The covering volume of lattices, and nearly uniform coverings (1/3)

Dear all,

Professor Barak Weiss from Tel Aviv University will give a series of three
lectures. The details are below.

                          Time and Venue

Lecture 1: 10:00 am to 11:00 am October 18, AG–66
Lecture 2:  4:00 pm to  5:00 pm October 19, AG–69
Lecture 3:  4:00 pm to  5:00 pm October 21, AG–77


Title: The covering volume of lattices, and nearly uniform coverings

Abstract: Let L in R^n be a lattice and let K be a convex body. The
covering volume of K w.r.t. L is the minimal volume of a dilate rK such
that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small
covering volume correspond to efficient coverings of space by copies of K,
translated by elements of L. Finding upper bounds on the covering volume
as the dimension n goes to infinity, is a well-studied problem, with
connections to practical issues arising in computer science and electrical
engineering. In a recent paper with Ordentlich (EE, Hebrew U) and Regev
(CS, NYU), we obtain substantial improvements to bounds of Rogers from the
1950’s. In another recent paper, we obtain bounds on the minimal volume of
nearly uniform covers, where a pair (L, K) give an epsilon-nearly uniform
cover if the ratio between max_x |{l in L : x in l+K}| and min_y |{l in L
: y in l+K}| is at most 1+epsilon. The key to these results are recent
breakthroughs due to Dvir and others regarding the discrete Kakeya
problem. I will give three lectures about these results, including
history, applications, and some ideas of the proofs. No prerequisites
beyond undergraduate material (measures, volumes, vector spaces over
finite fields) will be assumed.

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YouTube live link:

18-10-2022 --> https://youtu.be/029xTLyiNbw

19-10-2022 --> https://youtu.be/jW3ZTdpQPl4

21-10-2022 --> https://youtu.be/iwQAOaM5Mbw

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