Winter 2024 Projects

Applications for Winter 2024 are closed. Selected students will be notified in about 10 days following the application deadline.

STABILITY ANALYSIS OF AN AGE-STRUCTURED MODEL
OF CANCER CELLS AND HPV DYNAMICS

Supervisor: Vitalii Akimenko

Level:  3000/4000

Prerequisites:  MATH 1300, MATH 1700, MATH 3440 

Programming Skills: MathType, LaTex, MS Equation 

Project Description: Stability analysis of an autonomous epidemic model of age-structured sub-classes of susceptible, infected, precancerous and cancer cells and unstructured sub-population of human papilloma virus (HPV) (SIPCV epidemic model) aims to gain an insight into the features of cervical cancer disease. We consider the initial-boundary value problem for system of semi-linear hyperbolic PDE. Cell population dynamics is described by the initial-boundary value problems for the semi-linear hyperbolic equations with age-dependent coefficients and HPV dynamics is described by the initial problem for nonlinear ODE. We study conditions of existence of the disease-free and the endemic equilibria of system and conditions of their local asymptotic stability. This study allows us to get: (a) the conditions of cancer tumor localization (asymptotically stable dynamical regimes), (b) outbreak of cancer cell population (that may correspond to metastasis), (c) outbreak of dysplasia (precancerous cells) which induces the outbreak of cancer cells (that may correspond to metastasis). In cases (b), (c) the conditions of existence of endemic equilibrium do not hold. It is preferred to illustrate all dynamical regimes obtained in stability analysis of cells-HPV population dynamics model by numerical experiments.

Keywords: SIPCV epidemic model. Age-structured model. System of semi-linear hyperbolic PDE. Stability analysis.

We expect further development of the results obtained in papers:

Expected Outcome: Journal/undergraduate journal paper

ELEMENTARY SYNTHETIC ORTHOGONALITY THEORY

Supervisor: Robert Craigen

Level: 2000+

Prerequisites: MATH 1240 (B or better), MATH 1220, 1300 or other Linear Algebra I (B or better), MATH 2020 or 2030 or 2400 or 2170 or approved course in axiomatic geometry (C or better)

Maturity: This project will be the equivalent of (at least) a 2000+ level "honours-level" course because of the strong reliance on the student understanding how ideas are pursued in mathematics and how to articulate and think through complex (though elementary) questions. Accordingly, candidates must show strong mathematical aptitude and an affinity for applying original thinking in solving abstract problems. For example experience with mathematical competition or significant recreational problem solving would be good indicators of this.

Programming Skills: Experience with programming language in either mathematical software such as Maple/Mathematica/Magma/similar or general coding experience in Python/C/similar; LaTeX proficiency desired but training will be provided in any case

Project Description: We are all familiar with how of the term "orthogonal" is used in geometry.  We have also seen it used, somewhat differently, in an abstract algebraic framework (such as in linear algebra).  And combinatorists speak of "orthogonal" latin squares and other such configurations.  What, at root, does "orthogonality" mean at its most basic level that unifies the use of this idea across such disparate settings? How does a concept of orthogonality arise from some first principles, and what exactly are those principles? The questions are abstract and may strike one as vague and philosophical.  Yet they can be addressed concretely and turn out to be rich, deeply practical and amenable to direct attack using elementary tools by any adventurous spirit and creative mind.

Fifteen years ago De Launey and Flannery, in their groundbreaking book Algebraic Design theory, introduced a purely combinatorial framework within which to understand combinatorial designs relative to what they call "orthogonality sets" in the most generality possible, without assuming combinatorial, algebraic or geometric ideas at the start.  Then they go on to develop an "Ambient ring" and pursue the topic through the lens of the resulting algebra and implications for designs.  I had proposed from the early days of their work on this subject that someone take their essential axiomatic framework and, rather than develop it in a specific desired setting, study its consequences in their natural generality, categorizing and understanding the nature of all the objects this theory canonizes as "orthogonal" in a systematic manner, remaining agnostic as to whether or not algebra per se is a necessary ambient setting for it. Over a couple of decades I have explored this idea and produced a small body of work, but it is clear that there is a huge amount of elementary case exploration their program has left left undone.

In this project we will pursue this challenge, building on some foundational work I have begun, drawing on little beyond combinatorial thinking, clearheaded logic, understanding of development of theory from axiomatic systems, creative thinking, and problem solving skills.  Some coding experience is almost certainly an asset and is strongly desired.  There are no advanced ideas in the setup of this project but the student’s maturity will determine their aptitude for it.

Expected Outcome: Presentation project, generation of a collection of elementary examples of specified object types, and contribution to a  journal article

ON STARS IN TREES AND STRONGLY REGULAR GRAPHS

Supervisor: Mahsa N. Shirazi

Level: 3000+

Prerequisites: MATH 2030, or MATH 2070, or MATH 3360, or MATH 3370

Programming Skills: LaTeX

Project Description: A family F of subsets of the set {1, 2, . . . , n} is called t-intersecting if any two sets in F have at least t elements in common. Erdős, Ko, and Rado started a research area by proving that if F is a t-intersecting family of k-subsets of {1, 2, . . . , n}, then (n-t)!/((k-t)!(n-k)!) is a tight upper bound on the size of F with n sufficiently large. Also they showed that if equality holds, then F consists of all k-subsets containing a fixed t-subset of {1, 2, . . . , n}. This is an active area and there have been interesting recent works on the analogs of EKR theorem on various mathematical objects. Such results would give the size and structure of the largest set of intersecting objects. One of the main goals of this project is to find extensions of the famous Erdős-Ko-Rado (EKR) theorem to 1-intersecting families of k-cocliques in a graph.

In graph theory, a coclique (independent set) in a graph X is a set of vertices in which no two are adjacent. A set of cocliques of size k (k-coclique) in X, containing a fixed vertex v is called a star, and is denoted by Ikx(v). By definition, any star is an intersecting set of k-cocliques, and they are related to k-EKR graphs which makes them interesting. A graph is k-EKR if no intersecting family of k-cocliques is larger than a largest star. So one can ask when, and for which vertex a star is of maximum size? In 2011, Hurbert and Kamat conjectured that in trees, the largest stars are centered at leaves. The conjecture was shown to be false independently by Baber, Borg, and Feghali, Johnson and Thomas. The next question is when a graph satisfies the Hurbert’s and Kamat’s conjecture or simply HK property? In 2016 and 2022, Hulbert and Kamat showed that Spiders and Pendant trees satisfy HK property. Also in 2021, Estrugo and Pastine proved that Caterpillars satisfy HK property, and lobsters almost satisfy HK property.

The goal of this project is to study some classes of graphs, mostly trees and strongly regular graphs to see when they satisfy HK property.

References:

Expected Outcome: Journal paper, or undergraduate journal paper