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!
Moduli spaces of Higgs bundles and the Kobayashi-Hitchin correspondence Higgs bundles first emerged in 1987 in Hitchin’s study of the self-duality equations on a Riemann surface. The “moduli space” of such bundles forms a manifold with extraordinarily rich geometry and plays a role in many different areas including gauge theory, Kähler and hyperkähler geometry, surface group representations, integrable systems, nonabelian Hodge theory, the Deligne-Simpson problem on products of matrices, and (most recently) mirror symmetry and Langlands duality. This course will begin with the differential geometry underlying the study of Higgs bundles, following lecture notes by Guichard. We will develop the essential tools for studying vector bundles on real and complex manifolds, including curvature and characteristic classes, and the relationship between (the space of) “flat” bundles—those with vanishing curvature—and (the space of) representations of the fundamental group. One goal will be to understand the “Kobayashi-Hitchin” correspondence between flat bundles and the Higgs bundle moduli space, with many examples along the way. The first part of the reading course will be an overview of the necessary background, which could be useful for many graduate students interested in differential topology and geometry. We will follow Guichard’s notes, which we will complement as appropriate. Organizers: Sara Maloni and Tom Mark
Invariants of 4-manifolds from Khovanov homology We will study a new invariant, informally known as the “skein lasagne module,” that relies on ideas from representation theory and categorification, as well as some algebraic topology (blob homology, a close cousin of factorization homology) to construct invariants of smooth 4-dimensional manifolds.
We’ll start with some basic material on Khovanov homology of links in 3-space, and the goal of this reading course is to go through the 2019 preprint of Morrison-Walker-Wedrich defining the skein lasagna modules, and the 2020 preprint of Manolescu-Neithalath computing it in some instances, thus arriving at the research frontier of the subject.
All participants will be expected to present some material in this seminar. Organizer: Slava Krushkal
Anosov representations Anosov representations are discrete, faithful representations of word hyperbolic groups into semisimple Lie groups, with strong dynamical properties. They were introduced by Labourie in 2006 for fundamental groups of closed negatively curved manifolds and generalized by Guichard and Wienhard in 2012. They have been studied a lot in the past few years and play an important role in higher Teichmüller-Thurston theory and in recent developments in the theory of discrete subgroups of Lie groups. In particular, these representations can be thought as a generalization of convex-cocompact representations in rank one groups to Lie group of higher real rank. We will introduce these representations, give examples, and discuss some characterizations. We will follow Canary’s lecture notes, which we will complement as appropriate.
Teichmueller geometry and Nielsen-Thurston Classification
Heegaard Floer homology
Lie Algebras in Homotopy Theory <!–A main goal for the course is understanding Gijs Heuts’ preprint “Lie Algebras and Periodic Spaces”, also surveyed in his article “Lie algebra models for unstable homotopy theory”.
In brief, in 1969 Quillen showed that the rational homotopy of (simply connected) spaces could be modeled by differential graded Lie algebras over the rationals. The new results say that the ‘$v_n$-periodic’ homotopy of spaces can similarly be modeled by algebras over the Lie operad in ‘$v_n$ local spectra’.
A suggested ‘Reading Course Schedule’ is posted under Resources/Fall19ReadingSeminar on the topology seminar collab site, and some key papers are also posted there. If you are planning to come to this seminar (and this hopefully includes alg top students) please look at this!
I (Nick) am planning to give a two talk overview, but after that we need (student) volunteers!! Let me know what you might wish to talk (and thus learn) about; we can also do some planning at the first talk.
Organizers: Nick Kuhn and Julie Bergner–>
The mapping class group, Teichmueller space and the space of measured foliations
An overview of Thurston geometrization