Abstract:

In this talk, I will sketch a surprising proof due to Gyorgy Elekes on a non-trivial lower bound for the sums-versus-product problem in combinatorial number theory.

  Abstract:

In this talk, I'll consider a type of "perfect-information" game where two players take turns making moves, each hoping to achieve some goal first. Before playing the game, how can we tell who should have the winning strategy? One answer simply enumerates all possible ways the game could be played: If Player 1 makes some move, what are all the possible ways Player 2 could respond? What are all the ways that Player 1 could respond to Player 2's move? and so on. This is a bit mathematically unsatisfying and also inefficient. I will talk about techniques to analyze certain classes of games, which will sometimes lead to efficient algorithms for deciding which player should win if both are playing optimally. In particular, I will focus on games where both players have the same set of options for each move.

  Abstract:

I will explain the Hausdorff-Banach-Tarski paradox: one can decompose a sphere into a finite number of pieces which can be reassembled into two spheres of the same size as the original.

  Abstract:

Can every positive integer be written as the sum of 4 squares? In how many ways can one do this? These simple arithmetic questions turn out to be a vista into some very interesting mathematics. I will explain a classic argument that answers the above questions using modular forms. Modular forms are certain complex analytic functions that have many symmetries. Modular forms, and their generalizations, are at the heart of much of current number theory.

  Abstract:

The basic question of enumerative geometry can be simply stated as: How many geometric objects of a given type satisfy given geometric conditions? For instance, one may ask for the number of lines through 2 points in the plane, or the number of conics through 5 points in the plane. The purpose of this talk is to give an introduction to counting problems in algebraic geometry.

  Abstract:

I will discuss the isoperimetric problem of determining a plane figure of the largest possible area whose boundary has a specified length.

  Abstract:

Can you find infinite groups with slow volume growth but not polynomial? This was asked by Milnor back in 1960s. It turns out to be quite difficult to construct such groups. I will explain a few examples and some reasons why we still don't understand these groups well.

  Abstract:

We will discuss three problems with number theoretic and geometric nature and how tools from dynamics of group actions can be used to study these problems.

  Abstract:

Suppose we color each natural number either red or blue. Can you find 100 natural numbers all equally spaced (i.e., an arithmetic progression) that all have the same color? If so, how far do you have to look?* I'll explain how to solve this question, and talk a bit about the state of the art in arithmetic Ramsey theory.

* Spoiler: the answer is "probably further than you think".

  Abstract:

The heat equation, also known as the classical diffusion equation, describes the spreading of a quantity like heat within a given region. In a broader sense, a diffusion process can be understood as a process where the variable under consideration tries to return to its surrounding average. This discussion will introduce nonstandard diffusion processes that have found widespread applications in recent years.