We will have dinner together from Thai Market in the Cantor Lounge at 6:45. Feel free to come meet others interested in mathematics!

We investigate the following question: if A and A′ are products of finite cyclic groups, when does there exist an isomorphism f:A→A′ which preserves the union of coordinate hyperplanes (equivalently, so that f(x) has some coordinate zero if and only if x has some coordinate zero)? We show that if such an isomorphism exists, then A and A′ have the same cyclic factors; if all cyclic factors have order larger than 2, the map f is diagonal up to permutation, hence sends coordinate hyperplanes to coordinate hyperplanes. Thus one can recover the coordinate hyperplanes from knowledge of their union.
In this talk, I’ll give a brief introduction to tensor (monoidal) categories using the important example of finite-dimensional vector spaces. I’ll show how a diagrammatic calculus known as string diagrams can be used to understand and generalize many constructions from linear algebra such as the tensor product and trace, and how these can be extended to categories like representations and manifolds.

We will discuss how geometric properties of an algebraic variety determine its arithmetic behaviour such as existence and count of rational points. We will talk about some guiding conjectures and results in this direction.

Graphs, which consist of vertices and of edges between them, are well-studied in combinatorics. In this talk, we will discuss some ways in which graphs can be used to study groups. We will focus on the Cayley graph of a group, which is defined using a set of chosen generators of the group, and see how properties of the Cayley graph relate to properties of the group.

The Mathematics department will be having an open house from 6:30-7:30 pm this week in the Cantor lounge:

All Columbia and Barnard current and prospective mathematics majors, joint majors, and concentrators are invited to meet faculty and other students who can answer questions about the Mathematics Department, the courses it offers and the major.  We will outline the major for prospective students, discuss the various joint majors which are currently offered or are in the planning stages, and talk about more advanced courses and graduate schools with the junior and senior majors.

I will talk about my current project, that is concerned with constructing the kind of holomorphic curves that obstruct symplectic embeddings. I will begin with an introduction to the symplectic world, and try to explain how the obstructions work.

In the 1990s, a remarkable correspondence was discovered between the geometry of algebraic curves and infinite-dimensional systems of differential equations. The correspondence has its origin in the study of two-dimensional quantum field theories and is related to many different areas of mathematics. After introducing the relevant objects, I will then state the first result in this story, which was conjectured by Witten and proved by Kontsevich.

 

Carleson's theorem is a fundamental result establishing the almost everywhere convergence of square integrable functions’ Fourier series. In this talk, we'll discuss probabilistic stopping time arguments from a gambler's perspective by utilizing this theorem. I'll give a brief introduction to these ideas in Fourier analysis, their applications, and focus on how this result is the missing piece to the puzzle of the Law of Large Numbers being satisfied for these Fourier sums alongside other probabilistic priniciples.

Directed polymers in random environments (DPRE) have long been an interesting model to study for statistical physicists and mathematicians for their phase transition properties and path behaviours under different environments. In this talk, I will give a brief introduction to this topic and discuss some of the conjectures for these models.

Starting from the elementary question of what is the shortest distance between two given points in the plane, we will move onto more complex minimization problems from which so-called free boundary problems arise. We will discuss in particular a problem arising in 2-dimensional fluid dynamics and try to understand the fundamental questions of existence, uniqueness, and regularity of solutions to such problems.

Many techniques have been adopted in order to prove whether or not a system of homogeneous polynomials (over a number field) admits a non-zero solution. We will focus on invariants whose non-vanishing gives an obstruction to the existence of solutions. In particular we will discuss the local-global principle and extension of coefficients, explaining positive results, counterexamples as well as open problems.

Seniors and other undergraduates will present their senior theses and other research projects. Come to see what research other students are doing! Topics will include mathematical physics and gauge theory, partial differential equations, dimensionality reduction, financial mathematics, monodromy of covers, and a proposal to optimize NYC's trash collection system.