Date

Speaker

Title & Abstract

Title: Stochastic Differential Equations and other topics>

Abstract: In this talk I will present some of the research I did for my PhD explained in a basic manner. My studies were based on uniqueness of solutions to stochastic differential equations both ordinary and partial differential equations and some elementary result on binary matrices.

Title: Blow-up problem of Stochastic Differential Equations

Abstract:In this talk, I will describe the blow-up problem of Stochastic Differential Equations. The main part of this talk is to introduce the Feller's test of Explosion which provides a precise criteria to determine whether solutions explode with probability zero or positive.

Title: A Ham Sandwich Proof of the Spencer-Szemerdi-Trotter Theorem

Abstract: The classic Ham Sandwich theorem states that three bodies in $\mathbb{R}^3$ of finite volume can be bisected simultaneously by a single plane. Guth and Katz used a generalized version of the Ham Sandwich theorem to create cell decompositions of $\mathbb{R}^n$ with polynomial boundaries. These cell decompositions played a key role in their 2010 proof of a nearly optimal lower bound on the number of distinct distances in the plane. We will use Guth and Katz's theorem to prove the Spencer-Szemerdi-Trotter theorem, which says that $n$ circles and $n$ points in $\mathbb{R}^2$ have at most $O(n^{4/3} )$ incidences.

Title: A concentration of measure result for the Betti numbers of the Rip complex of Erdos-Renyi model.

Abstract: For the classical Erdos-Renyi model, we can consider the "dual" simpli- cial complex and hence its topological invariants in terms of probability. We will quickly identify the inequality of the Betti numbers of the so called Rip complex and move on for some combinatoric results, which is under a stronger condition. Then it is reasonable to make some concentration of measure guess for a weaker condition using the Azuma's theorem. The proofs are really elementary even we have a longer title.

Title: Representations of a free abelian group.

Abstract: We look at the space of representations of a free abelian group of finite rank. This is the same as studying the space of commuting elements in a Lie group $G$. We will consider some specific cases from the topological point of view. These spaces have been studied intensively in the last years and they have connections to quantum field theories such as the Chern-Simons theory.