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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

Maxwell's Equations and HMM (by Boo) and A flexible algorithm for low-dose region-of-interest tomography (by Cowal)

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Wei-Xi Boo and Peter Cowal

ABSTRACT (talk by Boo): Meta-materials have unique electromagnetic properties and possess transformative potential for optics. Electromagnetic waves are governed by a set of partial differential equations called Maxwell's equations. The material response to electromagnetic waves is modeled by constitutive laws. Accurate simulations of Maxwell's equations with dispersive constitutive laws can aid the design of such materials. However, materials with such unique electromagnetic properties often have nano-scale structures that pose a challenge in solving Maxwell's equations numerically due to the resulting high computational cost. We describe a novel numerical method enabled by Heterogeneous Multiscale Methods (HMM), designed to simulate Maxwell’s equations coupled with the constitutive laws efficiently. We also present an energy analysis and error analysis for this numerical method.ABSTRACT (talk by Cowal): Traditional algorithms for CT reconstruction require measurements to be taken… Read more.

The Method of Moving Spheres on Hyperbolic Space and Classification of Solutions to a Class of PDEs

Kidder 237
Analysis Seminar

Speaker: Jianxiong Wang

The classification of solutions for semilinear partial differential equations, as well as the classification ofcritical points of the corresponding functionals, have wide applications in the study of partial differential equationsand differential geometry. The classical moving plane method and the moving sphere method on $\mathbb{R}^n$ provide aneffective approach to capturing the symmetry of solutions. In this talk, we focus on the equation\begin{equation*} P_k u = f(u)\end{equation*}on hyperbolic spaces $\mathbb{H}^n$, where $P_k$ denotes the GJMS operators and $f : \mathbb{R} \to \mathbb{R}$ satisfies certain growth conditions. I will introduce a moving sphere approach on $\mathbb{H}^n$, to obtain the symmetry property as well as the classification result towards positive solutions. Our methods also rely on Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalities on hyperbolic spaces together with a newly introduced Kelvin-type transform on $\mathbb{H}^n$. Read more.

Cobordism Obstructions to Complex Sections

Kidd 280
Geometry and Topology Seminar

Speaker: Dennis Ngyuen

There is a classical problem to determine whether a manifold admits r linearly independent tangent vector fields. In the case of one everywhere non-zero vector field, this problem was solved by Hopf, and the obstruction is the Euler characteristic of the manifold. Bokstedt, Dupont and Svane approached this problem by instead determining the obstruction to finding a cobordant manifold with r vector fields. We extend their results by looking at obstructions to finding linearly independent complex sections of the tangent bundle of almost complex manifolds. In this case, we are able to describe the rational obstruction for almost complex manifolds. This obstruction is given in terms of Chern characteristic numbers. Moreover, we are able to give certain bounds for r under which the torsion obstruction vanishes. Read more.

Free abelian groups detected by the Weil height

STAG 263
Algebra and Number Theory Seminar

Speaker: Jeff Vaaler

In 1947 Skolem proved that the multiplicative group of an algebraic number field K modulo its torsion subgroup is a free abelian group. We outline a proof that this remains true for infinite algebraic extensions of the rationals provided the infinite extension satisfies the Bogomolov property. In contrast to these results, the multiplicative group of all nonzero algebraic numbers modulo its torsion subgroup is known to be a vector space over the rationals, and therefore it is a divisible abelian group. Read more.

The Moduli Space of Graphical Associative Submanifolds

Kidd 280
Geometry and Topology Seminar

Speaker: Emily Windes

In this talk, I discuss an infinite-dimensional Lagrange-multipliers problem that first appeared in Donaldson and Segal’s paper “Gauge Theory in Higher Dimensions II”. The longterm goal is to apply Floer theory to a functional whose critical points are generalizations of three (real) dimensional, special Lagrangian submanifolds. I will discuss a transversality theorem related to the moduli space of solutions to the Lagrange multiplers problem. Read more.

Numerical Solution of Double Saddle-Point Systems

TBA
Colloquium

Speaker: Chen Greif

Double saddle-point systems are drawing increasing attention in the past few years, due to the importance of multiphysics and other relevant applications and the challenge in developing efficient iterative numerical solvers. In this talk we describe some of the numerical properties of the matrices arising from these problems. We derive eigenvalue bounds and analyze the spectrum of preconditioned matrices, and it is shown that if Schur complements are effectively approximated, the eigenvalue structure gives rise to rapid convergence of Krylov subspace solvers. A few numerical experiments illustrate our findings. Read more.

Rankin-Cohen Type Differential Operators on Automorphic Forms

STAG 263
Algebra and Number Theory Seminar

Speaker: Francis Dunn

In the classical setting, the derivative of a holomorphic modular form of integral weight on the complex upper half-plane is not in general a modular form since the derivative fails to satisfy the correct transformation properties. However, R. A. Rankin and H. Cohen were able to construct particular bilinear differential operators sending modular forms to modular forms. These Rankin—Cohen operators have several interesting properties and have been studied by D. Zagier, Y. Choie, T. Ibukiyama, and others.In this talk I will discuss the classical Rankin—Cohen operators, and some of their generalizations to automorphic forms in higher dimension, including ​​constructing Rankin—Cohen​ type differential operators on Hermitian modular forms of signature (n,n). Read more.

Journey to the Center of the Earth

Colloquium

Speaker: Gunther Uhlmann

We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It also has several applications in optics and medical imaging among others.The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will survey some of the known results about this problem.No previous knowledge of differential geometry will be assumed. Read more.

Digital Twins for Time Dependent Problems

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Juan Restrepo

ABSTRACT: A digital twin is a set of algorithms that connect the virtual world to the physical worl in a fully bi-directional way: for example, a predictive digital twin will use physics models, machine learned models, constraints as well as observations to make forecasts. A digital twin used as a controller would yield a virtual prescription, taking into account observations, that prescribes changes in the real world aimed at obtaining a certain desired real world outcome. I will describe ongoing work on developing a digital twin that will become central to an artificial intelligence framework for large scale electric grid resilience via adaptation. BIO: Juan M. Restrepo is a Distinguished Member of the R&D staff and the section head of the mathematics in computation section at Oak Ridge National Laboratory. His research concerns foundational aspects of machine learning and the development of new artificial intelligence algorithms for science. He is a Fellow of the Society of… Read more.