Fall 2017 Schedule

October 13

Speaker: David Nguyen

Title: Sheaf Cohomology

Abstract: It has been said that "cohomology theory is applied mathematics: something to be appreciated and actively used more than studied and analyzed in its own right." In this talk, I will discuss what sheaf cohomology is and how it is used. The talk is meant to be expositional and is for those who are new to sheaves and cohomology, like me.

October 20

Speaker: Nadir Hajouji

Title: The Twisting Principle

Abstract: If X is some object defined over some field k, then a twist of X is another object Y defined over k, such that X and Y become isomorphic when we extend scalars to the separable closure of k. The problem of classifying all twists of a given object is often equivalent to computing a single Galois cohomology group. I will show how one can compute these cohomology groups in certain cases, and discuss some applications to number theory and algebraic geometry.

October 27

Speaker: David Wen

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

Abstract: Fuzzy mathematics was first formulated by Lotfi Zadeh in 1965 as a means to bridge the gap between the binary aspects of logic and set theory to the inherent vagueness of the real world. A mug that is half full would not be contained in the set of all mugs that are full, but it does satisfy some degree of membership of being in the set, just not quite there. Zadeh formulated these notion more rigorously and what resulted is fuzzy mathematics and it's applications resonate across many different fields of engineering and sciences. For this talk, I will introduce fuzzy sets, fuzzy groups and some fuzzy analogs of well known theorems of group theory and highlight some of the differences between group theory and it's fuzzy counterpart.

November 3

Speaker: Zach Blumenstein

Title: K3 surfaces via lattices

Abstract: A K3 surface is a simply-connected nonsingular complex surface with trivial canonical bundle. The intersection product turns H^2(X,Z) into a lattice, and every two such lattices are isometric. The positions of certain substructures inside H^2 give an enormous amount of information about the K3 surface. One notable example is the Global Torelli Theorem, which says that if H^2(X,Z) -> H^2(X',Z) is an isomorphism preserving certain substructures, it must be induced by a unique isomorphism X' -> X. We review this and other lattice-theoretic results.

November 10

No Seminar

November 17

Speaker: David Wen

Title: Schemes, Un-Sheaf-ed

Abstract: This will be somewhat of a follow up talk to David Nguyen's previous talk on sheaf cohomology. In that talk, David described a sheaf as a means to translating geometry into algebra. This talk will deal with the opposite direction of going from algebra to geometry via schemes. A scheme, as a mathematical object, is defined via algebra but what it "is" is a very deep question that wraps complex geometry, classical algebraic geometry and number theory into a (relative) nice package. It is because of this that the theory of schemes is usually the first foray into modern algebraic geometry.

November 24

No Seminar

December 1

Speaker: Nathan Schley

Title: Introduction to the Witt Ring

Abstract: I understand that many in the audience have not yet seen the Witt ring, so this talk will be an introduction to the Witt ring over a field with characteristic other than 2, which can be built from isometry classes of quadratic forms.

December 8

No Seminar