Representation stability is an exciting new area that combines ideas from commutative algebra and representation theory. The meta-idea is to combine a sequence of objects together using some newly defined algebraic structure, and then to translate abstract properties about this structure to concrete properties about the original object of study. Finite generation is a particularly important property, which translates to the existence of bounds on algebraic invariants, or some predictable behavior. I'll discuss some examples coming from combinatorial representation theory (Kronecker coefficients) and topology (configuration spaces).
Hilbert's 19th problem asks if minimizers of "natural" variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDEs. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, as well as outstanding open problems. Parts of this are joint works with A. Figalli and O. Savin.
Markov duality in spin chains and exclusion processes has found a wide variety of applications throughout probability theory. We review the duality of the asymmetric simple exclusion process (ASEP) and its underlying algebraic symmetry. We then explain how the algebraic structure leads to a wide generalization of models with duality, such as higher spin exclusion processes, zero range processes, stochastic vertex models, and their multi-species analogues.
This talk will illustrate some topological properties of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of the configuration spaces F_k(M) to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which these configuration spaces stabilize. In this talk I will explain these stability patterns, and describe a higher-order “secondary representation stability” phenomenon among the unstable homology classes. These results may be viewed as a representation-theoretic analogue of current work of Galatius–Kupers–Randal-Williams. The project is joint with Jeremy Miller.
A "critical" metric on a manifold is a metric which is critical for some natural geometric variational problem. Some important examples of critical metrics are Einstein metrics and extremal Kahler metrics, and such metrics typically come in families. I will discuss some aspects of the local theory of moduli spaces of critical metrics, and present some compactness results for critical metrics which say that, under certain geometric assumptions, a sequence of critical metrics has a subsequence which converges, in the Gromov-Hausdorff sense, to a singular space with orbifold singularities. I will also discuss some results regarding the reverse problem of desingularizing critical orbifolds to produce new examples of critical metrics on smooth manifolds.
Many real-world networks -- social, technological, biological -- have wonderful structures. Some structures may be apparent (such as trees) while others may be hidden (such as communities). How can we discover hidden structures? Known approaches to "structure mining" in networks come from a variety of disciplines, including probability, statistics, combinatorics, physics, optimization, theoretical computer science, signal processing and information theory. We will focus on new probabilistic approaches to structure mining. They bring together insights from random matrices, random graphs and semidefinite programming.
This is a joint applied math and probability seminar.
I will highlight some of the methods I've previously employed in undergraduate mathematics education. These methods were drawn from the research of others and some of my own inquires. I will give insight into how the methods are brought together so that they form a cohesive whole, while fitting the needs of the both the students and the department.
Central to the discussion is a particular enlightening experience I had with one of my students.
The experience has inspired my current methods. I will discuss this my vision for further developing curriculum and more so fostering a lively dynamic atmosphere around mathematics courses, both within and without the classroom walls.
Time permitting, I'll discuss some research that is underway with regards to these approaches.
We will discuss my experience and plans for teaching mathematics to students with increasing dependence on the internet. For example, we will discuss my use of online, hyperlinked lecture notes, the role of math.stackexchange.com and other websites for writing homeworks and exams, etc. Some new course proposals will be given, including an increased role of the math department for the UCI Data Science Initiative.
In this talk, I will consider quasilinear parabolic PDEs subject to stochastic or rough perturbation and explain how various assumptions on coefficients and roughness of the noise naturally ask for different notions of solution with different regularity properties and different techniques of the proofs. On the one hand, the problems under consideration will be stochastic second order parabolic PDEs with noise smooth in space, either with a possible degeneracy in the leading order operator, where only low regularity holds true, or under the uniform ellipticity assumption, where arbitrarily high regularity can be proved under suitable assumptions on the coefficients. On the other hand, I will discuss a rough pathwise approach towards these problems based on tools from paracontrolled calculus.