01/29/2021: Vivian Healey

Title: Scaling Limits and the Loewner Equation: From Brownian Motion to Multiple SLE

Abstract: Starting with a simple random walk, rescale (in space and time) and take the limit as the step size goes to zero; the result is Brownian motion, which is used to model random motion throughout the sciences and is one of the fundamental objects in the field of probability. But what limit would you get if you started by assuming that the random walk never intersected itself? The answer (to many versions of this question in two dimensions) is Schramm-Loewner evolution (SLE). In this talk, I will discuss SLE from a few perspectives: as the scaling limit of discrete models with links to physics, as the output of the Loewner differential equation, and as a random path in the disk. I will also discuss recent applications of these perspectives to the construction of multiple SLE curves. (Including joint work with Gregory Lawler.)


02/05/2021: Nik Weaver, Washington University, St. Louis

Title: A "quantum" Ramsey theorem for operator systems

Abstract: Let V be a linear subspace of M_n(C) which contains the identity matrix and is stable under the formation of Hermitian adjoints.  I claim that if n is sufficiently large then there exists a rank k orthogonal projection P such that dim(PVP) = 1 or k^2.

I will explain the statement of the theorem in more detail and talk about why it is "quantum" and how it relates to Ramsey's classic theorem about graphs.  Then I will describe some of the ideas that go into the proof.



02/26/2021: Gail Wolkowicz, McMaster University, Hamilton, Canada

Title: A Delay Model for Persistent Viral Infections in Replicating Cells

Abstract: Persistently infecting viruses remain within infected cells for a prolonged period of time without killing the cells and can reproduce via budding virus particles or passing on to daughter cells after division. The ability for populations of infected cells to be long-lived and replicate viral progeny through cell division may be critical for virus survival in examples such as HIV latent reservoirs, tumor oncolytic virotherapy, and non-virulent phages in microbial hosts. We consider a model for persistent viral infection within a replicating cell population with time delay modeling the length of time in the eclipse stage prior to infected cell replicative form. We obtain reproduction numbers that provide criteria for the existence and stability of the equilibria of the system and provide bifurcation diagrams illustrating {transcritical (backward and forward), saddle-node, and Hopf} bifurcations, and provide evidence of {\it homoclinic bifurcations} and a {Bogdanov-Takens bifurcation. We investigate the possibility of long-term survival of the infection (represented by chronically infected cells and free virus) in the cell population by using the mathematical concept of {robust uniform persistence}. Using numerical continuation software with parameter values estimated from phage-microbe systems, we obtain two-parameter bifurcation diagrams that divide parameter space into regions with different dynamical outcomes. We thus investigate how varying different parameters, including how the time spent in the eclipse phase, can influence whether the virus survives.


03/5/2021: David Pengelley, Oregon State University and NMSU

Title: How and Why did Fermat Discover his Theorem? A hands-on investigation

Abstract: We know why Fermat discovered his theorem: yes, that theorem, the one about power residues modulo a prime that underlies all of number theory and internet security today. We know because his letters written in 1640 reveal what problem from antiquity he was working on, why he was working on it, and what he wanted to know. But this doesn't tell us how he discovered his theorem, which is an unexpected amazement and wouldn't just be guessed. Participants will be invited to think like Fermat, look at the very problem and data he was studying, and conjecture the patterns he saw that led to his theorem. Absolutely no background is required! But this doesn't mean it won't be challenging. Please bring your pencil/paper/calculator!


03/12/2021: David Eisenbud, MSRI and University of California, Berkeley

Title: The Problem of Subtraction in Algebraic Geometry and Commutative Algebra

Abstract: Some curves in 3-space can be realized as the intersections of two surfaces; for example, the intersection of two quadric (= degree 2) hypersurfaces containing a line in common has another component, which can be thought of as the intersection minus the line. The invariants of that other component can be computed from this information: it must be a curve of degree 3 and genus 0. In commutative algebra, subtractions appear as ideal quotients, and raise other interesting questions, some very subtle.

Such problems have been studied for more than 100 years. I'll discuss the origins of this theory of "residual intersections", and some of the modern developments in algebraic geometry and commutative algebra.


03/19/2021: Fred Wehrung, University of Caen Normandy

Title: Intractability for images of certain functors

Abstract: There are various open problems asking for the description of certain classes of structures.

Examples are: the class of ordered K_0 groups of unit-regular rings; the class of (Zariski-like) spectra of abelian lattice-ordered groups; the class of submodule lattices of modules. All those classes are images of functors that preserve $\lambda$-directed colimits for large enough $\lambda$. I will present a general framework enabling to verify, in certain cases, a form of intractability for the given class. This intractability implies the failure of closure under elementary equivalence for any infinitary language. It is entailed by the existence of a certain type of non-commutative diagram, whose image under the given functor is commutative.



03/26/2021: Seth Sullivant, North Carolina State University

Title: Phylogenetic Algebraic Geometry

Abstract: The main problem in phylogenetics is to reconstruct evolutionary relationships between collections of species, typically represented by a phylogenetic tree. In the statistical approach to phylogenetics, a probabilistic model of mutation is used to reconstruct the tree that best explains the data (the data consisting of DNA sequences from homologous genes of the extant species). In algebraic statistics, we interpret these statistical models of evolution as geometric objects in a high-dimensional probability simplex. This connection arises because the functions that parametrize these models are polynomials, and hence we can consider statistical models as algebraic varieties. The goal of the talk is to introduce this connection and explain how the algebraic perspective leads to new theoretical advances in phylogenetics, and also provides new research directions in algebraic geometry. The talk material will be kept at an introductory level, with background on phylogenetics and algebraic geometry.



04/9/2021: Jonathan Montano, New Mexico State University

Title: Degrees and Multiplicities

Abstract: In how many points does a curve in the plane intersect a random straight line? The answer to this question is an invariant of the curve called "degree". The notion of degree extends to higher dimensional shapes that are defined in terms of polynomials (varieties) and it is a central invariant lying in the interplay between Algebraic Geometry and Commutative Algebra. In this talk I will give an overview of this theory and its generalization to varieties in multi-spaces. In particular, I will report on recent joint work with Castillo, Cid-Ruiz, Li, and Zhang.



04/23/2021: Ilya Shapirovskiy, New Mexico State University

Title: TBA

Abstract: TBA



04/30/2021: Feng Luo, Rutgers University

Title: Discrete Conformal Geometry of Polyhedral Surfaces and its Application

Abstract: Classical theory of Riemann surfaces is a pillar in mathematics and has many applications within and outside of mathematics. There have been many approaches to establish discrete versions of conformal geometry for polyhedral surfaces since the pioneer work of W. Thurston in 1978. In this talk, we will report some of the recent developments in this area. These include a notion of discrete conformality, a discrete uniformization theorem for polyhedral surfaces and their applications. This is a joint work with D. Gu, J. Sun and T. Wu. 



05/07/2021: Adina Oprisan, New Mexico State University

Title: Average and Diffusion Approximation Principle

Abstract: Weak convergence techniques provide paths in analyzing various stochastic approximations of dynamical systems subject to the effect of small random perturbations.
In both average and diffusion approximations, the smallness of the effect of the perturbations is ensured by quick oscillations of the random perturbation process. Limit theorems generalizing  classic types such as: the law of large numbers, the  central limit theorem, and large deviations, are developed for systems perturbed by ergodic Markov and semi-Markov processes.