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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

Exploring the Brownian loop measure

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

Speaker: Greg Lawler

I will review the Brownian loop measure as a limit of random walk loop measures, its relation to Schramm-Loewner evolutions, and discuss some decompositions into boundary bubbles. We give a formulation of the measure on the unit disk in terms of measure-driven Loewner evolutions and something we call “Brownian bubble tea”. This latter work is part of joint work with Frederik Viklund, Yilin Wang, and Catherine Wolfram. Read more.

Heegaard Floer homology, Dehn surgery and (2+1)-dimensional TQFT

Batcheller Hall 250
Geometry and Topology Seminar

Speaker: Ian Zemke

Abstract: Heegaard Floer homology is an invariant for 3-manifolds, as well as knots and links in 3-manifolds. A classical theorem of Lickorish-Wallace states that every 3-manifold can be constructed by Dehn surgery on a link in S^3. Heegaard Floer homology admits useful Dehn surgery formulas, which have played an important role in many applications of the theory. In this talk we will describe some interesting algebraic perspectives on these formulas, as well as some applications of these perspectives. Read more.

Dehn surgery and homology cobordism

KIDD 364
Colloquium

Speaker: Ian Zemke

One very important object of study in low-dimensional topology is the homology cobordism group. The structure of this group is largely unknown, although it is abelian and it is known that it is very large (it contains an infinite rank summand). In this talk, we will review the basics of this group, as well as more recent developments. We will end with some results which can be derived from Hendricks and Manolescu’s theory of involutive Heegaard Floer homology, focusing on a formula for Dehn surgery due to Hendricks, Hom, Stoffregen and myself. Read more.

Mathematical and Computational Aspects of Nonlinear Heat Conduction Models with Generalized Constitutive Laws

Kidder 274
Ph.D. Defense

Speaker: Madison Phelps

This dissertation is dedicated to the selected mathematical and computational challenges for nonlinear partial differential equation models of heat conduction. In particular, we illustrate the mathematical structure and develop robust computational techniques for a class of problems with generalized constitutive laws. The models we consider describe heat conduction with phase transitions in partially frozen to unfrozen soils for which the generalized constitutive laws may be expressed as multi-valued graphs. We show that these generalized constitutive laws can be framed in a single-valued monotone relationship in an appropriate product space. The solutions to these models may feature limited regularity or even discontinuities across the phase change interface and must be posed in the sense of distributions; we develop these formulations in detail. This framework also leads naturally to a new family of analytical solutions. However, the approximation of these solutions present… Read more.

Heap Birkhoff polytopes

Zoom
Algebra and Number Theory Seminar

Speaker: Emily Gunawan

Abstract: For each orientation c of a type A Dynkin quiver, we define a c-Birkhoff polytope Birk(H) and show that it is integrally equivalent to the order polytope for poset H, the heap of the c-sorting word of the longest permutation. A consequence of this result is that the volume of the c-Birkhoff polytope is the number of the longest chains in the type A c-Cambrian lattice. We will also discuss current work in type B and a generalization of our result to other Birkhoff subpolytopes Birk(H) corresponding to heaps H of other reduced words of an element in the symmetric group. Read more.

Loop-erased random walks and uniform spanning trees

KEC 1001
Colloquium

Speaker: Greg Lawler

I will discuss two related models that arise in statistical physics and give an overview of their properties and various methods used to understand them and their continuum limits. The methods are many: probabilistic, combinatorial, fractal geometry, Laplacians and their determinants, random geometry, conformal fields. This will be an introduction for a general mathematical audience: I do not expect people to have heard of these terms before. Read more.

Heterogeneous domain decomposition approach for coupled soil-snow model

STAG 110
Applied Mathematics and Computation Seminar

Speaker: Praveeni Mathangadeera

ABSTRACT: We present a heterogeneous domain decomposition approach for solving nonlinear heat conduction for a snow-soil model in Arctic environments. The snow and soil domains are treated as distinct physical domains, each described by a separate nonlinear heat equation with distinct nonlinear laws for material properties. These two models are coupled across their interface by ensuring the continuity of temperature and the conservation of heat flux. The coupled system is solved iteratively with a Dirichlet to Neumann approach, where the snow and soil models are solved independently, updating the interface conditions within a Richardson scheme until convergence is achieved. We show the basics of analysis, compare with Schur complement calculations, and with the monolithic approach. Read more.

The Arithmetic of Laurent Coefficients of Meromorphic Modular Forms

Zoom
Algebra and Number Theory Seminar

Speaker: Eleanor McSpirit

The arithmetic study of Taylor coefficients of modular forms at CM points has its roots in the classical theory of complex multiplication, beginning with the algebraicity the values of the j-function at CM points. The study of Taylor expansions of modular forms at CM points has since developed through work of Shimura on nearly holomorphic modular forms and became more explicit in later work of e.g. Rodriguez-Villegas and Zagier. Recently, Bogo, Li, and Schwagenscheidt studied Laurent expansions of meromorphic modular forms with poles at CM points and observed arithmetic patterns in examples. In this talk, I will discuss ongoing joint work with Rolen aimed at explaining and characterizing the arithmetic behavior of these Laurent coefficients. Read more.

A Comparative Analysis of Mathematical Knowledge for Teaching in Ghana’s Objective-Based and Standards-Based SHS Mathematics Curricula

Zoom
M.S. Presentation

Speaker: Philip Danso

Curriculum reform in mathematics education often seeks to improve students’ conceptual understanding, problem-solving abilities, and readiness for real-world application. In Ghana, the transition from the objective-based Senior High School (SHS) mathematics curriculum to the standards-based curriculum represents a significant shift in instructional expectations and learning goals. This study investigates these two curricula through the framework of Mathematical Knowledge for Teaching (MKT) in order to identify the types of teacher knowledge embedded within each and to examine how these have evolved over time. Using qualitative document analysis, the study codes both curriculum documents based on the MKT domains proposed by Ball, Thames, and Phelps (2008). The analysis focuses on key instructional components, including objectives, learning outcomes, and pedagogical guidance, to determine how each curriculum positions the role of the teacher in facilitating mathematical learning.… Read more.

A Stroll Through Geometric Ideas

KEC 1001
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.

A Stage-Structured Eco-Evolutionary Model with Pulsed Dynamics

Kidder 274
M.S. Presentation

Speaker: Abigail Adjei

Insecticide application is a major method of pest control in agriculture. Motivated by this, we develop a stage-structured mathematical model for population dynamics coupled with genetic evolution, in which the population is divided into developmental stages and the frequency of a resistant allele is tracked within each stage over time. Using this model, we study how repeated insecticide applications influence resistance evolution in pest populations. The baseline model combines continuous population dynamics, describing recruitment, maturation, and mortality, with allele-frequency dynamics for the resistant allele and discrete pulse updates representing repeated insecticide exposure. We establish basic qualitative properties of the model, including existence and uniqueness of solutions, positivity and invariance of the biologically relevant region, and we characterize how pulse events affect resistant-allele frequencies. The model is then extended to incorporate fitness trade-offs… Read more.

Geometry of Hidden & Broken Symmetries

Construction & Engineering Hall at LaSells Stewart Center
Lonseth Lecture

Speaker: Laura P. Schaposnik

Abstract: Many systems look complicated until we ask what stays the same - at both micro and macro scales. At the micro level, exchangeable interactions and local conservation laws simplify PDEs and networks; at the macro level, these invariances organize families of solutions and predict emergent behavior. Then we’ll break symmetry on purpose to see what drives reality: anisotropy in crystal growth and snowflakes, trust biases that trigger information cascades, and more. Symmetry and its selective breaking offer a way to reduce complexity, illuminate mechanisms, and connect fine-scale rules with large-scale patterns across geometry and the applied sciences. Read more.