Fall 2019

Speaker: Alex Semendinger
Title: An Incomplete Proof of Gödel's First Incompleteness Theorem
Abstract: Gödel's Incompleteness Theorems famously state that (1) any formal system powerful enough to perform arithmetic contains statements which cannot be proved or disproved within the system, and (2) any such system cannot be used to prove its own consistency. But what kind of black magic does it take to prove something like this? We will find out as we go through the main ideas of the proof of the first theorem, blithely glossing over the (copious) fiddly details.

Speaker: Mac Krumpak
Title: Gauge theory and topology
Abstract: I will discuss the general ideas of gauge theory and show some of its historical applications to questions in topology.

Speaker: Ian Montague
Title: Applications of Symplectic Geometry to Topology
Abstract: As a natural continuation of Mac's talk last week, I will discuss the general ideas of symplectic geometry and show some of its historical applications to questions in topology.

Speaker: Ray Maresca
Title: Quiver Representations in Matlab
Abstract: Often times pure and applied mathematics are interconnected in the sense that one can induce questions or help break barriers in the other. The main goal of this presentation is to explore the interconnectedness between coding algorithms in Matlab and the representation theory of quivers.

Speaker: Job Rock
Title: What is a representation?
Abstract: We'll explore a few different definitions of representation including a representation of a: group, quiver, and poset. We will then highlight the underlying structure they all have in common and present a general definition.

Speaker: Gary Dunkerley
Title: Introducing the Grassmannian
Abstract: The Grassmannian is a fundamental object in algebraic geometry, housing within several classical algebraic varieties of note. The speaker will present a few characterizations and some motivation for its study.

Speaker: Tianyu Ni
Title: The Weil conjectures for elliptic curves
Abstract: Weil conjectures were some influential proposals by Andre Weil, which led to the development of modern algebraic geometry and number theory. The case of elliptic curves is the first nontrivial example. There are three sections in this talk. First, we will introduce the definition of arithmetic zeta functions and the statement of Weil conjectures. We will see how to treat the problem “counting Fq-points of a variety'' as “counting the fixed points of 1-\phi'', where \phi is the Frobenius morphism. Second, we will briefly explain why counting fixed points is a relatively easy problem for elliptic curves. The essential tool is the Tate modules obtained from the geometric properties of the elliptic curve as an abelian variety. Finally, we will give the proof if time permits. There are no prerequisites for this talk except some basic definitions such as affine variety and projective variety.

Speaker: Rose Morris-Wright
Title: Generalizing hyperbolicity and what it can tell us about groups.
Abstract: The general principle behind geometric group theory is as follows. Start with an infinite order, but finitely presented group and construct a metric space from this group. If this metric space has certain hyperbolic like qualities then we can learn a lot about the the algebraic properties of the group. This talk will formally define a hyperbolic group and discuss some examples of questions that can be answered with this framework. If there's time I may also discuss further generalizations such as relative or acylindrical hyperbolicity.