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MTG6939 -- Topology and Geometry Seminar

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9/3:
Title: Graphs and Mapping Classes of Surfaces
Speaker: Eriko Hironaka

9/10:
Title: Comparing Curves and Surfaces
Speaker: Eric Klassen

Abstract: We wish to create a moduli space of surfaces (or curves) in Euclidean space, where we mod out by rigid motions, rescaling, and reparametrization. We a lso wish to put a Riemannian structure on this moduli space, so we can calculate distances (between two surfaces) as the length of a geodesic. I will suggest an approach to solving this problem.

9/17:
Title: Interplays between Algebraic Curves and Hyperbolic Geometry: the Holographic Principle
Speaker: Ettore Aldrovandi

Abstract: Given a Riemann surface, there is a map, defined by Deligne, which assigns to two line bundles equipped with a hermitian metric a one-dimensional complex hermitian vector space. Its hermitian form can be calculated in terms of the relevant geometric data, it is "bilinear" in an appropriate sense (in its line bundle variables), and it has an explicit expression as a singular integral over the Riemann surface. It is rather surprising that all these data, which are of algebraic geometric nature, can also be expressed in terms of the hyperbolic geometry of the handlebody whose boundary is the surface in question. In particular, the three dimensional hyperbolic volume will play an important role. This state of affairs is referred to as the "holographic principle." We will give an introduction to this topic.

9/24:
Title: Homotopy type of a topological stack
Speaker: Behrang Noohi

Abstract: We show how to associate a homotopy type to a topological stack. This allows one to associate various cohomology theories to topological stacks. In the case of the quotient stack of topological group action this recovers the corresponding equivariant cohomology theory obtaied via the Borel construction.

10/2: Change in time: 3:30 - 4:30
Title : Dynamics of Rational Surface Autoorphisms
Speaker: Kyounghee Kim

Abstract : We discuss a family of holomorphic automorphisms of a rational surface with positive entropy. We describe in detail which of these automorphisms have invariant curves; and we show that some do not. For those with invariant curves, we have a dichotomy : either (i) a mapping has a rotation domain centered at a fixed point or (ii) the mapping is real, and its restriction to the real plane has maximal entropy.

10/30: Change in time: 3:30 - 4:30
Title : Dynamics of Rational Surface Autoorphisms
continued

11/6
Title: Chern class identities and D-branes
Paolo Aluffi

Abstract: String theory considerations lead to conjectural identities involving the Euler characteristic of certain loci determined by an elliptic fibration. We formulate these identities in terms of Chern classes, and prove them by taking into account the contribution of singularities. This is joint work with physicist Mboyo Esole.