Jiange Li – Seminar
Friday, September 16, 2016, 2 p.m. , 36-156
The Network Coding and Reliable Communications Group Seminar
DATE: Friday 09/16/16 TIME:2:00 pm
University of Delaware
THREE TOPICS INTERTWINING PROBABILITY, COMBINATORICS AND INFORMATION THEORY.
I will introduce three questions and present some answers.
The first investigation, which I will spend the most time on, focuses on a theorem of Shannon-McMillan-Breiman type for a large class of finite-dimensional densities called s-concave densities. For such densities, not only can we dispense with stationarity and ergodicity requirements that are usually imposed, but we can obtain sharp results about rates that are sharp even in the constants. Along the way, we develop a sharp varentropy bound for such densities. The techniques are nontrivial and come from functional analysis and convex geometry. This part is joint work with S. Bobkov, M. Fradelizi and
The second investigation focuses on comparing the entropies of the sum and the difference of i.i.d. random variables taking values in a cyclic group. Connections are made to more-sums-than-differences questions in additive combinatorics, and applications are given to the design of polar codes. This part is joint work with E. Abbe and M. Madiman.
The third investigation, which we will only briefly outline, focuses on small ball inequalities for sums and differences, which we show are closely connected to extremal combinatorics.
A unifying theme of my talk will be the interconnections between different areas of mathematics, and the role of informa- tion theory and entropy ideas in all of them.
Jiange Li received his M.S. and Ph.D. degree in mathematics from University of Delaware in 2011 and 2016, respec- tively, and his B.S. degree in applied mathematics from Harbin Institute of Technology, China, in 2009. Dr. Li spent a semester as a long-term visitor at the Institute for Mathematics and its Applications, Minneapolis. Dr. Li’s research is primarily in the area of probability, but also interacts with information theory, combinatorics, and convex geometry. He was awarded the inaugural Wenbo Li Scholarship Prize for Graduate Research in 2015, given for outstanding re- search in the mathematical sciences at the University of Delaware.