Fall 2026
Guangbo Xu
Subtitle:
Cobordism and Characteristic Classes
Course Description:
This graduate course develops the foundations and applications of cobordism theory with an emphasis on its deep connections to characteristic classes and manifold topology. Beginning with basic differential topology, students learn to reinterpret cobordism groups in homotopy-theoretic terms and compute basic examples, including unoriented, oriented, and complex cobordism rings. The course then introduces characteristic classes—Stiefel–Whitney, Chern, and Pontryagin classes—and explains how characteristic numbers serve as complete invariants for cobordism classes in many settings. Central results such as the Hirzebruch signature theorem are presented to illustrate the powerful interplay between topology and geometry. Depending on time, topics may include the structure of the complex cobordism ring, formal group laws and Quillen’s theorem, as well as geometric applications such as exotic spheres and the role of characteristic classes in distinguishing smooth structures.
Text:
I will write my own notes. Good references include: Characteristic classes (Milnor-Stasheff), Notes on cobordism theory (Stong), Topological methods in algebraic geometry (Hirzebruch).
Prerequisites:
Concepts of smooth manifolds, homology and cohomology
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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.
Text:
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:
- Koebe circle packing theorem of for planar graphs
- Tutt embedding theorem of planar graphs
- Brunn-Minkowski inequality
- Minkowski theorem on convex polytopes
- Cauchy rigidity theorem for convex polytopes
- Bourgain’s embedding of finite metric spaces
- Johnson-Lindenstrass dimension reduction theorem
- Johns’ ellipsoid theorem
- Hadwiger's theorem
- Duality of convex polytopes
- Schramm’s rigidity theorem for infinite circle packing
- Discrete Schwarz lemma
- Discrete Laplacian and its applications
The topics on combinatorial topology include a brief introduction to simplicial complexes and
- Sperner's lemma, Brouwer fixed point theorem and applications
- Tucker's lemm, Borsuk–Ulam theorem and applications
- Chromatic and Tutt polynomials of graphs
- Jones polynomial of knots and an introduction topological quantum field theory (TQFT)
- Turaev-Viro invariant of 3-manifolds and the volume conjecture
Text:
None
Prerequisites:
Real analysis, complex analysis and linear algebra.