In this talk, we present some work from data science teams at Google to give
students an impression of how statistics and experimental design are used in
practice. Most examples will be drawn from the Search Ads Data Science team, but
many of them are representative of techniques used all around the tech industry.
One point of emphasis is that in data science techniques for creating meaningful
data sets can be as or more important than the statistical techniques that are
afterwards applied to them. Often doing a good job both in data generation and
evaluation are needed to answer interesting research questions. The talk is aimed
at graduate and undergraduate students with an interest in applied statistics.

We show that the collection of groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions is invariant under measure equivalence. In the case of orbit equivalence, the proof is mainly an application of a cocycle untwisting lemma of A. Furman and S. Popa, along with the fact that any orbit equivalence of free p.m.p. actions can be lifted to an orbit equivalence of the corresponding Bernoulli extensions of those actions.

Spherical Designs are finite sets of points
on the sphere with the property that the average of
low-degree polynomials over the sphere coincides
with the average over the finite set. These objects
are very beautiful, very symmetric and have been
studied since the 1970s. We use a completely
new approach that replaces delicate combinatorial
arguments with the a simple application of the heat
equation; this approach improves the known results
and extends to other manifolds. We also discuss
some related issues in Combinatorics, Irregularities
of Distribution and Fourier Analysis.

The distribution of descents in certain conjugacy classes of $S_n$ have been previously studied, and it is shown that its moments have interesting properties. This talk provides a bijective proof of the symmetry of the descents and major indices of matchings and uses a generating function approach to prove a central limit theorem for the number of descents in matchings. We also extend this result to derangements.

Given a Cayley graph of a finitely generated group, one can consider its growth function which counts how many elements are there in a ball of radius n on the graph. We will discuss two seminal results in the subject of growth of groups proved in early 1980s: Gromov’s polynomial growth theorem and Grigorchuk’s construction of groups of intermediate growth. We will illustrate how random walks on the Cayley graphs can help to study growth.

After a short introduction to topics in nonparametric curve estimation, covered in my new 2018
Chapman & Hall’s book with the same title as the talk, three specific problems will be considered.
The first one is nonparametric regression with missing at random (MAR) responses. It will be explained that a complete case approach is optimal in this case. The second problem is nonparametric regression with missing at random (MAR) predictors. It will be explained that in general a complete case approach is inconsistent for this type of missing and a special procedure is needed for efficient estimation. The last explored problem is devoted to estimation of hazard rate functions for truncated and censored data.

I will discuss a model of nonintersecting Brownian bridges on the unit circle, which produces quite a few universal determinantal processes as scaling limits. I will focus on the tacnode process, in which two groups of particles meet at a single point in space-time before separating, and introduce a new version of the tacnode process in which a finite number of particles ``switch sides'' before the two groups separate. We call this new process the k-tacnode process, and it is defined by a kernel expressed in terms of a system of tau-functions for the Painleve II equation. Technically, our model of nonintersecting Brownian bridges on the unit circle is studied using a system of discrete orthogonal polynomials with a complex (non-Hermitian) weight, so I'll also discuss some of the analytical obstacles to that analysis.

\noindent This is joint work with Dong Wang and Robert Buckingham

This talk will focus on my works on mean-field models for chemotaxis based on kinetic theory, including pathway based mean-field models, augmented Keller-Segel model for E. coli chemotaxis, and an asymmetric model for biological aggregation. I will give mathematical derivation of the mean-field models by taking some proper moment closure of kinetic biological systems. Building biological mechanism in the models are essential to capture some interesting swarming phenomena, for example, phase-delayed traveling wave (memory effect) and soliton solution (asymmetric sensing). Connections to the chemotaxis model proposed in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101] will be also discussed.

The Navier-Stokes regularity problem is well-known, but there are many related simpler equations, including equations in 1d, for which our knowledge is surprisingly incomplete. Several of them have been recently studied, and in this talk we will discuss some of the results, together with connections to the full problem.