Millions of Korean workers follow night or nonstandard shifts. A similar number of Koreans travel overseas each year. This leads to misalignment of the daily (circadian) clock in individuals: a clock that controls sleep, performance, and nearly every physiological process in our body. A mathematical model of this clock can be used to optimize schedules to maximize productivity and minimize jetlag. We simulate this model in a smartphone app, ENTRAIN (www.entrain.org), which has been installed in phones over 200,000 times in over 100 countries. I will discuss techniques we have been developing and clinically testing to determine sleep stage and circadian time from wearable data, for example, as collected by our app or used in many commercially available sleep trackers. This project has led us to develop new techniques to: 1) estimate phase from noisy data with gaps, 2) rapidly simulate of population densities from high dimensional models and 3) determine how mathematical models of sleep and circadian physiology can be used with machine learning techniques to improve predictions.

(This is a reading seminar for graduate students.)

Algebraic K-theory originated with the Grothendieck-Riemann-Roch theorem, a generalization of Riemann-Roch theorem to higher dimensional varieties. For this, we shall discuss the definitions of $K_0$-theory of a variety, its connection with intersection theory, $lambda$-operation, $gamma$-filtration, Chern classes and Adams operations.

I report on work with M. Gubinelli and T. Oh on the
renormalized nonlinear wave equation
in 2d with monomial nonlinearity and in 3d with quadratic nonlinearity.
Martin Hairer has developed an efficient machinery to handle elliptic
and parabolic problems with additive white noise, and many local
existence questions are by now well understood. In contrast not much is
known for hyperbolic equations. We study the simplest nontrivial
examples and prove local existence and weak universality, i.e. the
nonlinear wave equations with additive white noise occur as scaling
limits of wave equations with more regular noise.

It remains  to bound the terms on the expansion of the logarithm of the
transmission coefficient.

I report on joint work with Mihaela Ifrim, Xian Liao and Daniel Tataru.

A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph $G$ does not contain $K_{2,2}$ as an induced subgraph yet has at least $c(\frac{n}{2})$edges, then $G$ has a complete subgraph on at least $\frac{c^2}{10}n$vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced $K_{2,2}$, which allows us to extend their result to $k$-uniform hypergraphs. Our result also has interesting consequences in topological combinatorics and abstract convexity, where it can be used to answer questions by Bukh, Kalai, and several others.

The Gross-Pitaevskii equation is essentially the defocusing nonlinear
Schrödinger equation with a nonzero boundary condition at infinity. This
makes the phase space very nonlinear and we will need a good metric and
smooth structure on it.