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.
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.
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.
Title: Nonautonomous Dynamics
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.
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.