Algebra Qualifying Exam
Apr 24 @ 9:00 am – 1:00 pm
Logic seminar: David Webb
Apr 27 @ 2:30 pm – 3:30 pm
Colloquium: Eliot Fried (Okinawa Institute of Science and Technology) @ Keller 401
Apr 27 @ 3:30 pm – 4:30 pm
Speaker: Eliot Fried (Okinawa Institute of Science and Technology)

Title: Kaleidocycles and Möbius bands

Abstract: Many of Escher’s works have become mainstays of popular culture. Famous examples include his kaleidocycles, each consisting of six identical regular tetrahedra and being capable of undergoing a cyclic everting motion that brings different tesselations of the tetrahedra into view. Esher also provided memorable interpretations of Möbius bands. We will consider kaleidocycles made from seven or more identical twisted tetrahedra, or disphenoids, and expose a deep, and to our knowledge, previously unnoticed connection between kaleidocycles and the 3π-twist Möbius band.

Elliot Ossanna, Master’s presentation
Apr 30 @ 10:30 am – 11:30 am

Elliot Ossanna, Master’s presentation, Monday, April 30, 2018, 10:30 am, Keller 403

Fractal nature of generalized binomial triangles modulo $p$

Abstract: A well-known property of Pascal’s Triangle is that reducing entries modulo a prime yields a fractal, Sierpinski’s Triangle-like pattern. We generalize this to triangles generated by strong divisibility Lucas Sequences, and conclude that the generated fractal is uniquely determined by the prime modulus, not the underlying generating sequence.

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.