Summer Minicourses 2021

Abstract. This minicourse will offer an introduction to spectra, the basic objects of study of stable homotopy theory. Spectra have many lives: as (co)homology theories, as infinite loop spaces, as the stabilization under iterated suspension of pointed spaces, as the homotopical analogue of abelian groups, or as a distinguished stable symmetric monoidal infinity category. We will discuss all these different perspectives on spectra, and some of the things we can do with them, in the field that has been called “Brave New Algebra”.

Abstract. This minicourse will provide an introduction to the mathematical study of topological field theory. We will discuss some general theorems then focus on examples, including invertible field theories; finite path integrals and Dijkgraaf-Witten theory; and Turaev-Viro theory and state sums. We will also discuss an application to Hurwitz-like problems.

Abstract. In this minicourse, five speakers will discuss group cohomology, its interpretation, and its applications from several different perspectives including abstract algebra, algebraic topology, and hyperbolic geometry. Speakers will explain how they think about group cohomology and give examples of how they use it in their everyday lives.

Abstract. We will start by reviewing the definition and basic properties of weak convergence in a fairly abstract setting. We will then cover some foundational results like Skorohod's Representation Theorem and Prokhorov's Theorem. Finally, we will discuss some important applications. Depending on time and interests, these applications may include the construction of Brownian motion as the weak limit of rescaled random walks, the Wasserstein space, and weak convergence of empirical distributions.

Abstract. Abelian duality is a major theme in both math and physics, more specifically, representation theory and QFTs. In this mini-course, we will explore a homotopical version of this duality, via the formalism of topological field theory. Along the way, we will find higher categories, path integrals, sheaf theory, and lots of other good stuff. We will sketch a proof of the topological abelian duality theorem with emphasis on viewing topological field theories and path integrals via sheaf theory and pushforwards. Hope to see you there!

Abstract. This mini-course will aim to provide an “intuitive prelude” to a first course in algebraic geometry – a somewhat conversational overview of the basic language of varieties and schemes, with an emphasis on motivating the concepts at hand and illustrating some key principles through simple examples. We will try to keep prerequisites minimal so as to make the course accessible to a broad audience. The course will meet for two hours each weekday, so as to allows for some time for exercise sessions at the end. Some topics that we expect to cover include affine and projective varieties, schemes and sheaves over them, algebraic vector bundles, algebraic differential forms, the GAGA principle, and some basic moduli theory.

Abstract. This mini-course will be a tour of a three dimensional topological quantum field theory called Chern-Simons theory. This may seem like an obscure topic at first, but this TQFT appears again and again in math and physics. In math, diverse topics such as the Volume Conjecture in 3-manifold topology, symplectic geometry of character varieties of surfaces, and representation theory of affine Lie algebras can be viewed as aspects of Chern-Simons theory. There are many applications in physics, but one I vaguely understand is that Abelian Chern-Simons theory has been successful in explaining the fractional quantum Hall effect. Even though Chern-Simons theory was in a sense exactly solved soon after it was introduced in 1989, it still (in my opinion) defies unified mathematical understanding.

Instead of focusing on a single aspect of Chern-Simons theory, we will hop between topics and try to get an idea of the big picture. The tentative plan is as follows. First we will define the Chern-Simons action, and sketch its modern interpretation as an invertible field theory. Second we will try to make sense of the path integral, and find ourselves learning about Ray-Singer torsion. Third, we will learn about geometric quantization of character varieties of surfaces. Fourth, we will talk about the Jones polynomial and maybe quantum groups. We will skip a vital piece of the story because it is a whole complicated subject in itself: WZW conformal field theory and rep theory of affine Lie algebras. Tom's minicourse might fill this in to some extent.

Abstract. An introductory course in matroid theory with a focus on elementary open problems. In this course, we will cover the most common cryptomorphic definitions of matroids and discover the abundance of matroids hidden beneath familiar problems. Time will be spent developing the concepts of duality, deletion, and contraction as well as introducing the classes of matroids which are closed under these operations. As each concept is introduced, an open problem will accompany it- beginning with matroid enumeration and ending with recent questions posed this past year.

Abstract. This course will introduce basic examples and properties of contact structures on manifolds, especially 3-dimensional ones. Broadly, our goal is to discuss how contact geometric questions can be answered by studying knots and surfaces, especially highlighting where contact theoretic questions can be studied by topological techniques. In particular, we'll use these ideas to study the dichotomy: tight vs. overtwisted.

Abstract. This course will give a (topologically motivated) introduction to perverse sheaves and intersection (co)homology. The first lecture will introduce intersection homology (following Goresky and MacPherson) and review classical results on the topology of complex algebraic varieties. From there, we will discuss the relevant background in homological algebra (derived categories and derived functors) and the constructible derived category of sheaves in order to formulate intersection homology sheaf theoretically. In the third lecture, we will introduce and prove Verdier duality and consequently Poincare duality for singular spaces. The fourth lecture will finally define the perverse t-structure and establish the basic properties of perverse sheaves on complex varieties. To illustrate the general theory, we will study perverse sheaves on curves. Finally, in the last lecture, we will state the decomposition theorem, discuss its proof, and give some applications in algebraic geometry or representation theory (depending on the audience's taste). The presentation will be relatively elementary, but will assume a general familiarity with algebraic topology, algebraic geometry, and classical homological algebra.

Abstract. Affine Lie algebras are a certain generalization of a finite dimensional Lie algebra 𝔤 which is used, roughly speaking, to capture the representation theory of 𝔤((t)). These algebras have importance in representation theory and string theory, although this course will not cover much of the latter. We'll start by quickly reviewing the vocabulary of representation theory of compact Lie groups, including a brief discussion on why semisimple Lie algebras are classified by their Cartan matrix. We will then define the notion of a Kac-Moody algebra using the notion of a generalized Cartan matrix. We will then specialize to the case of an affine Kac-Moody algebra, and explain some of the basics of its representation theory. Time permitting, I will take requests for topics to cover (feel free to let me know in advance if there's something you want to see!)–one possible instance is how this story fits into the local Geometric Langlands correspondence.