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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

Landis' conjecture in the plane

Kidder 237
Analysis Seminar

Speaker: Blair Davey

In the late 1960s, E.M. Landis made the following conjecture: If u and V are bounded functions, and u is a solution to the Schr\"odinger equation in Euclidean space that decays faster than exponential, then u must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded, complex-valued functions u and V that solve the Schr\"odinger equation in the plane and satisfy |u(x)| \le c \exp(- C |x|^{4/3}). The examples of Meshkov were accompanied by qualitative unique continuation estimates for solutions in any dimension. Meshkov's estimates were quantified in 2005 by J. Bourgain and C. Kenig. These results, and the generalizations that followed, have led to a fairly complete understanding of these unique continuation properties in the complex-valued setting. However, Landis' conjecture remained open in the real-valued setting. We will discuss a recent result of A. Logunov, E. Malinnikova, N. Nadirashvili, and F. Nazarov that resolves the real-valued… 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.

Distinctness of Two Pseudo-Anosov Homeomorphisms

Kidder 274
Ph.D. Defense

Speaker: Mesa Walker

In 1981, Arnoux and Yoccoz constructed the first known family of odd degree pseudo-Anosov homeomorphisms. In 1985, David Fried constructed another family of pseudo-Anosov homeomorphisms using completely different methods. Fried’s genus 3 homeomorphism and Arnoux and Yoccoz’s genus 3 homeomorphism both have the same stretch factor, and Fried asked if these maps are distinct or the same. We show that these maps are distinct. We prove the distinctness by blowing up Arnoux and Yoccoz's surface at its cone singularities and applying Fried's method of studying cross sections of mapping tori. We find in our analysis that there is no torus with two points blown up as a cross section of the mapping torus of Arnoux and Yoccoz's blown up homeomorphism. However, Fried's example similarly treated must lead to such a cross section. We can therefore conclude that the pseudo-Anosov homeomorphism found by Arnoux and Yoccoz is distinct from the one found by Fried, as the mapping tori are distinct. Read more.

Fast-spreading pathogens affecting small populations: Why does FMDV persist in African Buffalo populations?

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Ricardo Reyes Grimaldo

ABSTRACT: The rise of interactions between wildlife and humans increases the need to understand the relationship between different ecological and epidemiological processes within and among different interacting populations. Foot-and-mouth disease viruses (FMDV) are among the most infectious pathogens known to man, and Wild buffalo (Syncerus caffer) act as a reservoir host for FMDV in the sub-Saharan region of the African continent. Because of the impact on international livestock trade that Foot-and-mouth disease can cause, and its ubiquitous characteristic of persisting in wild populations of African buffalo while remaining highly contagious, raises the question of how this pathogen persists in the wild. In this talk, we will discuss current modeling work concerning the propagation of FMDVs in their reservoir host, through a modified McKendrick–von Foerster equation solved through the Crank-Nicolson method. We consider the dynamics for the loss of acquired immunity, its role in the… Read more.

Effects of Mixing Flows on Aggregative Behavior

Kidder 237
Analysis Seminar

Speaker: Nicholas Harrison

In this talk, I'll consider the scenario described by the Patlak-Keller-Segel (PKS) Model of chemotactic aggregation together with the presence of some strong ambient advective forcing fluid, say wind or turbulent waters. The PKS model is well-known to exhibit finite-time blow-up behavior in many relevant settings, but it turns out that if the advective flow is sufficiently well-mixing, then there is a regularization effect which suppresses this blow-up (and in fact induces convergence to the mean). I'll discuss an analytic measure of mixing, introduce what are known as Relaxation Enhancing (RE) flows, and how RE flows can suppress the blow-up described as above in the parabolic-elliptic PKS Model. If there is time, I'll pose the question of "anomalous" blow-up suppression in the case of the zero diffusivity limit to the hyperbolic-elliptic PKS, for which there appears to be many similarities with recent results around scalar turbulence. Read more.

Reducible Surgeries on Slice and Almost L-Space Knots

Kidd 280
Geometry and Topology Seminar

Speaker: Holt Bodish

A celebrated theorem of low dimensional topology states that any 3-manifold can be obtained from S^3 by Dehn surgery on a link. However, it is still an open question which 3-manifolds arise as Dehn surgery on a knot (a link with one component). We use tools from Heegaard Floer homology to investigate a special case of this question: when does Dehn surgery on a knot produce a reducible 3-manifold (a non-trivial connected sum of two 3-manifolds). We show that slice knots only admit reducible surgeries of a particular kind and that only certain slopes on almost L-space knots can produce reducible 3-manifolds. This is joint work with Robert DeYeso III. 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.

Lonseth Lecture: Inverse Problems and Harry Potter's Cloak

Construction & Engineering Hall at the LaSells Stewart Center
Lonseth Lecture

Speaker: Gunther Uhlmann

Abstract: Inverse problems arise in all fields of science and technology where causes for a desired or observed effect are to be determined. By solving an inverse problem is in fact how we obtain a large part of our information about the world. An example is human vision: from the measurements of scattered light that reaches our retinas, our brains construct a detailed three-dimensional map of the world around us. In the first part of the talk we will describe several inverse problems arising in different contexts.In the second part of the lecture we will discuss invisibility. Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last 20 years or so there have been several scientific proposals to achieve invisibility. We will describe in a non-technical fashion a simple and powerful proposal, the so-… 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.