# Graduate Colloquium Fall 2012

For further information please contact Corey Shanbrom cshanbro@ucsc.edu

**October 11, 2012**

**Burnside and double Burnside groups and rings**

**Burnside and double Burnside groups and rings**

**Philipp Perepelitsky **

In this talk, I shall introduce the notions of Burnside and double Burnside groups and rings. I shall present some classical results concerning these objects as well as some recent results obtained in joint work by Robert Boltje and myself.

**November 1, 2012**

**Quantum Dimensions and Quantum Galois Theory**

**Quantum Dimensions and Quantum Galois Theory**

**Xiangyu Jiao, Mathematics Graduate Student **

I will talk about "dimensions" of modules of a vertex operator algebra.

Quantum Galois theory,which is an analogue of classic Galois theory in VOA theory, will also be presented.

**November 8, 2012**

**Z-Graded Weak Modules and Regularity**

**Z-Graded Weak Modules and Regularity**

**Nina Yu**

Rationality, C_{2}-cofiniteness and regularity are probably the three most important concepts in represenation theory of vertex operator algebras. In my talk I am going to prove that if any \mathbb{Z}-graded weak module for vertex operator algebra V is completely reducible, then V is regular. This gives a natural characterization of regular verex operator algebras.

**November 15, 2012**

**Floer Homology and the Arnold Conjecture**

**Floer Homology and the Arnold Conjecture**

**Gabriel Martins, UCSC Graduate Student, Mathematics **

On this talk I'll discuss a bit of the history of the Arnold Conjecture and illustrate the construction of the Floer complex which was the major technique used to establish it in the general case.

**November 22, 2012**

**No Colloquium - Happy Holidays!!!**

**December 6, 2012**

**Psuedo VOAs: Complex, But Still Semi-Simple**

**Psuedo VOAs: Complex, But Still Semi-Simple**

**Robert Laber **

A vertex operator algebra (VOA) is a certain kind of non-associative, non-commutative "algebra," together with a distinguished element omega, called the conformal vector. Associated to this element is a linear operator L(0) that is required to be semisimple with integral spectrum. A psuedo vertex operator algebra (PVOA) generalizes the notion of VOA by relaxing these requirements on L(0). For a suitably "nice" VOA V, there is a finite dimensional, semisimple associative algebra A(V) associated to V with the property that the simple A(V)-modules are in bijective correspondence with the simple admissible V-modules. In this talk, I will (briefly) present the formal definitions of VOAs and PVOAs, and I will show how one can obtain a family of PVOAs from a given VOA by "shifting" the conformal vector. I will also discuss my current project, which is to find an analog of the associative algebra A(V) for a PVOA.