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.
 

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.
 

Teaching is a learning process, with both ups and downs. In this talk, I will discuss some of the trends I have observed as I progressed from being a student to being an instructor. These include the effects of technology on Mathematics education, approaches to teaching Mathematics, etc. I will also discuss some of the lessons I have learned along the way.

I will describe some recent work on the incompressible Euler equations and related partial differential equations specifically related to "Critical Phenomena". It is, by now, known that the incompressible Euler equation is ill-posed in most "critical classes" such as the class of Lipschitz continuous or C^1 velocity fields (even when the data is taken to be smooth away a single point). Despite this, we prove well-posedness (global in 2d and local in 3d) for merely Lipschitz data which is smooth away from the origin and satisfies a mild symmetry assumption. To do this requires a deep understanding of the nature of unboundedness of singular integrals on $L^\infty$. Through this understanding, we define a well-adapted class of critical function spaces in which we prove well-posedness. After this, we extract a simplified equation which is satisfied by "scale invariant" solutions which lie within the setting of our local/global well-posedness theory. These scale-invariant solutions, in the 2d Euler setting, can be shown to have very interesting dynamical properties such as time-quasiperiodic behavior. Moreover, these scale-invariant solutions (while having infinite energy) can be used to prove the existence of finite-energy solutions with the "same" dynamical properties.This is joint work with In-Jee Jeong.