# Graduate Colloquium Spring 2016

**Wednesday April 6, 2016***Unit Groups of Ghost Rings as Inflation Functors***Rob Carman, University of California, Santa Cruz**There are several important rings used in the study of the representation theory of finite groups: the Burnside ring, character ring, trivial source ring, Brauer character ring, etc. The additive structures of each of these representation rings form a biset functor. Each of these rings is also embedded inside an associated ghost ring, where the additive structures also form a biset functor. Multiplicatively, the unit groups of these representation rings form inflation functors. With a focus on the trivial source ring, I will describe how the unit groups (and more importantly their torsion parts) of these ghost rings also form inflation functors.

**Wednesday April 13, 2016***Future directions: A conversation with Robert Boltje***Professor Robert Boltje, University of California, Santa Cruz**This week we will have an informational session with our graduate vice-chair Robert Boltje. He will elaborate on the plans for the future of our department and will talk about the issues we had this year and the plans the department has to address them. The intention is twofold, first to ease any concerns graduate students might have because of the changes we had this year and keep us a little better informed on what's behind the curtain, and second to let the department know of our perspective on where things are going.

**Wednesday April 20, 2016****No Colloquium**

**Wednesday April 27, 2016***Representation Theory of sl_2***Charles Petersen, University of California, Santa Cruz**The Lie algebra sl_2(C) is a building block of Lie theory. The representation theory of sl_2(C) is fundamental in uncovering the structure of finite dimensional semisimple Lie algebras and understanding their representations. The purpose of this talk is to classify all finite dimensional sl_2(C)-modules. We'll start with a few preliminaries on Lie algebras, their representations and the abstract Jordan decomposition. Next we'll outline the proof of Weyl's Theorem on complete reducibility. To finish the classification we'll construct a list of finite dimensional irreducible sl_2(C)-modules and show that every such module is isomorphic to a module in this list.

**Wednesday May 4th, 2016***Magnetism as a twisted geodesic flow***Gabriel Martins, University of California, Santa Cruz**This talk will be centered on an interesting computation that has a good mix of Riemannian and Symplectic geometry. The end goal will be to write the Hamilton-Jacobi equations for the Hamiltonian H=|v|^2/2 on the tangent bundle using different twisted symplectic forms. We will start by defining the canonical Symplectic form on the cotangent bundle of a manifold. Then using a Riemannian metric we will pullback this form to the tangent bundle and find a formula for it (call this form \omega_0). The twisting comes from choosing a 2-form B on the base manifold (representing a magnetic field) and considering the twisted Symplectic form \omega_B=\omega_0+\pi^*B.

**Wednesday May 11, 2016****TBA**

**Wednesday May 18, 2016****TBA**

**Wednesday May 25, 2016****TBA**

**Wednesday June 2, 2016****"Sums of Four Squares and Modular Forms"****Richard Gottesman, University of California, Santa Cruz**Every positive integer is a sum of four perfect squares. How the hecke can modular forms be used to prove this fact? I will explain this connection tomorrow and give a proof that every positive integer is a sum of four squares. No background in modular forms will be assumed.