Lecture Series on Instanton Floer Theory

Basic Information

Dates/Day/Time: 09Jan - 20Jan, 2023, MWF, 10:30-12:00 PM

Location: Lecture Hall 3 in the Dept. of Math.

Speaker: Deeparaj Bhat (MIT)

Pre-requisites for students: While no hard pre-requisites will be imposed, and the lectures will be open to all, you are recommended to have a working knowledge of differentiable manifolds at the level of MA 235 , and algebraic topology at the level of MA 232 . Some familiarity with basic Riemannian geometry will also be useful.

Tentative schedule, titles and abstracts

For online access: Please click on the topic names for online access to the talks.
Number Date (Day) Topic (Meeting link) Abstract (+ video link) Lecture notes
1 09/01/2023 (Monday) A Crash Course on Morse Theory This talk will be a brief introduction to the Morse-Homology of compact manifolds. We will start with the definition of Morse functions and recover the Euler characteristic as a warm-up. We then define the Morse-chain complex and sketch how it recovers singular homology. Finally, we will recast the definition of the Morse-chain complex using gradient trajectories as a prequel to the formalism of Floer homology, which will be discussed in more detail in the third lecture. video Lecture-1
2 11/01/2023 (Wednesday) Chern-Simons Invariant and Representation Spaces We briefly review some gauge theory and then define the Chern-Simons invariant. The Chern-Simons invariant will play the role of an infinite dimensional morse function. With this in mind, we compute the set of critical points and critical values of the Chern-Simons functional for two classes of examples: Lens spaces and Brieskorn homology spheres. We then give a heuristic as to how the count of critical points defines an invariant for integer homology spheres called the Casson invariant. Time permitting, we state a few results regarding the computation of Casson invariants for Seifert fibred spaces and links of some isolated complex surface singularities. video Lecture-2
3 13/01/2023 (Friday) Floer Homology: An Infinite-dimensional Morse Theory We will attempt to mimic the outline from finite morse theory in the setting of the Chern-Simons functional defined on the configuration space of connections to define Floer Homology. Along the way, we will encounter issues that are specific to the case of instanton floer theory. Imposing some restrictive hypotheses, we will see how to overcome these. Finally, we will explain how Fintushel and Stern computed the instanton floer homology of Brieskorn homology spheres. Time permitting, we will comment on their work in the case of general Seifert homology spheres. video Lecture-3
4 16/01/2023 (Monday) Construction and Properties of Instanton Floer Homology We continue from last time discussing spectral flow, compactness and gluing. After this, we use topological constructions to remedy the issues we encountered earlier. With this in place, we then discuss invariance of floer homology and how cobordisms define maps on these floer groups. Time permitting we state some sample computations. video Lecture-4
5 18/01/2023 (Wednesday) Knots and Singular Instantons We will talk about an extension of instanton theory for knots where we allow mild singularities of connections along codimension 2 submanifolds. After a brief discussion of how this fits in the earlier framework, we introduce some topological constructions analogous to the 3 manifold case and state some examples. Finally, we talk about cobordism maps which we intended to discuss in the last talk.
6 20/01/2023 (Friday) Exact Triangles in Instanton theory In the lectures so far, we have defined the instanton floer groups but have not given any systematic way to go about computing them. In this last lecture, we discuss one technique towards a systematic way to compute certain flavours of these groups: exact triangles. Two immediate applications are the existence of spectral sequences from more combinatorially defined homology theories like Khovanov homology abutting to instanton floer groups. We then state the main homological algebra lemma that these results rest on and show how it applies in the 3-manifold context. Time permitting, we sketch the case of knots. video Lecture-6

Last modified: Mon Feb 13 06:49:47 IST 2023