Fall 2013 Schedule

October 9

Speaker: Cindy Tsang

Title: Galois modules and realizable classes

Abstract: Let K be a number field with ring of integers O and G an abelian group. Given a Galois G-extension L, its ring of integers O_L and square root of the inverse different A_L/K are OG-modules. I will discuss how their module structure (global and local freeness) is related to the ramification of L/K. I will also talk about the group structure of the classes realizable by the O_L and A_L/K in the locally free class group Cl(OG), and if time allows, those realizable by the A_L/K in the unitary class group UCl(OG).

October 16

Speaker: Nathan Saritzky

Title: Cohomology and elementary school math

Abstract: Back in our youths, we learned an algorithm for adding numbers that involved "carrying" (as in "carry the 1"). To exhibit this, we can look at addition in Z_100 in terms of the "tens" place and the "ones" place. If we vary how we carry (e.g., carry a "2" whenever we would normally carry a "1"), we can get different groups. This will lead us into the theory of group extensions, and ultimately into cohomology of groups. (Reference: "A cohomological viewpoint on elementary school arithmetic" by Daniel C. Isaksen, The American Mathematical Monthly, Vol. 109, No.9. (Nov., 2002), pp. 796-805)

October 23

Speaker: Julia Galstad

Title: Modules over finite dimensional algebras Part 1

Abstract: This will be an introduction to thinking about modules over finite dimensional algebras using graphical methods. Modules over certain rings are familiar objects to most math students. For example, modules over the integers are abelian groups and modules over a field k are k-vector spaces. This talk will emphasize some techniques of thinking of modules that are generalizations of thinking about groups and vector spaces, so we can feel related to the objects in question. These techniques will also be related to graphs of modules, which we will motivate a definition for and give plenty of examples.

October 30

Speaker: Julia Galstad

Title: Modules over finite dimensional algebras Part 2

Abstract: Continuation of the talk from last week.

November 6

Speaker: Joseph Ricii

Title: Quandles

Abstract: A quandle is a set with a binary operation whose axioms correspond to the Reidemeister moves for planar knot diagrams. Quandles arise in the definitions of certain knot invariants but are interesting in a purely algebraic way as well. Many examples of quandles will come from group and module theory.

November 13

Speaker: Kevin Lui

Title: Introduction to Category Theory

Abstract: It is often the case in mathematics that many similar concepts and ideas appear in different forms in different branches of mathematics. For example, in group theory, two groups are isomorphic if there exists a bijective map that preservers group structure; similarly in topology, two topological spaces are homeomorphic if there is a bijective map preserving the topology in both directions. Category theory provides a language for us to describe these similarity in a more abstract setting. In this talk, I will give an introduction to the concepts of category theory and provide some examples. And if time permits, we'll talk about a applications of category to Physics and Computer Science.

November 20

Speaker: Matthew Tucker-Simmons

Title: Monoids and Monoidal Categories

Abstract: I am going to talk about monoids and groups, and how to make sense of these ideas in arbitrary categories. This will lead to the notion of monoidal categories. The most familiar example is probably the category of vector spaces over a field, but there are others. I hope to discuss Maclane's Coherence Theorem and the Eckmann-Hilton argument, time permitting.

November 27

No seminar

December 4

Speaker: David Wen

Title: Fuzzy Sets and Groups and Cosets. Oh my!

Abstract: Currently set theory lies at the foundation of mathematics, but the binary aspects of sets are at times inadequate in modeling the nuances of real life. An example would be a bottle half full, while it is not in the set of bottles that are full, we have that to some degree it does fulfill the criteria of the set. This inadequacy of set theory led to the development of Fuzzy set theory which captures degree of membership of an element to a set and just as set theory corresponds to mathematics as we know it, fuzzy set theory corresponds to fuzzy mathematics. In this talk I will introduce Fuzzy sets, Fuzzy groups and prove fuzzy versions of some familiar theorems of Group Theory.