Week of February 17, 2019

Tue Feb 19, 2019
11:00am to 11:50am - 306 RH - Probability
Roman Vershynin - (UCI)
Hashing

Hashing is a technique widely used in coding theory (an area of computer science) and in cryptography. Although hashing is an interesting mathematical object, it is surprisingly little known to the "mainstream" mathematicians. I will focus on one specific result on hasing, namely the Leftover Hash Lemma. We will state it as a result in extremal combinatorics, give a probabilistic proof of it, and relate it to another fundamental result in extremal combinatorics, the Sauer-Shelah Lemma.

4:00pm - RH 306 - Differential Geometry
Man-Chun Lee - (UBC)
Hermitian manifolds with non-positive curvature

A recent breakthrough of Wu and Yau asserts that a compact projective Kahler 
manifold with negative holomorphic sectional curvature must have ample 
canonical line bundle. In the talk, we will talk about some of the recent 
advances along this direction. In particular, we will discuss the case 
where the manifold is a noncompact Kahler manifold. We will also discuss 
the case when the Kahlerity is a priori unknown. Part of these are joint 
work with S. Huang, L.-F. Tam, F. Tong.

Thu Feb 21, 2019
12:00pm to 1:00pm - RH 340N - Probability Preprint Seminar
Mike Cranston - (UCI)
Stochastic localization II

Stochastic localization is a powerful probabilistic tool invented recently by Ronen Eldan. It has found several applications to challenging problems in probability and geometry, see e.g. https://arxiv.org/abs/1612.01507

4:00pm to 5:00pm - RH 306 - Colloquium
Lars Hesselholt - (Nagoya University and Copenhagen University)
Higher algebra and arithmetic

This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.

Fri Feb 22, 2019
11:00am to 11:50am - PSCB 220 - Cryptography
Travis Scholl - (University of California, Irvine)
Altug-Chen Candidate Group with Infeasible Inversion

A group where finding inverses is hard can be used for many cryptographic constructions. In this talk we will discuss a candidate construction by Salim Ali Altug and Yilei Chen of a group where finding inverses is supposed to be difficult, see https://eprint.iacr.org/2018/926. Their construction works by representing the class group of a quadratic imaginary field as an isogeny graph of an elliptic curve over the ring Z/NZ where N is a product of two primes. After going over the basic constructions, we will discuss some open problems relating to the security of the construction.

3:00pm to 4:00pm - RH 440R - Nonlinear PDEs
Chenchen Mu - (UCLA)
Mean field games on graphs

Mean field game theory is the study of the limit of Nash
equilibria of differential games when the number of players tends to infinity. It
was introduced by J.-M. Lasry and P.-L. Lions, and independently by P.
Caines, M. Huang and R. Malhame. A fundamental object in the theory is the
master equation, which fully characterizes the limit equilibrium. In this
talk, we will introduce Mean field game and master equations on graphs. We will
construct solutions to both equations and link them to the solution to
a Hamilton-Jacobi equation on graph.

4:00pm - PSCB 140 - Graduate Seminar
Li-Sheng Tseng - (UC Irvine)
What is cohomology?