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To celebrate PIMS 25th Anniversary we have convened a special Network Wide Colloquium featuring distinguished speakers with mathematical connections throughout the PIMS network. Join us as we continue this fall with a lecture from Rafe Mazzeo, Professor at Stanford University. Registration is open.

Z_2 Harmonic Spinors in Gauge Theory Rafe Mazzeo, Stanford University

Gauge-theoretic moduli spaces are often noncompact, and various techniques have been introduced to study their asymptotic features. Seminal work by Taubes shows that in many situations where the failure of compactness for sequences of solutions is due to the noncompactness of the gauge group, diverging sequences of solutions lead to what he called Z_2 harmonic spinors. These are multivalued solutions of a twisted Dirac equation which are branched along a codimension two subset. This leads to a number of new problems related to these Z_2 harmonic spinors as interesting geometric objects in their own right. I will survey this subject and talk about some recent work in progress with Haydys and Takahashi to compute the index of the associated deformation problem.

Speaker Biography: Rafe Mazzeo is an expert in PDEs and Microlocal analysis. He did his Ph.D. at MIT and was then appointed as Szegő Assistant Professor at Stanford University, where he is now Professor and Chair of the Department of Mathematics. He has served the mathematical community in many important ways, including as Director of the Park City Mathematics Institute.

Register here.

Recent progress in model theory is changing our understanding of how the finite and infinite interact. One aspect of this story has to do with the emerging appearance of complexity in so-called simple unstable theories, which provide a model theoretic framework for studying certain families of "random" objects, such as the theories of random graphs or hypergraphs or pseudofinite fields.

Speaker Biography: Maryanthe Malliaris received her Ph.D. from the University of California at Berkeley in model theory, and is currently Professor of Mathematics at the University of Chicago. She was a winner of the 2010 Kurt Gödel Research Prize Fellowship, and in 2017 was awarded the Hausdorff medal for joint work with Saharon Shelah. Malliaris was an invited speaker at the 2018 International Congress of Mathematicians.

Abstract: In this talk I will discuss activated random walk (ARW) and the stochastic sandpile (SSM), two interacting particle systems that exhibit a phase transition on infinite domains and self-organized criticality on finite domains. ARW and SSM both consist of identical particles that perform simple random walk, initially with average density μ particles per site, but they follow different local interaction rules. Depending on the value of μ, particles may eventually stop moving, or remain active forever. Current research is focused on determining where the transition between these two phenomena occurs, and many questions still remain – even on Z. I will present some recent results and discuss the novel tools involved.

3:00 pm Coffee Meet and Greet
3:30 pm Lecture

Abstract: Recursive distributional equations (RDEs) are ubiquitous in probability. For example, the standard Gaussian distribution can be characterized as the unique fixed point of the following RDE

X = (X_1 + X_2)/sqrt(2),

among the class of centered random variables with standard deviation of 1. (The equality in the equation is in distribution; the random variables X,X_1 and X_2 must all be identically distributed; and X_1 and X_2 must be independent.)

Recently, it has been discovered that the dynamics of certain recursive distributional equations can be solved using by using tools from numerical analysis, on the convergence of approximation schemes for PDEs. In particular, the framework for studying stability and convergence for viscosity solutions of nonlinear second order equations, due to Crandall-Lions, Barles-Souganidis, and others, can be used to prove distributional convergence for certain families of RDEs, which can be interpreted as tree-valued stochastic processes. I will survey some of these results, as well as the (current) limitations of the method, and our hope for further interplay between these two research areas.

Based on joint work with Erin Beckman, Hannah Cairns, Luc Devroye, Celine Kerriou, Jessica Lin, and Rivka Maclaine Mitchell.​

Download poster (PDF)

For more information and to register see https://www.pims.math.ca/scientific-event/210929-tppfsec.

For more information and to register see https://www.pims.math.ca/scientific-event/210923-pnwcan.

For more information and to register see https://www.pims.math.ca/scientific-event/210915-tppfshh.