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Department of Mathematics

Upcoming Events

Branwen Purdy at her stall during OMSI meet-a-scientist day.

Branwen Purdy prepares hands-on activities for kids at the OMSI Meet-A-Scientist Day in Portland, to share hands-on learning experiences about her research in topological data analysis.

Join us for these events hosted by the Department of Mathematics, including colloquia, seminars, graduate student defenses and outreach, or of interest to Mathematicians hosted by other groups on campus.

Access our archive of events

Data-Driven Kernel Matrix Computations: Geometric Analysis and Scalable Algorithms

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Cai Difeng

ABSTRACT: Dense kernel matrices arise in a broad range of disciplines, such as potential theory, molecular biology, statistical machine learning, etc. To reduce the computational cost, low-rank or hierarchical low-rank techniques are often used to construct an economical approximation to the original matrix. In this talk, we consider general kernel matrices associated with possibly high dimensional data. We perform analysis to provide a straightforward geometric interpretation that answers a central question: what kind of subset is preferable for skeleton low-rank approximations. Based on the theoretical findings, we present scalable and robust algorithms for black-box dense kernel matrix computations. The efficiency and robustness will be demonstrated through experiments for various datasets, kernels, and dimensions, including benchmark comparison to the state-of-the-art packages for N-body simulations.BIO: Difeng Cai received his BS in math from University of Science and Technology… Read more.

Inverse Iteration for the Laplace Eigenvalue Problem with Robin and Mixed Boundary Conditions

STAG 110
Analysis Seminar

Speaker: Nicholas Zitzelberger

Inverse iteration is a tool used in operator theory to investigate the spectral properties of a linear operator. We apply the method of inverse iteration to the Laplace eigenvalue problem with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove convergence of the iterates to a non-trivial principal eigenfunction and show that the corresponding Rayleigh quotients converge to the principal eigenvalue. We also propose a related iterative method for an eigenvalue problem arising from a model for optimal insulation and provide some partial results.This presentation is based on a project that I conducted with Benjamin Lyons (Rose-Hulman Institute of Technology), and Ephraim Ruttenberg (University of Maryland Baltimore County), under the mentorship of Farhan Abedin (Lafayette College) and Jun Kitagawa (Michigan State University). Read more.

Maximal boundary rigidity for Alexandrov spaces

111 STAG
Geometry and Topology Seminar

Speaker: Qin Deng

Given an Alexandrov space with curvature bounded below by $K$, dimension $m$ and radius $r$, one may ask how large the boundary of X can be. In the case where $K=1$ and $r = \pi/2$, this is known as Lytchak’s problem and was answered by Petrunin, who showed the sharp upper bound $\mathcal{H}^{m-1}(\partial X) \leq \mathcal{H}^{m-1}(\mathbb{S}^{m-1})$. Rigidity was later analyzed by Grove-Petersen, who showed that in the case of equality $X$ must be a hemisphere or the intersection of two hemispheres. In this talk, I will address both the bound and rigidity for arbitrary $K$ and $r$ Read more.

Non-branching on spaces with Ricci curvature bounded from below

OWEN 101
Colloquium

Speaker: Qin Deng

On a smooth Riemannian manifold, the uniqueness of a geodesic given initial conditions follows from standard ODE theory. In this talk, I will extend a version of this result to the setting of RCD(K,N) spaces, which are metric measure spaces satisfying a synthetic notion of Ricci curvature bounded below first introduced by Sturm-Lott-Villani. Read more.

Modularity and Resurgence

Zoom
Algebra and Number Theory Seminar

Speaker: Eleanor McSpirit

The study of asymptotics as q approaches roots of unity is central to the theories of mock and quantum modular forms. In a collection of works, Gukov, Pei, Putrov, and Vafa proposed a candidate for a q-series invariant of closed 3-manifolds coming from physics. Many of these invariants are known to be mock and quantum modular forms, and this modularity has been integral to their study. Resurgent analysis is a natural tool to study this invariant from the perspective of physics, and is a theory centrally concerned with the relationship of functions to their asymptotic series. This has led to several questions on the interrelationship of resurgence and modularity. While this has been discussed across the subject, many questions remain. This talk will discuss ongoing work to make this connection explicit and natural from the perspective of number theory. Read more.

Math Education Seminar

Strand Agriculture Hall (STAG) 211
Mathematical Education Seminar

Speaker: Kristen Vroom

Dr. Kristen Vroom of Michigan State University will join us to present on her work as we continue discussion from last week. We will also discuss another of her papers:Vroom, K., & Ellis, B. (2024). Sociomathematical scaffolding as students engage in revising draft definitions, conjectures, and proofs. Educational Studies in Mathematics, 1-21. Read more.

Geometric inference and estimating the reach

111 STAG
Geometry and Topology Seminar

Speaker: John Harvey

In topological data analysis, the "manifold hypothesis" is that a dataset lying in Euclidean space in fact lies on, or near, some submanifold of that space. The submanifold is understood to provide information about the non-linear relationships between different variables.The goal of statistical inference is to use some set of sample data to understand aspects of an entire population and express a mathematically justifiable level of confidence in the statements made. In this talk I will introduce geometric inference, which aims to understand the geometry of a submanifold of Euclidean space using a finite sample of points drawn from the submanifold.I will introduce some of the philosophy and tools of inference which are probably not familiar to most practicing geometers and survey some results in this field. I will have a particular focus on the reach, where I proved new bounds with Clément Berenfeld, Marc Hoffmann and Krishnan Shankar, which Berenfeld and others recently showed were… Read more.

Generalized geometries for space-times

Colloquium

Speaker: John Harvey

The geometry of space-time used for the theory of relativity is Lorentzian geometry. Gravity is expressed through the curvature of space-time. I will introduce Lorentzian geometry and describe some of its key features. I will explain some physical motivation for generalizing this geometry and demonstrate some of the potentially undesirable phenomena that can arise as we do this. Recently Kunzinger and Sämann proposed Lorentzian length spaces as a synthetic generalization of Lorentzian geometry. Generally, a synthetic geometry is a theory which takes certain theorems from a well-established smooth geometric theory and uses them instead as axioms. I will discuss these spaces and, in particular, ways of defining curvature on them.This will include recent work I have done with Tobias Beran, Lewis Napper and Felix Rott showing that curvature bounds, which are essentially local, can have implications on the global geometry of the space. Read more.

Modular functions and the monstrous exponents

WNGR 201
Algebra and Number Theory Seminar

Speaker: Holly Swisher

In 1973 Ogg initiated the study of monstrous moonshine with the observation that the prime divisors of the monster group are exactly those for which the Fricke quotient X_0(p)+p of the modular curve X_0(p) has genus zero. Here, motivated by Deligne's theorem on the p-adic rigidity of the elliptic modular j-invariant, we present a modular function-based approach to explaining some of the exponents that appear in the prime decomposition of the order of the monster.This is joint work with John Duncan. Read more.

SIAMinar - Dr. Jane MacDonald

Kidder 274
Student Event

Speaker: Jane MacDonald

Title: The Postdoctoral Journey: Not Unlike the Ups and Downs of Cross-Country SkiingAbstract: Join me as a I share my story of the ups and downs of life as a post-doctoral researcher. I’ll recount how I chose my path, lessons I have learnt along the way, and what’s on my mind now as I navigate this phase of my career. Alongside these personal reflections, I’ll introduce one of my recent research interests — a mathematical model I developed for cross-country skiing — where I will draw some parallels between navigating a cross-country ski track and my science career. If you join us early at 3:30 pm in the lounge (Kidd 302), we'll have free food and conversation! Read more.

Sink or Soar: the interplay between buoyant bubbles and sinking sediments inenergizing turbulence near the ice-ocean boundary

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Megan Wengrove

ABSTRACT: At the terminus of tidewater glaciers an interplay of connected processes result in the melt of ice. From both field and laboratory observations, it has been suggested that both bubbles and sediments could be important yet neglected contributors to ice melt at the submarine tidewater glacier terminus. In the laboratory it has been shown that as glacier ice melts, air trapped in pores inside of the ice is released creating flow transpiration at the boundary and buoyant bubble rise at the ice-ocean interface, leading to increased melt (Wengrove et al.,2023). Additionally, during separate laboratory experiments, sediments entrained in the subglacial discharge plume are shown to increase the entrainment of warm ocean water toward the ice leading to higher melt rates (McConnochie andCenedese, 2023). In July 2024, we made the first ever video observations of both bubbles rising and sediments mixing and falling from a stationary-bolted platform to an Alaskan tidewater glacier… Read more.

Finite element methods for the Landau-de Gennes minimization problem of nematic liquid crystals

Colloquium

Speaker: Ruma Maity

Nematic liquid crystals represent a transitional state of matter between liquid and crystalline phases that combine the fluidity of liquids with the ordered structure of crystalline solids. These materials are widely utilized in various practical applications, such as display devices, sensors, thermometers, nanoparticle organizations, proteins, and cell membranes. In this talk, we discuss finite element approximation of the nonlinear elliptic partial differential equations associated with the Landau-de Gennes model for nematic liquid crystals. We establish the existence and local uniqueness of the discrete solutions, a priori error estimates, and a posteriori error estimates that steer the adaptive refinement process. Additionally, we explore Ball and Majumdar's modifications of the Landau-de Gennes Q- tensor model that enforces the physically realistic values of the Q tensor eigenvalues. We discuss some numerical experiments that corroborate the theoretical estimates, and adaptive… Read more.