Georgia Institute of Technology College of Sciences

The rotor-router model, also known as the Propp machine, is a deterministic process analogous to a simple random walk on a graph. In this talk, we are concerned with a generalized model, functional-router model, which imitates a Markov chain possibly containing irrational transition probabilities. We investigate the discrepancy of the number of tokens between the functional-router model and its corresponding Markov chain, and give some upper bounds in terms of the mixing time of the Markov chain.

We consider the problem of how to control the measures of false scientific discoveries in high-dimensional models. Towards this goal, we focus on the uncertainty assessment for low dimensional components in high-dimensional models. Specifically, we propose a novel decorrelated likelihood based framework to obtain valid p-values for generic penalized M-estimators. Unlike most existing inferential methods which are tailored for individual models, our method provides a general framework for high-dimensional inference and is applicable to a wide variety of applications, including generalized linear models, graphical models, classifications and survival analysis. The proposed method provides optimal tests and confidence intervals. The extensions to general estimating equations are discussed. Finally, we show that the p-values can be combined to control the false discovery rate in multiple hypothesis testing.

We study the construction of exponential bases and exponential frames on general $L^2$ space with the measures supported on self-affine fractals. This problem dates back to the conjecture of Fuglede. It lies at the interface between analysis, geometry and number theory and it relates to translational tilings. In this talk, we give an introduction to this topic, and report on some of the recent advances. In particular, the possibility of constructing exponential frames on fractal measures without exponential bases will be discussed.

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of $K_5$. This conjecture was proved by Ma and Yu for graphs containing $K_4^-$. Recently, we proved this entire Kelmans-Seymour conjecture. In this talk, I will give a sketch of our proof, and discuss related problems. This is joint work with Dawei He and Xingxing Yu.

Hybrid particle/grid numerical methods have been around for a long time, andtheir usage is common in some fields, from plasma physics to artist-directedfluids. I will explore the use of hybrid methods to simulate many differentcomplex phenomena occurring all around you, from wine to shaving foam and fromsand to the snow in Disney's Frozen. I will also talk about some of thepractical advantages and disadvantages of hybrid methods and how one of theweaknesses that has long plagued them can now be fixed.

Detecting change-points from high-dimensional streaming data is a fundamental problem that arises in many big-data applications such as video processing, sensor networks, and social networks. Challenges herein include developing algorithms that have low computational complexity and good statistical power, that can exploit structures to detecting weak signals, and that can provide reliable results over larger classes of data distributions. I will present two aspects of our recent work that tackle these challenges: (1) developing kernel-based methods based on nonparametric statistics; and (2) using sketching of high-dimensional data vectors to reduce data dimensionality. We also provide theoretical performance bounds and demonstrate the performance of the algorithms using simulated and real data.

We show that there exists an absolute constant c>0 with the following property. Let A be a set in a finite field with q elements. If |A|>q^{2/3-c}, then the set (A-A)(A-A) consisting of products of pairwise differences of elements of A contains at least q/2 elements. It appears that this is the first instance in the literature where such a conclusion is reached for such type sum-product-in-finite-fileds questions for sets of smaller cardinality than q^{2/3}. Similar questions have been investigated by Hart-Iosevich-Solymosi and Balog.