representation ρℓ:Gal(¯K/K)→G(Qℓ). We will show that for a place of good reduction v∤ℓ of A, the conjugacy class of Frobenius ρℓ(Frobv) does not depend on ℓ. This is joint work with Rong Zhou.
The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. It is known in certain cases that the marked length spectrum determines the metric up to isometry, and this is conjectured to be true in general. In this talk, we explore to what extent the marked length spectrum on a sufficiently large finite set approximately determines the metric.
Abstract: In this talk, I will discuss adaptive statistical inference based on variational Bayes. Although a number of studies have been conducted to analyze theoretical properties such as posterior contraction properties of variational posteriors, there is still a lack of general and computationally tractable variational Bayes methods that can achieve adaptive inference. To fill this gap, we propose a novel adaptive variational Bayes framework, which can operate on a collection of models. The proposed framework first computes a variational posterior over each individual model separately and then combines them with certain weights to produce a variational posterior over the entire model. It turns out that this combined variational posterior is the closest member to the posterior over the entire model in a predefined family of approximating distributions. We show that the proposed variational posterior achieves optimal contraction rates adaptively under very general conditions and attains model selection consistency when the true model structure exists. We apply the general results obtained for the adaptive variational Bayes to a large class of statistical models including deep learning models and derive some new and adaptive inference results.
Abstract: Let A be an abelian variety over ¯Q. In this talk I will consider the following conjecture of Mocz. Conjecture: Let c>0. In the isogeny class of A, there are only finitely many isomorphism classes of abelian varieties of height c. I will sketch a proof of the conjecture when the Mumford-Tate conjecture - which is known in many cases - holds for A. This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field. This is joint work with Lucia Mocz.
Abstract: Many machine learning algorithms have been developed to elicit information from large datasets, however, techniques for quantifying the uncertainty in the estimates and for conducting statistical inference are not as well developed. We discuss resampling inferential techniques for these tasks, and present a unified way of studying the theoretical properties of these techniques using a framework involving random resampling weights. Depending on distributional properties of these random weights, the framework we propose can be used for consistent inference under very general conditions involving dependent data, high or infinite dimensional parameters and estimators obtained as approximate solutions to optimization problems, or alternatively for very precise and accurate inference in several other problems. Thus, our framework can balance the desirable robustness and efficiency goals of statistical inference. We further extend the framework for computational efficiency in big data applications, and prove that the extended framework is both uniformly consistent as well as computationally efficient under mild conditions. Illustrative examples from climate sciences, biomedical applications and ethical data sciences are discussed.
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