Geometry/Topology Reading Courses

Each semester one or more geometry/topology faculty or postdocs organize a seminar-style reading course for graduate students. Usually these courses center around a book or collection of articles, which students study on their own and present in course meetings. Current courses are listed below; more information can be found by contacting the organizer.

Students are encouraged to suggest topics for future reading courses!

Spring 2022

Stable Homotopy and Low-dimensional Topology For some time now, there have been increasing interactions between low-dimensional topology and stable homotopy theory. Smooth topology in low dimensions is in some sense the opposite of the “stable regime”, so this interaction is somewhat indirect. This seminar will focus on the path between the two fields that passes through gauge theory, particularly Seiberg-Witten theory, which involves the study of vector bundles, connections, and curvature. Roughly, one uses the differential geometry \to construct a homotopy-theoretic object associated to a given smooth 3- or 4-dimensional manifold, such as an element of a stable homotopy group, or a spectrum in the sense of homotopy theory, whose properties reflect subtle aspects of the differential topology of the original manifold.

As this involves ideas from a wide swath of topology and geometry, a goal of the seminar will be to introduce participants to topics they are not familiar with; it is not anticipated that anyone will be expert, or even conversant, with everything mentioned above.

As a starting point, we will study the paper of Bauer and Furuta that introduces the stable homotopy refinement of the Seiberg-Witten invariant for 4-manifolds; then progress to the work of Manolescu constructing a ``Seiberg-Witten-Floer homotopy type’’ for 3-manifolds.
Options abound for further reading, including a variety of quite recent results and applications.

Organizer: Tom Mark

Past Courses
Fall 2021

Moduli spaces of Higgs bundles and the Kobayashi-Hitchin correspondence Followed lecture notes by Guichard. Organizers: Sara Maloni and Tom Mark

Invariants of 4-manifolds from Khovanov homology Studied a new invariant, informally known as the “skein lasagne module,” giving rise to invariants of smooth 4-dimensional manifolds. Organizer: Slava Krushkal

Fall 2020–Spring 2021

Anosov representations Organizer: Sara Maloni

Summmer–Fall 2020

The Z-hat invariant and quantum modular forms Studied the invariants of 3-manifolds recently introduced by Gukov-Pei-Putrov-Vafa based on ideas of supersymmetry in physics, and their relation to quantum invariants of 3-manifolds and to quantum modular forms. Organizer: Slava Krushkal

Summmer 2020

Teichmueller geometry and Nielsen-Thurston Classification Organizer: Sara Maloni

Spring 2020

Heegaard Floer homology Organizer: Thomas Mark

Fall 2019

Floer Theory Organizer: Tom Mark.

Lie Algebras in Homotopy Theory Organizers: Nick Kuhn and Julie Bergner

Fall 2018

The mapping class group, Teichmueller space and the space of measured foliations Organizer: Sara Maloni

Spring 2018

An overview of Thurston geometrization Organizers: Sara Maloni and Tom Mark