Nick Addington - A Fun Long Exact Sequence
We review the definition of a fiber bundle, the clutching construction, pullback bundles, and associated bundles. We define H^0 and H^1 of a space with coefficients in a non-abelian group, but we cannot define H^2. The proof of the long exact sequence is unexpectedly fun, partly because the sequence isn't very long. As an application, we give a concrete interpretation of the first and second Stiefel-Whitney classes, building on our discussion of G-structures on vector bundles last semester. We might need two lectures to do all of this.
Noah Kieserman - Basics of Cech-de Rham Theory
Following Bott and Tu, the Mayer-Vietoris principle generalizes to a relation between the deRham complex and the Cech complex for sheaf cohomology. We organize our information in a double complex, prove isomorphism of deRham and Cech cohomologies for manifolds, and start a motivation for spectral sequences.
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Noah Kieserman - Some foliation cohomologies and related spectral sequences
We see a natural motivation for spectral sequences within the context of the Cech-deRham double complex, then explore what they might tell us in another, mostly unrelated example, namely the foliation spectral sequence. We will meet there some natural cohomology groups which give funny results.
Garrett Alston - The Cousin Problems on a Stein Manifold
I will talk about the first Cousin problem, also known as the Mittag-Leffler problem. This problem can be stated in terms of sheaf cohomology. It can always be solved in C, but in C^n for n>1 additional hypothesis are needed on the domain. The generalization of these hypotheses leads to the notion of a Stein manifold. I will state Hormander's famous L2 existence theorem for the d-bar operator and show how this leads to the solution of the problem. I will also state some interesting properties of Stein manfolds.
Junwu Tu - A-Infinity Algebras
No abstract.
Stefan Müller - J-holomorphic Curves in Symplectic Manifolds
I will explain how the study of (the moduli spaces of) J-holomorphic curves naturally leads to the bubbling phenomenon (which will be discussed in detail in part II). After introducing the main objects of the theory, I will explain why one studies J-holomorphic curves to study symplectic manifolds, and give examples of some of their powerful applications to symplectic topology. I will state the relevant main results about above moduli spaces, followed by some simple examples. Emphasis will be put on explaining these results and their hypotheses rather than discussing their proofs. We will briefly discuss Gromov-Witten invariants. Towards the end of the talk, we will see the phenomenon of spheres bubbling off emerge. (This program seems ambitious, and may have to be continued the following week.)
Stefan Müller - The Bubbling Phenomenon
I will continue the motivation from part I if necessary. Then I will explicitly construct a J-holomorphic sphere that bubbles off. The main ingredients are a uniform bound on the energy for a sequence of J-holomorphic curves, conformal invariance of the energy in two dimensions, and the removal of singularities theorem. If time allows, I will add some remarks concerning suitable compactifications of the moduli spaces of J-holomorphic curves. In particular, spheres bubbling off is essentially the only source of noncompactness.
Christine Lien - Feeling Steenrod Squarish
We will define Steenrod Squares and Powers, the properties that they satisfy, and discuss why they're better than the cup product in some respects. We will use Steenrod squares and powers to look at some explicit examples and to show that certain spaces are not the wedge of two spaces. Given time, we may explicitly construct the Steenrod Squares.
Song Sun - Topological K-theory
We will first review some facts about complex vector bundles, and then define K groups and use Bott periodicity theorem to show it extends to a generalized cohomology theory with a homotopic description. If we have time, we will state the Atiyah-Singer index theorem in the language of K-theory (without proof).
Antonio Ache - The Hodge Decomposition Theorem
We will go over some ideas involved in the proof of the classical Hodge Theorem as well as some consequences like Poincaré duality for compact orientable differentiable manifolds. If time permits, we will discuss some results of the Bochner Technique applied to the study of harmonic 1-forms.
Yongqiang Zhao - Small and Big Picard Theorems
We will talk about two kinds of geometric proof of the Picard Theorems: one is topological in nature and the other using some differential geometry. In the former, a modular function will be explicitly constructed and used, and the connection between hyperbolic metric and holomorphic functions will be discussed in the latter.
Sarah Matz - Toroidal Categories and Low Genus M-Surfaces
We will survey certain objects of recent interest, these days known as M-Surfaces. These are best discussed in the slightly more general context of toroidal categories. We will analyze symmetries and surface features, the important cutting construction, and sketch the celebrated devouring lemma, which is widely considered the first landmark of the theory. For simplicity we restrict to genus less than or equal to three. If time permits, we will discuss a 'recipe' for the explicit construction of M-Surfaces.