More than 200 years ago, the Gaussian distribution was discovered by De Moivre (and again by Laplace) from analyzing the binomial distribution arising from coin flips. This distribution is the basis for classical statistics and arises is many different types of physical and mathematical systems. There are, however, many systems for which classical statistics fails and new distributions arise. These distributions are also universal within certain classes of systems. In this talk we will begin to describe where these universal probability distributions come from, what they describe and why they are important. We will touch on many applications such as big data, random growth, traffic flow and more.

The technique of solving the d-bar equation and using Hormander's L2 estimates has been used to obtain many interesting results in complex and algebraic geometry. In this talk, we will solve the one dimensional d-bar equation for a domain in the complex plane.

In linear algebra, one is tasked with the problem of diagonalizing an n x n matrix, or a linear transformation from a finite-dimensional vector space to itself. Diagonalizing a linear transformation on an infinite-dimensional vector space is far more involved, and a spectral theorem is a result about when can a linear transformation be diagonalized. In this talk, I will state and prove the spectral theorem for compact, self-adjoint operators on a Hilbert space. Familiarity with linear algebra and modern analysis is assumed.

Most of the major open problems in topology involve understanding how many smooth manifolds there are of a given topological type. This problem is particularly difficult in dimension 4, the only dimension in which the smooth Poincare conjecture remains unsolved. I will discuss several of these problems, as well as the role of invariants in shedding light on them.

In the absence of coordinates, how does one define the dimension of a topological space? In this talk, we'll give a leisurely introduction to dimension theory, which attempts to answer this question. After motivating the field with several examples, we'll give three possible definitions of topological definition. We'll then use one of these to prove that R^n is homeomorphic to R^m iff n = m.

Since Gromov's groundbreaking non-squeezing theorem, a central question in symplectic geometry is when can a subset of even dimensional Euclidean space be embedded into another symplectically. I will explain what this means and talk about work by McDuff and Schlenk, who showed a remarkable relation between embeddings of 4-dimensional ellipsoids into 4-dimensional balls and Fibonacci numbers. I will not assume any knowledge of symplectic geometry, but familiarity with Fibonacci numbers will be useful.

You may have seen this problem before: can a checkerboard with two opposite corners removed be tiled with dominoes?  The answer is no, and one can prove that no tiling exists by counting the number of light and dark squares on the board.  I will explain how some similar tiling problems can be solved by "counting" with nonabelian groups instead of the integers.

I will state and prove the Combinatorial Nullstellensatz, and show some of the very diverse areas in which it can be applied, including number theory, graph theory, as well as contest problems. 

Geometric flows and energy estimates have been used to obtain many results in the theory of PDE and geometry. In this talk, we will survey some of the aforementioned techniques and explore the interplay between geometry and analysis.

In this talk, we will introduce the formalism of category theory. The emphasis will be on explaining a basic principle of categories that illuminates one of the reasons they are useful and interesting. To this end, we will discuss two objects: topological spaces and algebraic varieties over a field k. These will naturally define categories, we will show how the formalism of category theory allows us to take properties that a priori will only defined for one of these objects and get a corresponding notion  it in the other case. In particular, we will see how the notions of the Hausdorf and proper can be generalized to the context of algebraic varieties. Time permitting an advanced topic might be briefly discussed: the K-Theory of a field. Knowledge of Point-Set Topology and Ring Theory is necessary.

Homological algebra has many applications in Topology, Algebraic Geometry, and Abstract Algebra, as well as being interesting in its own right. One of the most helpful techniques in certain situations is a Spectral Sequence. I will introduce spectral sequences and apply them to prove some basic results in homological algebra such as the Snake Lemma and the Five lemma. If there is still time, I will explain how we can apply spectral sequences to proving the equivalence of cellular and singular homology.

We set up the basics for doing algebraic geometry over finite fields. We give an elementary proof of the Chevalley–Warning theorem, which states that low degree polynomials over a finite field always have solutions. We end with a wild set of conjectures (the Weil conjectures, proven in the 70s by Deligne et al.) relating counting formulas with some homological information. No prior knowledge of algebraic geometry or algebraic topology is assumed. Some prior exposure to ring theory is useful, but not required.