Join us for an upcoming seminar featuring mathematics faculty and invited speakers on one of our seven research topics. You may also see upcoming seminars by topic:

Join us at the for the first P&D seminar meeting! We will have brief introductions and we’ll discuss plans for the rest of the term. This is great opportunity to get to know the probability group. Read more.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.