Research

Graduate Seminar Series
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Friday, November 17, 2023

Title: A Search to Decision Reduction for Code Equivalence
Speaker: Phillip Waitkevich
Time: 3:00pm–4:00pm
Place: CMC 108

Abstract

Finding an algorithm that efficiently determines if two codes are equivalent is an open problem in coding theory. It is important to know whether such an algorithm exists because if it does not, we can use code equivalence as a way to safely encrypt information, potentially even against quantum computers. In this talk, I will show a way to efficiently construct an equivalence between 2 codes by asking an oracle multiple yes or no questions. As a disclaimer, this process does not yet solve the code equivalence problem, but instead reduces it to a different problem. No prior knowledge of coding theory is necessary for this talk, but basic linear algebra (basis, vectors, matrices) is recommended.

Friday, November 10, 2023

Due to the Veteran's Day holiday, there will be no seminar this week.

Friday, November 3, 2023

Title: Modules over Commutative Rings
Speaker: Zachary Dorson
Time: 3:00pm–4:00pm
Place: CMC 108

Abstract

Basic introduction to modules over commutative rings, and some similarities and differences between modules and vector spaces.

Friday, October 27, 2023

Title: Eventual Stability of Pure Polynomials over the Rational Field
Speaker: Mohamed Osama Hafez Darwish Mohamed
Time: 3:00pm–4:00pm
Place: CMC 108

Abstract

A (discrete) dynamical system \((A,\phi)\) is a set \(A\) together with a self-map \(\phi:A\to A\). The \(n\)th-iterate of the map \(\phi\) is defined by \[ \phi^n:=\underbrace{\phi\circ\phi\ldots\circ\phi}_\text{$n$-times} \] Conventionally, \(\phi^0\) is the identity map on \(A\).

A polynomial with rational coefficients is said to be pure with respect to a rational prime \(p\) if its Newton polygon has one slope. In this talk, we prove that the number of irreducible factors of the \(n\)-th iterate of a pure polynomial over the rational field \(\mathbb{Q}\) is bounded independent of \(n\). In other words, we show that pure polynomials are eventually stable. Consequently, several eventual stability results available in literature follow; including the eventual stability of the polynomial \(x^d+c\in\mathbb{Q}[x]\), where \(c\ne 0,1\), is not a reciprocal of an integer. In addition, we establish the dynamical irreducibility, i.e., the irreducibility of all iterates, of a subfamily of pure polynomials, namely Dumas polynomials with respect to a rational prime \(p\) under a mild condition on the degree. This provides iterative techniques to produce irreducible polynomials in \(\mathbb{Q}[x]\) by composing pure polynomials of different degrees. In addition, we characterize all polynomials whose degrees are large enough that are not pure, yet they possess pure iterates. This implies the existence of polynomials in \(\mathbb{Z}[x]\) whose shifts are all dynamically irreducible.

This is a joint work with Mohammad Sadek (Sabanci University).

Friday, October 20, 2023

Title: Hackenbush, A Window into Combinatorial Game Theory, Part II
Speaker: Logan Nott
Time: 3:00pm–4:00pm
Place: CMC 108

Friday, October 13, 2023

Title: Hackenbush, A Window into Combinatorial Game Theory
Speaker: Logan Nott
Time: 3:00pm–4:00pm
Place: CMC 108

Abstract

Combinatorial game theory is a branch of mathematics that studies turn-based games of perfect information, partisan and non-partisan alike.

For the sake of a simple introduction, we will restrict ourselves primarily to the game of Hackenbush, which is a game played on a graph of coloured edges in any configuration, such that all edges are connected to a 'ground.'

Technicalities aside, we shall look for and find: partisan and non-partisan variants; an infinite field of characteristic two; numbers large and small and in-between; and new numbers that stretch the very definition of what it means to be a number.

Friday, October 6, 2023

Title: Category Theory, the Conclusion
Speaker: Bryce Virgin
Time: 3:00pm–4:00pm
Place: CMC 108

Friday, September 29, 2023

Title: An Introduction to Category Theory, Part II
Speaker: Bryce Virgin
Time: 2:00pm–3:00pm
Place: CMC 108

Friday, September 22, 2023

Title: An Introduction to Category Theory
Speaker: Bryce Virgin
Time: 2:00pm–3:00pm
Place: CMC 108

Abstract

Category theory is a field of mathematics whose development underlies many important mathematical discoveries of the last century. It, like set theory, is both a language used to define and investigate other mathematical objects, as well as an object of study in its own right. However, quite unlike set theory, one will rarely find an introductory course on category theory in the catalogue of their university math department. This talk will be an introduction to the primary definitions of the subject, with an emphasis on how many seemingly disparate ideas in different subjects can all be unified from the categorical perspective. An understanding of basic set theory is all that will be necessary to follow the talk, but some knowledge of abstract algebra and topology are recommended.

Friday, September 15, 2023

Title: Painlevé Equations and Isomonodromic Deformation Method, Part II
Speaker: Louis Arenas
Time: 2:00pm–3:00pm
Place: CMC 013

Friday, September 8, 2023

Title: Painlevé Equations and Isomonodromic Deformation Method
Speaker: Louis Arenas
Time: 2:00pm–3:00pm
Place: CMC 013

Abstract

From solutions to percussions on a drum to field intensities of light passing through an opaque object, special functions dominated the attention of 19th-century physicists. Today, special functions have an unavoidable presence in several applications in Math, Physics, and Engineering. Almost simultaneously in the early 20th century, Paul Painlevé and his school completed the classification of a special class of second-order non-linear differential equations. It was during this classification that Painlevé and his school landed on six second-order ODEs whose solutions are not “solvable” in terms of elementary functions. The solutions to these six ODEs were later characterized in the late 1970s as nonlinear special functions due to their roles in describing nonlinear PDEs like KdV, Sine-Gordon, and the Non-linear Schrödinger equation.

In the early 1980s, building on the ideas of Richard Fuchs and Ludwig Schlesinger, two groups of Mathematicians and Physicists created a technique that would eventually give “connections” and asymptotic formulas for the Painlevé equations and associated PDEs. This technique is called the Isomonodromic Deformation Method (or IDM).

The goal of this series of talks is to introduce IDM for the purposes of analyzing the asymptotics of a few Painlevé equations. A good portion of the talk will be spent on reviewing known results of systems of ODEs with rational coefficients; for the audience members studying complex analysis, some important ideas from the subject will be reiterated in the first talk. The talks aim to be as self-contained as possible; no tremendous background is needed.