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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

How do you lift a modular representation of a Galois group?

KEC 1001
Colloquium

Speaker: Ashwin Iyengar

This talk is about the role of Galois representations in the context of the Langlands program. First I will give a historical overview of the types of questions that come up in this area, and how they relate to number theory. Then I'll introduce the deformation theory of Galois representations and present a result, obtained jointly with G. Böckle and V. Paškūnas, which describes the space of lifts of modular representations of the Galois group of a p-adic local field. I will aim to be as non-technical and possible, and attempt to illustrate the broader ideas which motivate my research area. Read more.

Using Symmetry Reduction in Optimization to Investigate Stability of Shear Flows

Bexell Hall 321
Analysis Seminar

Speaker: Elizabeth Carlson

Abstract: Determining nonlinear stability of steady states for complex dynamical systems are notoriously difficult problems, even for the seemingly simplest cases. For example, it is expected that the standard steady state shear profile, 2D planar Couette flow, is globally stable for all Reynolds numbers, however the state-of-the-art analysis is decades old and only proves the stability for relatively low Reynolds numbers. In recent years, a promising computational approach uses polynomial sum-of-squares optimization to find Lyapunov functions based on low-mode projections onto an orthogonal basis of $L^2 \cap H^1$. Critically, physical symmetries inherent in the system can be exploited to reduce the complexity of the resulting optimization problem. We will present on rigorous and practical extensions of this work. Read more.

An introduction to the Langlands program: Understanding symmetry through number theory

OWEN 101
Colloquium

Speaker: Peter Dillery

The Langlands program predicts a deep connection between representationtheory and number theory. At its heart is the idea that spaces offunctions that are highly symmetric can be broken into "atomic" pieces,and that number theory governs how this decomposition works. This idea isa vast generalization of the decomposition of periodic functions intofrequencies using Fourier analysis.In this talk, I will start by discussing how to organize representationsusing linear algebra and harmonic analysis. I will then explain howintroducing number theory into the picture leads to the proof of Fermat'sLast Theorem, sheds light on the Ramanujan Conjecture, and culminates inthe Arthur multiplicity formula. This last conjecture completely describesa spectral decomposition for representations of Lie groups defined overthe rational numbers. Along the way, I will illustrate how modern numbertheory provides a unifying framework linking these seemingly differentproblems and conclude by discussing my… Read more.

The distribution of prime values of random polynomials

online
Algebra and Number Theory Seminar

Speaker: Katherine (Katy) Woo

The Bateman--Horn Conjecture predicts how often an irreducible polynomial assumes prime values. We will discuss how with sufficient averaging in the coefficients of the polynomial (exponential in the size of the inputs), one can not only prove Bateman--Horn results on average but also pin down precise information about the distribution of prime values at finite but growing scales. We will prove that 100% of polynomials satisfy the appropriate analogue of the Poisson Tail Conjecture, in the sense that the distribution of the gaps between consecutive prime values around the average spacing is Poisson. We will also study the frequencies of sign patterns of the Liouville function evaluated at the consecutive outputs of f; viewing f as a random variable, we establish the limiting distribution for every sign pattern. A key input behind all of our arguments is Leng's recent quantitative work on the higher-order Fourier uniformity of the von Mangoldt and M\"obius functions (in turn relying on… Read more.

Robustness of solvers and uncertainty quantification for heat conduction in permafrost soil

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Madison Phelps

We study heat conduction in permafrost soil at the Darcy scale. These involve nonlinear parabolic PDEs accounting for the phase transitions between frozen and unfrozen water phase transitions. We extend previous work of [Bigler, Vohra, Slugg, Mathangadeera and Peszynska] on the Stefan (ST) problem and adapted permafrost (P) models to treat the most general non-adapted case called (P*). Mathematically and computationally, (P*) is most challenging, while it is also most flexible from the applications point of view as it allows, in particular, immobile pockets of air and macro-pores. The model is only posed in the sense of distributions, and we develop a new analytical solution to the (P*)-model. The main challenge of the (P*)-model is that it involves the enthalpy-temperature relationship, $w \in \alpha(\theta)$, which is a multivalued graph at the freezing point. We show that it can be reformulated using a single-valued bijective map from $(\theta,\chi)$ to $w$, where $\chi$ is the… Read more.

Birkhoff sums: Recent research into irrational rotations

STAG 112
Dynamical Systems Seminar, Mathematical Biology Seminar, Probability and Data Science Seminar

Speaker: J.J.P. Veerman

How well distributed are $\{i\rho\}_{i=1}^\infty$ mod 1?Arrays of well distributed points are an important tool in Numerical Analysis. Irrational rotations play a central role in Ergodic Theory, Dynamical Systems, and Number Theory. Discrepancy (Pisot, Van Der Corput, 1930's) characterizes how evenly distributed a sequence of numbers in $[0,1)$ is. We study the discrepancy of $\{x_0+i\rho\}_{i=1}^n$.The Birkhoff measure $\nu(\rho,n,z) dz$ associated to ${\rm frac}(x_0+i\rho)$ for $i=1$ to $n$ is the probability that $\sum_{i=1}^n[{{\rm frac}}(x_0+i\rho)-1/2]$ is in $[z,z+dz)$ if the distribution of $x_0$ is uniform on the circle.New results: the graph of the Birkhoff measure $\nu(z)$ is a tile. If $n$ is a continued fraction denominator of $\rho$, then that graph is an isosceles trapezoid. The length of the support of $\nu$ equals the discrepancy (up to scaling).We also give new and much more efficient proofs of two classical - but largely forgotten - results that allow one to compute… Read more.

Fracture-controlled Organization of Mixing and Reaction Hotspots in Porous Media

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Lazaro Perez

Abstract:The talk will include a tutorial on modeling reactive transport. Fractures are ubiquitous features in porous media and exert a strong control on subsurface flow, mixing, and reactive transport by introducing preferential pathways and strong velocity contrasts that fundamentally reorganize how fluids spread, mix, and react. Despite their recognized importance, the mechanisms by which fracture connectivity controls the spatial localization of reactions and the emergence of reaction hotspots at the pore scale remain incompletely understood.Here, we investigate how increasing fracture connectivity reorganizes pore-scale flow, mixing and reaction dynamics using direct numerical simulations of flow coupled with reactive random walk particle tracking. We consider a bimolecular irreversible reaction occurring during the displacement of one reactant by another in two-dimensional heterogeneous porous media with progressively developed fracture networks. Reaction localization is… Read more.

Coupled Nonlinear Boundary Conditions in PDE-ODE Models discretized with Finite Elements: Analysis and Implementation 

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Tyler Fara

We study a nonlinear PDE-ODE system arising in bioheat modeling of localized cold exposure in which a parabolic temperature equation is coupled through thermoregulatory exchange to a lumped core-temperature variable, with nonlinear boundary fluxes modeling radiation, convection, and evaporation. We discuss a fully implicit finite element discretization and develop a nonlinear elliptic projection operator that accommodates the boundary coupling and provides uniform stability and approximation properties. These results yield optimal-order a priori error estimates for the backward Euler–Galerkin scheme. In the talk we focus on the construction and analysis of the projection operator, illustrate its behavior computationally, and apply it to the derivation of a priori error estimates for the numerical method.We illustrate with numerical results including those on multiscale modeling of the exchange terms. Read more.

Multiscale Modeling, Simulation, and Analysis of Microstructure Evolution in Polycrystalline Materials

Colloquium

Speaker: Yekaterina Epshteyn

In this talk, we will present recent perspectives on mathematical modeling, numerical simulation, and mathematical analysis of the evolution of the grain boundary network in polycrystalline microstructures. This evolution is a very complex, multiscale, and multiphysics process. Our efforts support the solutions of the central problems in materials science, the design of technologies delivering an arrangement of grains that produces a desired set of material properties.  Relevant recent experiments, along with current and future research, will be discussed as well.  Most technologically useful materials–spanning the length scale from meters to nanometers, from aircraft to microprocessors–are polycrystalline. Polycrystals are composed of small monocrystalline grains that are separated by grain boundaries of crystallites with different lattice orientations. The changes in the grain and grain boundary structure of polycrystalline materials highly influence their properties, including, but… Read more.

Structure-Preserving Algorithms for Hyperbolic Balance Laws and Related PDE-Based Models

STAG 210 (note unusual day/location)
Applied Mathematics and Computation Seminar

Speaker: Yekaterina Epshteyn

ABSTRACT: Hyperbolic conservation/balance laws and related PDE-based models are essential mathematical apparatus for modeling a variety of complex physical phenomena, including but not limited to wave propagation, fluid flow, biological and materials science phenomena. Over the past few decades there has been enormous progress in designing stable, robust, structure-preserving numerical schemes (such as positivity-preserving and/or well-balanced schemes) that have enabled high-fidelity simulation of phenomena described by nonlinear hyperbolic PDEs. The main goal of our recent work is to extend these capabilities to systems with a stochastic component, which are relevant models in practical real-world situations since precise knowledge of an environment or operating conditions is frequently absent in such scenarios, and naturally results in models where randomness is used to describe the ignorance.In this talk, we will discuss progress in the design of structure-preserving numerical… Read more.

A Stroll Through Geometric Ideas

Colloquium

Speaker: Laura Schaposnik

During the first half of the talk, we will introduce Higgs bundles, their integrable system, and motivate why they become useful tools to further our understanding in different geometric settings. After describing some dualities they satisfy (not only from mirror symmetry but also via other correspondences such as low-rank isogenies), we will then focus on different methods to understand the Hitchin fibration and branes it contains, and especially its singular fibers (monodromy, transitional geometries, Cayley correspondences). Then, we shall move on to more applied realms and look at how geometric insights can be used to classify viruses, understand the spread of fake news, examine the relationship between COVID and dengue, and address other questions about the world. Read more.