2019 Graduate Summer School Course Descriptions

Lecturer: Søren Galatius, University of Copenhagen. Title: "Bordism categories, invertible field theories, and characteristic classes of smooth bundles.”

"This course will introduce bordism categories, some of their many variants, and some of their usages.

The fact that compact smooth manifolds may be glued along parts of their boundary, and that this operation is associative up to diffeomorphism, leads to the idea that they should be regarded as morphisms in a category. This idea may be implemented in several ways, and this course will introduce some of them.

We will then discuss invertible field theories, and the closely related notion of classifying space of a category. The first goal of the lectures will be to state the classification theorem for invertible field theories, and sketch some steps in the proof.

Many interesting applications of bordism categories and field theories require modified versions of the basic idea. For example, we might restrict attention to connected manifolds, or to simply connected ones. Perhaps surprisingly, it seems more difficult to classify invertible field theories based on such restricted bordism categories. The second goal of the lectures will be to discuss this, including some non-trivial circumstances where the classification does not change upon passing to restricted bordism categories.

The final goal of the lectures will be an application of these ideas to understanding characteristic classes of manifold bundles. We will not have time for detailed proofs, but I hope to explain the important role played by restricted bordism categories."

Prerequisites: "Algebraic topology, basic category theory, and comfort with smooth manifolds. Comfort with homotopy theory (simplicial objects etc) will also be useful."

Lecturer: Andriy Haydys, University of Freiburg. Title: Introduction to gauge theory

Mathematical gauge theory in a broad sense is the theory of vector bundles, which are ubiquitous in geometry, topology, and physics. An object of particular interest in the theory of vector bundles is a connection, which provides a way to relate vectors lying in different fibers. Properties of the space of all connections as well as some particular subspaces, also known as “moduli spaces,” will be in the focus of these lectures. A prototypical example of a gauge theory is Maxwell’s theory of electromagnetism, which will be our starting point.

The main aim of these lectures is to outline common techniques and methods used in gauge theory. This includes in particular construction of certain particularly interesting moduli spaces, studies of their properties (compactness, boundary, singularities) and how these properties can be used to study topology and geometry of manifolds.

Prerequisites: I will assume acquaintance with manifolds, vector bundles, and Lie groups as described, for example, in the book of Barden and Thomas “An introduction to differentiable manifolds.”. Although I will discuss briefly basic facts from the theory of elliptic PDEs and characteristic classes, it may be helpful to get familiar with these concepts beforehand.

Suggested reading:

Ozsvath and Szabo, An introduction to Heegaard Floer homology,

http://math.mit.edu/~petero/Introduction.pdf

Ozsvath and Szabo, Lectures on Heegaard Floer homology,

http://math.mit.edu/~petero/Lectures.pdf

Ozsvath and Szabo, Heegaard diagrams and holomorphic disks,

https://arxiv.org/pdf/math/0403029.pdf

Lecturer: Tom Mrowka, MIT. Title: Floer homology

Floer homology has proved to be a powerful tool to attack problems in low dimensional topology. This course will focus on applications of instanton Floer homology to various problems in the topology of three- and four-dimensional manifolds as well as knot theory. Topics to be covered include a quick introduction to instanton homology and its generalization to sutured manifolds and knots. Some topological consequences of instanton Floer homology will be sketched including applications to representing 2-dimensional homology classes by surfaces of minimal genus. The surgery exact triangle and skein exact triangle will be developed and we will discuss a spectral sequence relating instanton Floer homology and Khovanov homology.

Lecturer: Andy Neitzke, The University of Texas at Austin. Title: BPS states and spectral networks

Given a supersymmetric quantum field theory of the right sort, one obtains a graded vector space of "one-particle BPS states." Decomposing this space under the action of the supersymmetry algebra one obtains integers called "BPS degeneracies," much studied by physicists. In various examples of field theories, the BPS degeneracies are concretely computable and turn out to coincide with invariants studied in algebraic or differential geometry. In particular, there is a class of examples ("N=2 theories in four dimensions") in which the BPS degeneracies are expected to coincide with the "generalized Donaldson-Thomas invariants" studied by some algebraic geometers.

A rich source of examples is the "class S" construction, which produces a field theory starting from the data of a Riemann surface C and an ADE Dynkin diagram. In this case the BPS degeneracies can be computed using the geometry of "spectral networks," certain networks of trajectories on the surface C. Moreover the BPS degeneracies turn out to be connected to many other geometric phenomena associated to C, including the hyperkahler geometry of Hitchin's integrable system, the WKB method for ordinary differential equations defined on C, and the symplectic geometry of a Calabi-Yau threefold constructed from C.

A rough lecture plan follows.

Lecture 1: BPS states in supersymmetric field theory

Lecture 2: Class S and spectral networks

Lecture 3: The WKB method and ODEs on Riemann surfaces

Lecture 4: The hyperkahler geometry of the Hitchin system

Prerequisites: The main prerequisite is some familiarity with complex geometry. At various points some other background could be helpful, but is not strictly necessary: e.g., for lecture 2 some familiarity with Lie algebras, for lecture 4 some Kahler geometry, for lecture 2 some familiarity with Calabi-Yau 3-folds. I won't assume any background in quantum field theory, though of course that won't hurt, either.

Lecturer: Pavel Putrov, International Centre for Theoretical Physics. Title: Topological Quantum Field Theory, knots and BPS states

In my lectures I plan to cover the following topics: general notion of Topological Quantum Field Theory (TQFT); Witten-Reshetikhin-Turaev (WRT) TQFT and the corresponding invariant of 3-manifolds; perturbative invariants of 3-manifolds; knot invariants and BPS states; invariants of (plumbed) 3-manifolds valued in q-series; relation between perturbative invariants and q-series via resurgence theory; BPS states and categorification of WRT invariant.

Prerequisites: basic algebraic topology and basic gauge theory. Also helpful: knot theory, 3-manifold topology, category theory, quantum mechanics (all at basic level).

Lecturer: Jake Rasmussen, University of Cambridge. Title: Introduction to knot theory

This course will give an introduction to polynomial invariants of knots and their categorifications (Khovanov homology). Knot polynomials have a long history, dating back to work of Alexander in the 20's, but the modern history of the subject starts with the discovery the Jones polynomial in 1984. As Witten explained, these invariants are best understood in the context of 2+1 dimensional TQFT's. From a topologist's perspective, categorification can be viewed as a technique for upgrading these 2+1 dimensional TQFT's to 2+1+1 dimensional extended TQFT's. My goal is to define the invariants, describe some of their properties, and give some topological applications.

I'll start by discussing the Alexander polynomial and its relation to cyclic coverings and the Seifert genus. (Knot Floer homology, which categorifies the Alexander polynomial, will be discussed in Hom's lectures.) I'll then move on to the Jones polynomial and Khovanov homology, first for links in $S^3$, and then for tangles.

The HOMFLY-PT polynomial is a common generalization of the Alexander and Jones polynomials. I'll discuss its definition via the Jones-Ocneanu trace, and its categorification, due to Khovanov and Rozansky. The sl(n) knot polynomials are specializations of the HOMFLY-PT polynomial. I'll describe the categorified analog (spectral sequences) and applications to the slice genus of knots. In the final lecture, I'll talk about colored polynomials, Jones-Wenzl projectors, and their categorifications.

Course Outline:

Lecture 1: The Alexander polynomial

Lecture 2: The Jones polynomial and Khovanov homology.

Lecture 3: Khovanov homology for tangles

Lecture 4: HOMFLY-PT homology

Lecture 5: Deformations and Spectral Sequences

Lecture 6: Colored homologies

Prerequisites: Basic algebraic topology, including covering spaces and homology. Some experience with representation theory (for either finite groups or Lie groups) would be helpful for context, but is not strictly necessary.

Suggested reading (overlaps with the first two lectures):

W.B.R. Lickorish: Introduction to Knot Theory, Springer (chapters on Jones and Alexander polynomials)

Dror Bar-Natan: "On Khovanov's categorification of the Jones polynomial"

arXiv:math/0201043

Lecturer: Laura Schaposnik, University of Illinois at Chicago. Title: Advanced topics in (mathematical) gauge theory / Higgs bundles

Higgs bundles provide a unifying structure for diverse branches of mathematics and physics. The Dolbeault Moduli space of Higgs bundles has a hyperkahler structure, and through different complex structures it can be understood as different moduli spaces: Via the non-abelian Hodge correspondence the moduli space is diffeomorphic as a real manifold to the De Rahm moduli space of flat connections. Via the Riemann-Hilbert correspondence there is an analytic correspondence with the Betti moduli space of conjugacy classes of surface group representations. We will begin by introducing Higgs bundles and their main properties (Lecture 1), and then we will discuss the Hitchin fibration and its different uses (Lecture 2).

The second half of the course will be dedicated to studying different types of subspaces (branes) of the moduli space of Higgs bundles, their appearances in terms of flat connections and representations (Lecture 3), as well as correspondences between them (Lecture 4).

A rough lecture plan is as follows:

Lecture 1: An introduction to Higgs bundles.

Lecture 2: The Hitchin fibration and Langlands duality

Lecture 3: Branes in the moduli space of Higgs bundles

Lecture 4: Brane correspondences

Prerequisites: The main prerequisite is some familiarity with complex and differential geometry, as well as basic Lie theory.

Suggested reading (overlaps with the first two lectures):

Lecture 1 & 2 (and references therein):

** N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. LMS 55 3 (1987)

** N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 1 (1987) 91-114.

** N.J. Hitchin, Langlands duality and G2 spectral curves, Quat. J. Math (2007).

Lecture 3 & 4 (and references therein):

** N.J. Hitchin, Higgs bundles and characteristic classes, arXiv:1308.4603 (2013).

** D. Baraglia, L.P. Schaposnik, Higgs bundles and (A,B,A)-branes, Communications in Mathematical Physics, 331 3, pp 1271-1300 (2014).

** D. Baraglia, L.P. Schaposnik, Real structures on moduli spaces of Higgs bundles, Adv. in Theo. and Math. Physics., Vol 20 No. 3, 525–551 (2016).

** L.P. Schaposnik, Higgs bundles and applications, OBW Reports, 31, (2015).

** S. Bradlow, L.P. Schaposnik, Higgs bundles and exceptional isogenies, Research in the Mathematical Sciences, DOI:10.1186/s40687-016-0062-0 (2016).

** S. Heller, L.P. Schaposnik, Branes through finite group actions, to appear in Journal of Geometry and Physics, (2017).