Garrett Alston - The Euler Class of Orbibundles
I will begin with some basic theorems about Lie groups acting on manifolds. Then I will define orbifolds and orbibundles, give some examples, and use the theorems from the first part to talk about the structure of an orbifold. Then I will recall the classical Euler class, do an example, and show how to modify the definition for orbibundles. I will end by computing the Euler class of the tangent bundle of the tear drop oribfold. Time permitting, I will also talk about singularities of 2-orbifolds.
Garrett Alston - Kuranishi Structures and Virtual Cycles of Moduli Spaces
This is a continuation of last week's talk.
Links
- Schedule
- Current Abstracts
- Abstracts, Fall 2008
- Abstracts, Spring 2008
- Abstracts, Fall 2007
- Abstracts, Spring 2007
Antonio Ache - Warm-Up for the Real Seminar
He will say what he's going to say at 1:20, only with more mistakes. The audience will ask questions they couldn't ask in front of the faculty.
Nick Addington - Deformations of Complex Manifolds
We will define deformations of a complex manifold and give examples. After a crash course in sheaf cohomology, we will show how a deformation of X gives a class in H1(TX), where TX is the tangent bundle. We will interpret H1(TX) as the (Zariski) tangent space at X to the moduli space of complex manifolds and say what it means to be "unobstructed". Last we will compute H1(TX) for Pn, hypersurfaces in Pn, and Riemann surfaces.
Nick Addington - Deformations of Vector Bundles
In parallel with last week's talk, we will define deformations of a vector bundle E, give examples, and say that they are parametrized by H1(End E). Our examples will be line bundles on Riemann surfaces and the tangent bundle of Pn. Then we will say how this generalizes to sheaves and study the example of a skyscraper sheaf.
Jo Nelson - The Maslov Index and its Applications
I will spend a few minutes explaining the basics of symplectic topology that we need. Then I will define the Maslov index and give a few examples of how to compute it. I will also draw some pictures at this point. Then I will state an important theorem that relates the Maslov index of a curve to its cohomology class. Time permitting, I will briefly talk about Spectral Flow and its relationship to the Maslov index. This talk will be accessible to anyone that knows what a manifold is.
Michael Childers - Clifford Algebras
No abstract.
Patrick Curran - Some Category Theory
I'm going to give an overview and some examples of natural transformations, universal properties and adjoint functors.
Christine Lien - Gaussian Curvature
We start with the definition of the curvature of a plane curve, which is introduced in 222 here. We state the plane curve classification theorem along with the total curvature theorem for simple closed curves. We will use these ideas to generalize to the Gaussian curvature for surfaces in R3 and compute a few, i.e. 2, examples. We will also mention without proof the Uniformization Theorem. Then we generalize again to higher dimensions and introduce briefly the Riemannian curvature tensor, the Ricci curvature tensor, and the scalar curvature, with the assumption that the audience is familiar with vector fields and a covariant derivative. We will finish with the Gauss-Bonnet theorem, and mention the Chern-Gauss-Bonnet theorem, without really explaining what the Pfaffian is, and thereby giving Nick basis for another talk in the seminar. In this talk, we aim to tell a pretty story, give examples, and prove nothing. The talk may or may not be 50 minutes; much depends on the mood of the speaker.
Diane Holcomb - Geometry on the Complex Unit Disc
There are close connections between geometric and analytic approaches to exploring the complex plane, and understanding these connections can lead to a greater understanding of both approaches. After a brief review of important results in complex analysis and differential geometry we will explore connections between the two topics. This will include discussion of applications of various metrics, and curvature to complex analysis, as well as alternate geometric proofs and statements of major complex analytic results.
Jason Murcko - Connections and Characteristic Classes
Following some notes of Joel Robbin, we explain how a connection gives us characteristic classes. The trace gives us the first Chern class, the determinant the top Chern class, and the Pfaffian the Euler class, as Christine alluded to two weeks ago.
Nick Addington - The Pfaffian and Spin
There are many square roots associated with SO(n). Spin(n) is a double cover of SO(n). The Dirac operator is the square root of the Laplacian. The Pfaffian is the square root of the determinant. For an oriented vector bundle of even rank, the Euler class is the square root of the top Pontryagin class. The embedding line bundle O(1) on the ordinary Grassmannian has a square root when restricted to the orthogonal Grassmannian, which sometimes has one connected component and sometimes has two. We will say what all these things are and try to see if they're related.