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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.)

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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.

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02/12/2021: 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.

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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 \textit{transcritical (backward and forward), saddle-node, and Hopf} bifurcations, and provide evidence of {\it homoclinic bifurcations} and a    \textit{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 \textit{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.