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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

TBA

STAG 112
Mathematical Biology Seminar

Speaker: Taranjot Kaur

In an ever-changing natural world, both plants and pollinators are continually confronted by perturbations. Responses to such perturbations can ripple from populations to communities through networks of interacting species. Additionally, the response to perturbations can unfold at various timescales ranging from short-term behavioral processes at the individual level to long-term population persistence. The goal of the work I will present in my talk is to evaluate how responses to perturbations propagate through timescales in plant-pollinator communities. Specifically, we use mathematical tools of non-linear averaging and stochastic processes to investigate how disturbances at shorter timescale of nectar regeneration scale to long-term outcomes of abundances of plants and pollinators. Furthermore, we study the impact of temporal correlation in stochasticity, network structure, and adaptive foraging dynamics on this scale transition framework. Read more.

Fokas Diagonalization

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Dave Smith

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included. BIO: Dave Smith is an Applied Mathematician working at Yale-NUS College, Singapore since 2016. He completed his… Read more.

Asymptotic properties and separation rates for local energy solutions to the Navier-Stokes equations

BEXL 322
Analysis Seminar

Speaker: Patrick Phelps

We present recent results on spatial decay and properties of non-uniqueness for the 3D Navier-Stokes equations. We show asymptotics for the ‘non-linear’ part of scaling invariant flows with data in subcritical classes. Motivated by recent work on non-uniqueness, we investigate how non-uniqueness of the velocity field would evolve in time in the local energy class. Specifically, by extending our subcritical asymptotics to approximations by Picard iterates, we may bound the rate at which two solutions, evolving from the same data, may separate pointwise. We conclude by extending this separation rate to solutions with no scaling assumption. Joint work with Zachary Bradshaw. Read more.

Drawing and Morphing Graphs on Surfaces

Kidder Hall 237
Geometry and Topology Seminar

Speaker: Yanwen Luo

In his famous paper ``How to draw a graph" in 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte's spring theorem. This construction provides not only one embedding of a planar graph, but infinite many distinct embeddings of the given graph. This observation leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984 stating that the space of geodesic triangulations of a convex polygon is contractible. In this talk, we will introduce spaces of geodesic triangulations of surfaces, review Tutte's spring theorem, and present this short proof. We will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of more complicated surfaces. This is joint work with Tianqi Wu and Xiaoping Zhu. Read more.

Math Circle

Franklin K-8 School
Outreach

Professor Gibson and several math graduate students will be offering two Math Circles this fall term. The elementary school math circle is a chance for 3rd-5th grade students to do group math work on fun and interesting problems that compliment the elementary math curriculum. The middle school math circle will provide engaging activities for students in grades 6-8. We will meet weekly for 8 weeks after school. Read more.

Computations in Support of the Paramodular Conjecture

STAG 212
Number Theory Seminar

Speaker: Jerry Shurman

Please note the atypical date, time, and room. Abstract: Just as the modularity theorem states that all elliptic curves over Q arise from classical modular forms, the paramodular conjecture states that abelian surfaces over Q arise from paramodular forms. Paramodular forms are far more unwieldy than their classical counterparts. This talk will try to convey some sense of these matters, and then some sense of how computer calculations helped to establish that one particular abelian surface is indeed paramodular. Read more.

Quantizing the hyperbolic volume

Kidder Hall 237
Geometry and Topology Seminar

Speaker: Calvin McPhail-Snyder

Most prime knots are hyperbolic. A knot K is hyperbolic if its complement admits a complete metric of constant negative curvature, which by Mostow-Prasad rigidity is uniquely determined by K. The volume of this metric (the hyperbolic volume) is an important invariant of K and it admits a natural complexification called the complex volume. Another way to get invariants of knots is to use the Reshetikhin-Turaev construction to interpret a diagram of K as a morphism between representations of a quantum group; this leads to quantum knot invariants like the Jones polynomial. This seems to have little to do with hyperbolic volume, but there are a number of conjectured relationships such as the Volume Conjecture of Kashaev and Murakami-Murakami. Recently I (joint with N. Reshetikhin) have defined a new family of knot invariants that quantize the complex volume: in some ways they behave like the complex volume, and in others they behave like Jones polynomials. In my talk I will expand… Read more.

The Knave's Cosmological Theorem

Zoom
Number Theory Seminar

Speaker: Tamsyn Morrill

Abstract: The Look-Say sequence is a classic example of recursion. Its terms are verbalized descriptions of their predecessors --- initialized at 1 --- 11, 21, 1211, and so on. Conway demonstrated that the asymptotic growth rate of this sequence is the unique real root of a degree 71 monic polynomial. The general idea is to recast the problem in linear algebra through use of his Cosmological Theorem. Today I present a variation of this problem. A knave (of Smullyan's famed door-keeper puzzle) now controls the recursion. After working through some small examples, we will remake the Cosmological Theorem in the knave's image. Read more.

Serifert-Torres Formulas for the Alexander Polynomial of Links from Quantum sl2

Kidder Hall 237
Geometry and Topology Seminar

Speaker: Matthew Harper

In this talk, I will recall how the Alexander polynomial, a classical knot invariant, can be constructed as a quantum invariant from quantum sl2 at a fourth root of unity. I will then discuss the development of a diagrammatic calculus based on further investigation of quantum sl2 representations. Applying this calculus in the context of the Alexander polynomial allows us to compute the invariant for certain families of links using quantum algebraic methods, rather than using methods of classical topology. Read more.

Global Sensitivity Analysis of Plasma Instabilities via Active Subspaces 

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Stephen Pankavich

The dynamics of laboratory and space plasmas are often driven by potentially uncertain values of physical parameters. For this reason, the utilization of computational methods to quantify such uncertainty represents an important tool to understand how certain physical phenomena depend upon fluctuations in the values of these parameters. In this direction, I'll discuss the construction and implementation of new computational methods, called active subspace methods, to quantify the induced uncertainty within the (linear) stability/instability rates generated by perturbations in a collisionless plasma near spatially homogeneous equilibria. BIO: Steve Pankavich is a Professor in the Department of Applied Mathematics and Statistics at the Colorado School of Mines, where he has served as a faculty member for 11 years. He earned a PhD in Mathematical Sciences from Carnegie Mellon University and was a Zorn Postdoctoral Fellow at Indiana University. Prior to joining Mines, he also held a… Read more.

Direct methods for inverse problems in the shear wave elastography field

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Olalekan Babanyi

Shear wave elastography is a technique used to noninvasively estimate the mechanical properties of tissue from propagating mechanical waves. These mechanical properties can be used to noninvasively diagnose and help with the treatment of various diseases like cancer, and liver fibrosis. The mechanical properties can also be used to understand various biological processes like wound healing, and cell division. To compute the mechanical properties, one needs to solve an inverse problem governed by differential equation models. I will present several direct variational formulations that can be used to efficiently solve the inverse problem. I will discuss some of the mathematical properties of these variational formulations, and compare their performance on simulated data. BIO: Olalekan Babaniyi is currently an assistant professor in the School of Mathematics and Statistics at Rochester Institute of Technology (RIT). Prior to joining RIT, he was a post-doctoral scholar at the… Read more.

Regularization Methods for Inverse Problems in Imaging

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Malena Espanol

Discrete linear and nonlinear inverse problems arise from many different imaging systems, exhibiting inherent ill-posedness wherein solution sensitivity to data perturbations prevails. This sensitivity is exacerbated by errors arising from imaging system components (e.g., cameras, sensors, etc.), necessitating the development of robust regularization methods to attain meaningful solutions. Our presentation commences with the exposition of distinct imaging systems, and their mathematical formalism, and subsequently introduces regularization techniques tailored for linear inverse problems. Then, we delve into the variable projection method, a powerful tool to address separable nonlinear least squares problems. BIO: Malena Español is an Assistant Professor in the School of Mathematical and Statistical Sciences at Arizona State University. She has a Bachelor's in Applied Mathematics from the University of Buenos Aires and a Master's and PhD in Mathematics from Tufts University; she was… Read more.

Development of the Arctic Coastal Erosion Model with a Demonstration at Drew Point, AK

STAG 112
Applied Mathematics and Computation Seminar

Speaker: Jennifer Frederick

Erosion is accelerating along many stretches of the coastal Arctic, putting critical infrastructure at risk and threatening local communities. These permafrost-laden coastlines are increasing vulnerable to erosion due to declining sea ice and increasing duration of open water, more frequent storms during ice-free periods, and warming permafrost soils. However, predicting shoreline erosion rate remains extremely challenging because of the highly non-linear behavior of the coupled and changing environmental system. Although the Arctic comprises one-third of the global coastline and has some of the fastest eroding coasts, current tools for quantifying permafrost erosion are unable to explain the episodic, storm-driven erosion events. In this talk I will present the details of the development and calibration efforts for the Arctic Coastal Erosion (ACE) Model at Sandia National Laboratories. The ACE Model is a multi-physics numerical tool that couples oceanographic and atmospheric… Read more.