Alejandro Guillen’s PhD defense
May 1 @ 1:00 pm – May 1 @ 3:00 pm

Keller Hall 403

Dissertation draft (department only)

Alejandro Guillen, PhD defense, Tuesday, May 1, 2018, 12 noon,
        Title:  On the Generalized Word Problem for Finitely Presented Lattices”

        Abstract: The generalized word problem for a lattice L in a variety V asks if, given a finite
        subset Y of L and an element d in L, there is an algorithm to determine if d is in the subalgebra 
        of L generated by Y. Freese and Nation showed that the generalized word problem for finitely 
        presented lattices is solvable. This algorithm, though effective, is potentially exponential. We 
        present a polynomial time algorithm for the generalized word problem for free lattices, but 
        explain the complications which can arise when trying to adapt this algorithm to the generalized 
        word problem for finitely presented lattices. Though some of the results for free lattices are 
        shown to transfer over for finitely presented lattices, we give a potential syntactic algorithm 
        for the generalized word problem for finitely presented lattices. Finally, we give a new proof 
        that the generalized word problem for finitely presented lattices is solvable, relying on the 
        partial completion, PC(P), of a partially defined lattice P.

Kyle Dailey, Master’s presentation
May 4 @ 1:00 pm – May 4 @ 2:00 pm

Draft of Master’s report

Kyle Dailey, Master’s presentation, 
Friday, May 4, 2018, 1:00, Keller 404
Title:  Rahman Polynomials and Lie Algebra Representations

Abstract:  In this paper, we examine a close connection between a certain class of orthogonal polynomials and Lie algebra representations. We use a specific inner product to observe how polynomials arise within the structure of representations of special linear algebras. We also use this inner product to give proofs of properties of the polynomials, such as orthogonality and recurrence relations. Our paper is guided by previous results on the topic, in this paper we outline a new approach to  achieve these results, as well as the difference between our approach and previous papers on the topic. 

Colloquium: Christian Pötzsche (Alpen-Adria Universität Klagenfurt, Austria) @ Keller 401
May 9 @ 3:30 pm – 4:30 pm

Title: Nonautonomous Dynamics

Christian Pötzsche
Alpen-Adria Universität Klagenfurt, Austria

The theory of dynamical systems has seen a remarkable progress over the last 100 years, beginning with the contributions of Poincaré and Lyapunov to a contemporary detailed understanding of the attractor for various infinite-dimensional systems. This success is partly due to the restriction to autonomous systems. However, many real-world problems are actually nonautonomous. That is, they involve time-dependent parameters, controls, modulation and various other effects. Special cases include periodically or almost periodically forced systems, but in principle the time dependence can be arbitrary. As a consequence, many of the now well-established concepts, methods and results for autonomous systems are no longer applicable and require appropriate extensions.

We discuss several basic ingredients from the theory of nonautonomous dynamical systems. Among them are (pullback) convergence, the dichotomy spectrum (to indicate stability) and approaches to understand nonautonomous bifurcations.

Number Theory Seminar: Jacob Tsimerman (Toronto) @ Keller 403
May 17 @ 2:00 pm – 3:00 pm

Title: Transcendence results and applications in number theory

Abstract: In a pioneering paper, Pila and Zannier showed how one can prove arithmetic results (the Manin–Mumford Conjecture) using transcendental methods (the Ax–Lindemann conjecture). Their approach has since been greatly developed, and is a major ingredient in the Andre-Oort conjecture for Shimura varieties as well as the more general Zilber–Pink conjecture, that serves as a sort of flagship for the field of unlikely intersections. We’ll explain this story, focusing on the classical case of (C^times)^n and the transcendence of the exponential function.

Algebras and Lattices in Hawaii
May 22 – May 24 all-day
Dissertation defense – TJ Combs
Jul 6 @ 2:00 pm – 4:00 pm

Time: Friday, July 6 from 2:00 – 4:00 pm
Location: Keller 401

Draft of dissertation

Abstract: \
We provide some general tools that can be used for polynomials in any degree to show $G_\infty = \text{Aut}(T_\infty)$.  We introduce the idea of Newton irreducibility to help push us closer to a proof to Odoni’s conjecture for monic integer polynomials when $d=4$.  We also show that current techniques used in the literature will not work in proving Odoni’s conjecture for monic quartic polynomials.  Finally, we look at how certain behaviors of the critical points of a polynomial $f(x) \in \mathbb{Q}[x]$ force it to not have full Galois image.

Dissertation defense, John Robertson @ Keller 302
Jul 19 @ 3:00 pm – 4:30 pm

Draft of dissertation:

In this paper we study properties of groupoids by looking at their $C^∗$-algebras. We introduce a notion of rapid decay for transformation groupoids and we show that this is equivalent to the underlying group having the property of rapid decay. We show that our definition is equivalent to a number of other properties which are in direct correspondence to the group case. Additionally, given two bilipschitz equivalent discrete groups we construct an isomorphism of the corresponding transformation groupoids and are able to reformulate the open problem of showing invariance of rapid decay under quasi-isometry.
We then begin to examine various notions of amenability when abstracted to measured ́etale groupoids. In the group case, the following properties are equivalent:
1) $G$ is amenable
2) $C_r^∗(G)=C^∗(G)$
3) The trivial representation decends from ^C^∗(G)$ to $C_r^∗(G)$.
In the groupoid, $G$, case we have 1) ⇒ 2) ⇒ 3), but it was shown by Rufus Willett that $C_r^∗(G) = C^∗(G)$ is not enough in general to give amenability of G. In this paper we study property 3) for groupoids, formulate some equivalent statements and show that 3) ⇒ 2) is also false in general.

AMS Spring Central and Western Joint Sectional Meeting
Mar 22 – Mar 24 all-day