Date |
## Spring 2016 Schedule |

April 7 |
Abstract: Let G be a group. A G-module is a module (over the integers usually), equipped with an action of G. This generalizes the familiar notion of a linear representation of G. The study of the category of G-modules leads very naturally to group cohomology: the nth group cohomology of G is a functor from the category of G-modules to the category of abelian groups. After defining group cohomology precisely, I will present several ways of thinking about them and computing them. Finally, I will present some important special cases of group cohomology that come up frequently, including Galois cohomology. |

April 14 |
Abstract: Riemann-Roch originally was a theorem of compact Riemann surfaces constraining rings of functions on the space by a topological invariant, the genus. Further generalizations eventually led to a far reaching deep abstraction called the Grothendieck-Riemann-Roch theorem. Grothendieck's approach was to change the narrative into a theorem not of spaces but of the morphisms between them, and to abstract the argument into what is essentially the beginnings of K-theory. This talk will be an exposition on setting up to understand the Grothendieck-Riemann-Roch theorem and it's reduction to the other ''Riemann-Roch'' theorems. |

April 21 |
Abstract: My talk will aim to answer those two questions. After presenting the classical statement of the theorem, I will outline some of the various ways the theorem has been generalized. I will then present an application to arithmetic geometry: specifically, I will use classical quadratic reciprocity to prove that the curve 2y^2 = x^4 - 17 violates the Hasse principle. |

April 28 |
Abstract: It is known that products exist in the category of Witt Rings. An important question is 'when does a given Witt Ring decompose into a direct product of other Witt Rings?' My talk will describe what the product in the category of Witt Rings is, and what it means to 'factor' a Witt Ring into it's direct factors. We will explore what it means to be an 'elementary indecomposable Witt Ring' and how these rings play a role in the general theory. We will define what it means for a Witt Ring to be of so-called elementary type, and conclude with the Elementary Type Conjecture: 'Is every Witt Ring of Elementary Type?'" |

May 5 |
Abstract: Let L/K be a Galois extension of number fields with group G. A normal basis of L over K is a basis of the form {\sigma(x):\sigma\in G}. Such a basis always exists by the normal basis theorem. It is then natural to ask whether the ring of integers O_L in L also has a basis of this form over the ring of integers O_K in K. This is a classical problem in number theory. More recently, instead of O_L, people have also looked at this question for other G-invariant ideals in L, one of which is called the square root of the inverse different A_{L/K}. This ideal is special because it is self-dual with respect to the trace, and one can ask whether it admits a self-dual normal basis {\sigma(x):\sigma\in G} over O_K. I will explain what all these mean in more detail and present some new results. |

May 12 |
Abstract: We introduce coalgebras, bialgebras, and Hopf Algebras and discuss the way they arise in the study of quantum groups and the search for solutions to the Yangâ€“Baxter Equation. |

May 19 |
Abstract: Let G be a finite group with generating set A and let d denote the diameter of the corresponding directed Cayley graph \Gamma. We are interested in the following question -- Is there a fast algorithm that finds a path in \Gamma whose length is roughly of the same order as d? This is the so-called navigation problem. In this talk, we will consider the group F_p with a fixed element \lambda\in F_p* and a slightly modified directed graph \Gamma. More specifically, the edges are now (y,y+1) and (y,\lambda y). I will show that if \lambda^2-b\lambda-1=0 for some positive integer b, then using generalized Fibonacci numbers, one can explicitly construct a path in \Gamma of length O(log p). This is a problem that a few other graduate students and I worked on at the Arizona Winter School this year. |

May 26 |
Abstract: This talk will include a short introduction to group cohomology, induced modules and Shapiro's lemma, followed by some computations and examples, including the long exact sequence (from the snake lemma) that results from a particular short exact sequence of G-modules. |

June 2 |
Abstract: This talk is an introduction to derived categories. I will go over the motivation for a derived category, explain the construction, and spend the rest of the time discussing open problems in algebraic geometry and number theory that are related to derived categories. |