"First cut" is a series of broad-brush video-lectures on topics in geometry and topology. The lectures might explain a concept or a theorem; they might survey an area; or they might trace the development of an idea. The subject can be brand new or classical. Speakers aim to convey, to an audience of graduate students in geometry and topology, as well as postdocs and faculty, their high-level understanding of the topic, without getting into technical details. These lectures are designed to be a pathway to enter the field. We plan to hold these lectures two to three times per semester, on Tuesdays at 12:30. Videos and slides for the lectures will appear on this page.
- Date: November 19th, 2013
- Subject: 4-manifolds and exotic smooth structures
- Abstract: Smoothing theory behaves pathologically in dimension 4. We will survey the classification theory of smooth structures on a given topological manifold, or how many diffeomorphism types occur within a given homeomorphism type, from the original high-dimensional theory to the most recent 4-dimensional results.
- Date: November 26th, 2013
- Subject: A primer on the Hitchin system
- Abstract: Let G be a complex Lie group, say G = SL(2,C). Let S be a Riemann surface. In 1987 Nigel Hitchin discovered an amazing correspondence between representations \pi_1(S) -- G and some holomorphic objects which he called "Higgs bundles" on S. This correspondence is a nonabelian generalization of the classical Hodge theorem. It is the jumping-off point for a number of further stories: in particular, the moduli space of these representations is an example of a complex integrable system, and even a Calabi-Yau manifold (in fact a hyperkahler manifold). This Calabi-Yau manifold provides an interesting and computable example of mirror symmetry, and plays a central role in the geometric Langlands correspondence. I will review this story, in as concrete a way as possible.