Summer Mini-Courses 2018

Abstract. This mini-course is an introduction to the basic properties of Dehn surgery on knots in 3-manifolds. We will closely follow the 2006 notes by Cameron Gordon. This course should be accessible to anyone comfortable with algebraic topology and differential topology prelim material. Topics will include the cabling conjecture, the Berge conjecture and L-space surgeries.

Abstract. Class field theory is a classical and fundamental part of algebraic number theory. It traces its origins to the very impetus of number theory in Gauss's celebrated Quadratic Reciprocity Law, while at the same time also providing the main prototype for the Langlands conjectures at the very forefront of current-day research.

In this minicourse, we will start by quickly reviewing the relevant key notions from number theory and group cohomology, followed by outlining the proofs of all the major results in class field theory. We will follow the standard approach of treating the case of local fields first, relying crucially on group cohomology, and then using the theory of adeles to derive results in the setting of global fields.

Depending on the interests of the attendees, we may conclude the minicourse with a brief foray into the Langlands program, of which class field theory is the G = GL_1 case.

The course will meet in RLM 8.136 There will be a break during the session.

Abstract.Derived algebraic geometry is a relatively new but highly active field. It extends usual algebraic geometry by building in homological algebra and/or homotopy theory into its foundations.

This minicourse is meant to offer a friendly introduction to DAG. We will work through the basic definitioms and work out some specific examples, instructive to understanding the extra features as compared to ordinary algebraic geometry. We will compute some cotangent complexes, some derived self-intersections, and maybe even a blowup or two.

Various unavoidable technicalities, such as infinity-categories, will be taken as a "black box". Prior knowledge of homotopy theory will not be required.

Abstract. The main target for this mini-course will be to shed some light on the result of Donaldson and Gompf that a 4-manifold admits a symplectic structure if and only if it admits a Lefschetz fibration.

We won't give a full proof of it, being one direction pretty involved, but we will hopefully give some intuition about the techniques involved. This result will let us introduce some basic concepts about symplectic manifolds, Kähler manifolds, blow-ups/down and Donaldson submanifolds.

Abstract. The first three days of this course will be lectures on the basics of homological algebra with an eye towards computations:

• diagram chasing
• working with chain complexes
• computing homology and cohomology
• using the Kunneth and universal coefficient theorems
• computing Ext and Tor groups (including group cohomology calculations)

On Thursday and Friday, we will discuss spectral sequences, focusing in particular on the Kunneth and Serre spectral sequences, which are also very useful for calculating homology / cohomology of topological spaces. There will be a problem session from 1-2pm.

Abstract. On Monday, there will be a motivation talk of tropical geometry towards its application to mirror symmetry and representation theory. Since Tuesday, we will study the basic of tropical geometry with the example of algebraic curves:

• Embedded tropicalization and abstrct tropicalization
• Tropicalization of algebraic curves
• Moduli space of tropical curves
• The relation of moduli space of tropical curves with the Deligne-Mumford moduli space of stable curves

Abstract. This course will be an introduction to the representation theory of Lie groups. We'll discuss what Lie groups are, and why their representation theory naturally leads into the representation theory of Lie algebras. We will then discuss the representation theory of Lie algebras by working through the extended examples of Lie(SL_2) and Lie(SL_3) to motivate the general theory. We will then discuss the general theory. Time and demand permitting, we will discuss the Weyl group of a Lie algebra, the Killing form, and/or the classification theorem of simple Lie algebras.

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 briefly discuss an application to condensed-matter physics.

Abstract.Differential Galois theory is an analogue of classical Galois theory. Whereas the latter can be instrumental in the study of solvability of polynomial equations, the former plays a similar role for differential equations. But just as Galois theory is mostly of little use in explicit numerical determinations of roots of polynomials, so does differential Galois theory not have much to contribute to the precise analytical properties of solutions of DEs, that a course on differential equations will usually focus on.

A prime application of differential Galois theory is instead to show that various differential equations admit no solutions in terms of elementary functions.We will build up to and prove some results of that sort in this minicourse, as well as outlining the general Galois correspondence in the differential setting. The only assumed prerequisite will be passing familiarity with classical Galois theory, on or below the level of that obtained in the prelim course.

Abstract.This course will be an introduction to the basic theory of hyperbolic groups (also called word-hyperbolic or Gromov-hyperbolic groups). Motivated by the study of fundamental groups of negatively curved manifolds, we'll start from the definitions of hyperbolic groups, and go through some of their basic properties. We'll then cover some of the computational results (e.g. solving the word problem), and discuss the construction of the boundary of a hyperbolic group (as well as some properties of the boundary). Finally we'll go over some generalizations of hyperbolic groups (relatively hyperbolic groups, semihyperbolic groups). No background is required, although passing familiarity with the hyperbolic plane might be helpful.