# Algebra and Number Theory Seminar Spring 2017

**April 7, 2017
No Seminar
**

**April 14, 2017
**

**Rob Carman, UCSC**

**Title: Unit Groups of Trivial Source Rings as a Biset Functor**

**Abstract: The theory of biset functors has been instrumental in studying the unit group of the Burnside ring of a finite group. I will explain a notion of tensor induction for modules of a group algebra and show how this can be used to define the biset functor structure for the unit group of the trivial source ring of a finite group.**

**April 21, 2017
**

**Markus Linckelmann, City University, London**

**Title: On Triangulated Categories in Representation Theory**

**Abstract: There are two types of triangulated categories which arise routinely in modular representation theory - derived categories, and stable module categories.
Derived categories determine many fundamental numerical invariants of algebras, such as the number of isomorphism classes of simple modules and irreducible characters.
By contrast, it is not known whether stable module categories determine any of the above mentioned numerical invariants in general. This is one of the major obstacles in modular representation theory. Unlike derived categories, stable module categories need not have any t-structures, and hence their stability spaces in the sense of Bridgeland may be empty. Still, a stable module category may have many abelian subcategories whose exact structure is compatible with the triangulated structure, and whose numerical invariants are, in some cases, those of the original algebra. We will describe a simple construction principle for abelian subcategories of stable module categories. **

**April 28, 2017
**

**Radha Kessar, City University, London**

**Title: Scalar symmetric algebras**

**Abstract: It is well known that the degree of any irreducible character of a finite group G divides the order of G. If one works over a field of characteristic p, then the dimension of any projective module is known to be divisible by the p-part of the order of G. I will explain how these and other arithmetic properties of group representations carry over to a larger class of symmetric algebras, called scalar symmetric algebras. Examples of scalar symmetric algebras include source algebras of blocks of finite groups and certain types of finite dimensional Hopf algebras (including group algebras). This is joint work with Florian Eisele, Michael Geline and Markus Linckelmann.**

**May 5, 2017
No Seminar**

**May 12, 2017
**

**Ander Steele, UC Santa Cruz**

**Title: ****Computing pairings on modular symbols**

**Abstract: Modular symbols give an explicit and computable method for understanding spaces of modular forms through group cohomology. In this talk, we will explain how to explicitly compute a bilinear pairing on spaces of modular symbols--this pairing corresponds to the usual Petersson inner product on modular forms. We will describe applications of this explicit pairing to computing L-values, congruence numbers, and understanding the geometry of the eigencurve.**

**May 19, 2017
No Seminar
**

**May 26, 2017
No Seminar
**

**June** **2, 2017**

**No Seminar**

**June 9, 2017
**

**Junecue Suh, UC Santa Cruz**

**Title: **To which algebraic tori can abelian varieties degenerate?

**Abstract: We will answer the question and 2 refinements. It is based on joint work with Kai-Wen Lan.**

**June 16, 2017
TBA
**

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