Course Descriptions

Spring 2024

Mallick Abhishek

Subtitle:

Floer homology and low-dimensional topology

Course Description:

This course will aim to bring graduate students to the forefront of recent research and developments in low-dimensional topology using Floer homology. We will start with a mild introduction to Morse theory and Morse homology. We will then introduce Heegaard Floer homology, an invariant for closed 3-manifolds and knot Floer homology, an invariant for knots, both defined by Ozsvath and Szabo. Afterwards the course will venture into several technical aspects of the theory such as invariance, functoriality and surgery formulas. This will be followed by various applications to low-dimensional topology pertaining to exotic smooth structures on 4-manifolds, homology cobordism in dimension 3, knot concordance and contact geometry. Discussing these applications will take the center stage of this course.

If time permits, we will also discuss some variants of Heegaard Floer homology such as bordered Heegaard Floer homology and involutive Heegaard Floer homology. The former is an invariant of 3-manifolds with boundary while the latter is another invariant of closed 3-manifold defined in the presence of a certain symmetry.

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NA

Prerequisites:

Some familiarity with Morse theory and Symplectic geometry will be helpful.

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Spring 2023

Feng Luo

Subtitle:

Topics in discrete geometry and combinatorial topology  

Course Description:

This course covers many topics in discrete geometry and combinatorial topology. 

In the discrete geometry part, we begin with some of the basics on convexity  and decompositions (separation theorem, Caratheodory, Helly, Minkowski’s lattice  theorems and Voronoi cells).  More advanced topics are:

1. Koebe circle packing theorem of for planar graphs
2. Tutt embedding theorem of planar graphs
3. Brunn-Minkowski inequality
4. Minkowski theorem on convex polytopes 
5. Cauchy rigidity theorem for convex polytopes 
6. Bourgain’s embedding of finite metric spaces 
7. Johnson-Lindenstrass dimension reduction theorem
8. Johns’ ellipsoid theorem 
9. Hadwiger's theorem 
10. Duality of convex polytopes
11. Schramm’s rigidity theorem for infinite circle packing 
12. Discrete Schwarz lemma 
13. Discrete Laplacian and its applications 

The topics on combinatorial topology include a brief introduction to simplicial complexes and 

14. Sperner's lemma, Brouwer fixed point theorem and applications
15. Tucker's lemm,  Borsuk–Ulam theorem and applications
16. Chromatic and Tutt polynomials of graphs 
17. Jones polynomial of knots and an introduction topological quantum field theory (TQFT)
18. Turaev-Viro invariant of 3-manifolds and the volume conjecture  

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None

Prerequisites:

Real analysis, complex analysis and linear algebra.