Math Calendar
Upcoming Events
PDE Seminar
Locat Global constructions of impulsive gravitational wave spacetimesion- SC1431- Please note the different date
Yannis Angelopoulos, California Institute of Technology.
In this talk I will try to describe an upcoming result (which is joint work with Jonathan Luk) on the construction of global geodesically complete spacetimes that satisfy the Einstein vacuum equations and that contain two colliding impulsive gravitational waves. I will start by reviewing some of the classical results in the literature of the general theory of propagation of singularities for wave equations, then I will also review the important work of Luk-Rodnianski on local generic constructions, and the extensions of these results to the semi-global setting, before presenting the global theory. Finally, I will mention some open problems.
Computational Analysis Seminar
Adaptive resolution of fine scales in modes of microstructured optical fibers- Location: SC1320
Gabriel Pinochet-Soto – Portland State University
An adaptive algorithm for computing eigenmodes and propagation constants of optical fibers is proposed. The algorithm is built using a dual-weighted residual error estimator. The residuals are based on the eigensystem for leaky hybrid modes obtained from Maxwell equations truncated to a finite domain after a transformation by a perfectly matched layer. The adaptive algorithm is then applied to compute practically interesting modes for multiple fiber microstructures. Emerging microstructured optical fibers are characterized by complex geometrical features in their transverse cross-section. Their leaky modes, useful for confining and propagating light in their cores, often exhibit fine scale features. The adaptive algorithm automatically captures these features without any expert input. The results also show that confinement losses of these modes are captured accurately on the adaptively found meshes.
Joint work with: J. Gopalakrishnan, J. Grosek, and P. Vandenberge.
Topology & Group Theory Seminar
Title-Finite quotients of four-punctured sphere bundle groups- Location SC-1432
Tam Cheetham-West Yale
The finite quotients of the fundamental group of a 3-manifold are the deck groups of its finite regular covers. We often pass to these finite-sheeted covers for different reasons, and these deck groups are organized into a topological group called the profinite completion of a 3-manifold group. In this talk, we will discuss how to leverage certain properties of the mapping class group of the four-punctured sphere to study the profinite completions of the fundamental groups of fibered hyperbolic four-punctured sphere bundles over the circle.
Colloquium
Non-vacuum initial data sets in general relativity- Location – SC 5211
David Maxwell -University of Alaska, Fairbanks
We describe a principled approach for constructing non-vacuum initial data sets for the Cauchy problem in general relativity. The core idea has an interesting history of having been known in the ’70s, forgotten by the mathematical relativity community for decades, and now independently rediscovered and rigorously demonstrated. We show how
it explains why certain techniques for generating initial data worked well in the past, but also how it leads to new equations with applealing physical properties when generating initial data containing fluids. The talk will be targeted at a broad audience.
Geometry and Topology Seminar
Asymptotic behaviors of solutions to homogeneous complex Monge-Ampere equations on ALE K\”ahler manifolds – SC 1312
Qi Yao – Stony Brook University
Initiated by Mabuchi, Semmes, Donaldson, homogeneous complex Monge-Ampere (HCMA) equations become
a central topic in understanding the uniqueness and existence of canonical metrics in K\”ahler classes. Under the setting of asymptotically locally Euclidean (ALE) K\”ahler manifolds, one of the main difficulties is the asymptotic behaviors of solutions to HCMA equations. In this talk, I will give an introduction to canonical metric problems under the setting of ALE K\”ahler manifolds and discuss the recent progress in studying asymptotic behaviors of solutions to HCMA equations. It is still an ongoing project.
Host: Ioana Suvaina.
Subfactor Seminar
Adelic structure of low complexity subshifts- Location- SC 1432
Darren Creutz, U.S. Naval Academy
A subshift is a closed shift-invariant subset of $\{ 0, 1 \}^{\mathbb{Z}}$ or more generally $\mathcal{A}^{\mathbb{Z}}$ for some finite set $\mathcal{A}$, the alphabet. Subshifts are an interesting class in their own right which also arise in a natural way from arbitrary probability preserving actions from the coding by a (reasonable) partition. I will present a somewhat surprising result, joint with R. Pavlov, that subshifts of very low complexity are all measurably isomorphic to rotations on one-dimensional adelic compact groups. Consequences of this include that Sarnak’s Conjecture holds for all low complexity subshifts and the resolution of an open question of Ferenczi about the minimal complexity for weak mixing.
Subfactor Seminar
Twisted Araki-Woods algebras: factoriality via conjugate variables – Location: SC 1432
Zhiyuan Yang – Texas A&M University
The q-deformed Gaussian von Neumann algebras or more generally the Yang-Baxter deformed Gaussian von Neumann algebras were introduced in 1994 by Marek Bozejko and Roland Speicher as a deformation of free Gaussian algebras (isomorphic to the free group factors). In 2023, da Silva and Lechner generalized this Yang-Baxter deformation construction to the nontracial cases and the resulting algebras are called the twisted Araki-Woods algebras. We explain how the existence of conjugate variables implies the factoriality of finitely generated twisted Araki-Woods algebras using the powerful results of Brent Nelson on nontracial finite free fisher information. This generalizes the corresponding results for q-Gaussians by A. Miyagawa and R. Speicher and for q-Araki-Woods by M. Kumar, A. Skalski and M. Wasilewski. Should time permit, we will also talk about certain sufficient conditions for those algebras to have the Akemann-Ostrand property.
Number Theory Seminar
Even values of Ramanujan’s tau-function – Location: SC 1117
Wei-Lun Tsai, University of South Carolina
Inspired by Lehmer’s conjecture on the non-vanishing of Ramanujan’s tau-function, it is natural to ask whether any given integer is a tau-value. Many recent works have identified explicit examples of odd integers which are not tau-values. In this talk, I will discuss the examples of even integers that are not tau-values and demonstrate such results for infinitely many even integers. This is joint work with Jennifer Balakrishnan and Ken Ono.
Computational Analysis Seminar
Generative Adversarial Networks: Dynamics and Mode Collapse – SC 1320
Matias Delgadino – UT Austin
Generative Adversarial Networks (GANs) was one of the first Machine Learning algorithms to be able to generate remarkably realistic synthetic images. In this presentation, we delve into the mechanics of the GAN algorithm and its profound relationship with optimal transport theory. Through a detailed exploration, we illuminate how GAN approximates a system of PDE, particularly evident in shallow network architectures. Furthermore, we investigate the phenomenon of mode collapse, a well-known pathological behavior in GANs, and elucidate its connection to the underlying PDE framework through an illustrative example.
Number Theory Seminar
Bianchi modular forms over $\mathbb{Q}(\sqrt{-17}$) – Location: SC 1117
Kalani Thalagoda – Tulane University
Bianchi Modular Forms are generalization of classical modular forms defined over Imaginary quadratic fields. Similar to the classical case, we can use the theory of modular symbols for computation. However, when the class group of the Imaginary quadratic field is non-trivial, we can only directly compute certain components. But we can still use some computational tricks to extract the Bianchi Modular forms as Hecke eigensystems. In this talk, I will go over some of these computational techniques for the field $\mathbb{Q}(\sqrt{-17}$ which has class number $4$. With explicit example, I will demonstrate interesting Hecke eigensystems we observed.
Geometry and Topology Seminar
Geometry and Topology – Title- TBA- Location: SC 1312
Bar Roytman – University of Michigan
Abstract – TBA