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Thu Sep 10

Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Kinetic theory of structured populations: demographics, cell size control, and stochastic hierarchies
Tom Chou, University of California, Los Angeles

We will briefly review, through two examples, classic deterministic PDE models of population dynamics structured according to attributes such as age and/or size. First, we describe how the original McKendrick model was used to motivate China's one-child policy, and generalize it to include an imposed, finite interbirth refractory period. We quantify the effectiveness of this softer, staggered birth policy and discuss its predicted effectiveness. We then review sizer-timer-adder-type models used to quantify proliferating cell populations. Here, blow-up in mean cell sizes can arise, which represents a challenging numerical problem. Finally, we extend these classic deterministic models to allow for both demographic and growth-rate stochasticity by developing a fully kinetic theory. Marginalization of the full density functions results in a set of coupled kinetic models similar to the BBGKY hierarchy. We map out the different combinations of stochastic descriptions and show how the classic age-dependent population models are connected to this hierarchy, the lowest order of which is a master equation for the total stochastic population. Differences in the stochastic description of birth through budding or splitting are explored.

Thu Sep 17

Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
25 years since Fermat's Last Theorem
Frank Calegari, University of Chicago

Wiles's proof of Fermat's Last Theorem was published 25 years ago. Wiles's paper introduced many new ideas and methods which have since shaped the field of algebraic number theory. This colloquium talk intends to give a (biased) tour of these developments, especially with regard to questions that might be of interest to non-specialists.

Thu Sep 24

Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Random walks in graph-based learning
Jeff Calder, University of Minnesota, Twin Cities

I will discuss several applications of random walks to graph-based learning, both for theoretical analysis and algorithm development. Graph-based learning is a field within machine learning that uses similarities between datapoints to create efficient representations of high-dimensional data for tasks like semi-supervised classification, clustering and dimension reduction. Our first application will be to use the random walk interpretation of the graph Laplacian to characterize the lowest label rate at which Laplacian regularized semi-supervised learning is well-posed. Second, we will show how analysis via random walks leads to a new algorithm that we call Poisson learning for semi-supervised learning at very low label rates. Finally, we will show how stochastic coupling of random walks can be used to prove Lipschitz estimates for graph Laplacian eigenfunctions on random geometric graphs, leading to new spectral convergence results.

This talk will cover joint work with many people, including Brendan Cook (UMN), Nicolas-Garcia Trillos (Wisconsin-Madison), Marta Lewicka (Pittsburgh), Dejan Slepcev (CMU), Matthew Thorpe (University Manchester).

Thu Oct 01

Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Gradient variational problems
Richard Kenyon, Yale University

This is joint work with Istvan Prause. Many well-known random tiling models such as domino tilings and square ice lead to variational problems for functions h:R^2->R which minimize a functional depending only on the gradient of h. Other examples of such variational problems include minimal surfaces and surfaces satisfying the "p-laplacian". We give a representation of solutions of such a problem in terms of kappa-harmonic functions: functions which are harmonic for a laplacian with a varying conductance kappa.

Thu Oct 08

Colloquium

3:30pm - Zoom ID 91514486597 (contact faculty for pw)
Mean-field disordered systems and Hamilton-Jacobi equations
Jean-Christophe Mourrat, Courant Institute of Mathematical Sciences, New York University

The goal of statistical mechanics is to describe the large-scale behavior of collections of simple elements, often called spins, that interact through locally simple rules and are influenced by some amount of noise. A celebrated model in this class is the Ising model, where spins can take the values +1 and -1, and the local interaction favors the alignement of the spins.

In this talk, I will mostly focus on the situation where the interactions are themselves disordered, with some pairs having a preference for alignement, and some for anti-alignement. These models, often called "spin glasses", are already surprisingly difficult to analyze when all spins directly interact with each other. I will describe a fundamental result of the theory called the Parisi formula. I will then explain how this result can be recast using suitable Hamilton-Jacobi equations, and what benefits this new point of view may bring to the topic.

Thu Oct 15

Colloquium

3:30pm - Via Zoom
Colloquium
Daniel Ruberman, Brandeis University
Thu Oct 22

Colloquium

3:30pm - Via Zoom
Colloquium
Edward Frenkel, University of California, Berkeley
Thu Oct 29

Colloquium

3:30pm - Via Zoom
Colloquium
Zhiwei Yun, Massachusetts Institute of Technology
Thu Nov 05

Colloquium

3:30pm - Via Zoom
Colloquium
David Gamarnik, Massachusetts Institute of Technology
Thu Nov 12

Colloquium

3:30pm - Via Zoom
Colloquium
Alexander Kiselev, Duke University
Thu Nov 19

Colloquium

10:00am - Via Zoom
Colloquium
Ana Caraiani, Imperial College London
Tue Dec 01

Colloquium

3:30pm - Via Zoom - Time TBA
Colloquium
Akshay Venkatesh, Institute for Advanced Study
Thu Dec 03

Colloquium

3:30pm - Via Zoom
Colloquium
Vlad Vicol, Courant Institute of Mathematical Sciences, New York University
Thu Dec 10

Colloquium

3:30pm - Via Zoom
Colloquium
Inna Zakharevich, Cornell University
Thu Jan 28

Colloquium

3:30pm - Via Zoom
Colloquium
Wieslawa Niziol, Sorbonne Universite'
Thu Feb 04

Colloquium

3:30pm - Via Zoom
Colloquium
Matthew Baker, Georgia Institute of Technology
Thu Feb 11

Colloquium

3:30pm - Via Zoom
Colloquium
Sijue Wu, University of Michigan, Ann Arbor
Thu Feb 18

Colloquium

5:00pm - Via Zoom
Colloquium
Takeshi Saito, University of Tokyo
Thu Feb 25

Colloquium

3:30pm - Via Zoom
Colloquium
Thomas Lam, University of Michigan, Ann Arbor
Thu Mar 04

Colloquium

3:30pm - Via Zoom
Colloquium
Xinwen Zhu, California Institute of Technology
Thu Mar 25

Colloquium

10:00am - Via Zoom (note special time)
Colloquium
Carola-Bibiane Schönlieb, Cambridge University
Thu Apr 01

Colloquium

3:30pm - Via Zoom
Colloquium
David Ben-Zvi, University of Texas, Austin
Thu Apr 08

Colloquium

3:30pm - Via Zoom
Colloquium
Kavita Ramanan, Brown University
Thu Apr 29

Colloquium

3:30pm - Via Zoom
Colloquium
Tatiana Toro, University of Washington