Summer Minicourses 2022

Abstract. Based on a portion of the following PhD thesis: https://arxiv.org/pdf/2112.13472.pdf.

Abstract. The aim of this mini-course is to serve as a brief survey and introduction to some classical ergodic theorems. We therefore completely omit topological and smooth dynamics. After some preliminaries, we will cover in depth two classical ergodic theorems: the Von Neumann and Birkhoff ergodic theorems. We follow this with a brief discussion on the space of invariant measures on a compact metric space, culminating in the unique ergodic theorem. If there is time, we will learn about the ergodic decomposition theorem. The final goal of this mini-course will be to discuss generalizations of the classical ergodic theorems to other group actions. We will finish by stating analogues of the mean and pointwise ergodic theorems for amenable group actions. In this mini-course, we will use without proof tools from measure theory and functional analysis, but typically we will define things when we need them.

Abstract. Extremal graph theory and Ramsey theory were among the early and fast developing branches of 20th century graph theory. The central theme of Turán type graph problems is to maximize the number of edges a graph can have under certain forbidding conditions. One of the earliest results, Mantel's theorem, states that a triangle-free graph on n vertices contains at most n^2/4 edges.

This problem has been studied for ordered graphs (when we order either the vertices or the edges of the graph), directed graphs, oriented graphs, geometric and topological graphs. Some of these results have found applications in combinatorial number theory, discrete geometry.

Abstract. I'd like to talk about Fukaya categories. Not sure yet what I want to cover but it's the perfect chance for me to study them. Clearly we will talk about basic definitions, bubbling, dehn twists... I'd like to at least mention some modern results like Abouzaid generation criterion.

Abstract. I will follow Audrey Terras' book. The possible topics include: the discrete Fourier transform on the finite circle, Cayley graphs and their spectra, random walks on Cayley graphs, the discrete Fourier transform on finite abelian groups. If time permits, I would also like to cover some of the non-abelian groups theory. This would include: Fourier transforms and representations of finite groups, and some particular classes of examples, like the Heisenberg group or the General Linear Group GL(2, F_q).

Abstract. Will survey various analytic techniques in algebraic geometry. We will first review the complex analytic case, and then spend the rest of the course discussing the non-Archimedean case. In particular we will discuss Berkovich analytic spaces and the relationship with other constructions such as rigid spaces and formal schemes.

Abstract. This course would provide an introduction to topological field theories, with the possibility of more specialized topics towards the end of the week. It would focus on the functorial approach to TFT.

Abstract. This will be a two week mini course. In the first week we will cover some basics of Riemannian geometry and lead to the development and utility of the Ricci flow. In week two we will discuss gradient flows in metric spaces, including existence and uniqueness, and finish by demonstrating how to adapt the Ricci flow into a gradient flow.

Abstract. A surface-knot is an isotopy class of a closed surface in 4-space. This mini-course will introduce the broken sheet approach to studying surface-knots with a focus on the triple point number, analogous to the crossing number of classical knots. In addition, this mini-course will introduce quandles which generalize 3-coloring of classical knots. Finally, I will talk about bridge trisections introduced by Meier-Zupan as a means to talk about bridge position for surface-knots.

Abstract. This is a 2-week survey course to build up the background needed to understand and motivate the statements of some of the conjectures connecting the statistics of zeroes of the Riemann zeta-function and eigenvalues of random matrices. The goal of the course will be to prove Montgomery's theorem and state the major conjectures in the field today.

Abstract. My plan is to cover a basic course on Algebraic Topology which includes fundamental groups, covering spaces, notion of simplicial complexes, deformation retract , van-kampen theorem. After that I will decide to throw a course on homology, co-homology.