Graduate Colloquium Spring 2014

It's tough to think about graduating and getting a job when you have things like prelims, orals, and never-ending TA duties on your plate, however, it is crucial to start to preparing yourself early for the competitive job market.  In this talk I'll discuss some things I have learned in my search for a job.  I'll discuss both academic and non-academic careers, and I'll share some things with you that I wish someone had shared with me when I was a new grad student.  All grads are welcome, and first and second year grad students are particularly encouraged to attend.

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The eigenspaces of the Laplacian on the two dimensional sphere consist of homogeneous polynomials and occur with increasing dimension as the eigenvalue grows. I'll explain how one can remove this high multiplicity by using arithmetic Hecke operators which arise from the Hamilton quaternions. The resulting Hecke eigenfunctions are subject to predictions arising from random function theory and quantum chaos, in particular concerning the topology of their zero sets. I'll discuss what is known in this area and how one can try to study these 'nodal lines'.

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May 1, 2014
The Tate-Shafarevich group, Coverings and Descent

Gabriel Martins, University of California, Santa Cruz

In this talk I'll explain how to understand and classify n-fold coverings of an elliptic curve. This will involve understanding cohomological obstructions to descent and will lead us to a nice geometric description of the n-th Selmer group and the Tate Shafarevich group of an elliptic curve.

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June 5, 2014

String Homology

Felicia Tabing, University of California, Santa Cruz

String topology is the study of the structure of the homology of free loop spaces. Chas and Sullivan defined an operation on the homology of the loop space of a closed, oriented manifold, called the loop product, that is a combination of the intersection product and loop concatenation product. The free loop space has a natural circle action given by the rotation of loops. String homology is the equivariant homology of the free loop space with respect to this circle action. Chas and Sullivan also defined a string bracket, which gives string homology the structure of a graded Lie algebra. For the first half of the hour, we will briefly discuss  the construction of the loop and string operations, and computations of string homology for spheres. For the second half of the hour, we will discuss the connection between string topology and Hochschild and cyclic homology.