The Department of Mathematics is hosting a regular seminar series. The goal of the seminar is to bring together students and faculty members to discuss interesting mathematics! Each speaker will present a topic of their choice with the intent that the talk will be accessible to undergraduate students. Typically a talk will be around 50 minutes, with 10 minutes reserved for questions. Talks may be a survey about a particular area of math, a tutorial about a specific topic, or an introduction to a research area with discussion about recent results. Everyone is welcome to attend!

If you are interested in speaking at the seminar, please contact one of the organizers, Cobus Swarts (jacobus.swarts (at) viu (dot) ca) and Melissa Huggan (melissa.huggan (at) viu (dot) ca).

Dates, times, and locations for upcoming seminars are below.

To be added to our mailing list, please email Melissa Huggan (melissa.huggan (at) viu (dot) ca).

## Upcoming Seminars

- April 12, 2024 at 4pm (Building 460, Room 324)
- Speaker:
**Matthias Hannesson** - Title:
**The Ternary Goldbach Conjecture: An Introduction to the Circle Method and the Large Sieve** - Abstract: The Goldbach conjecture states that every even number greater than 2 can be written as the sum of two primes. While an easy problem to state, it turns out to be one of the most infamously difficult problems in mathematics, and to this day remains unproven. The ternary Goldbach conjecture makes the weaker assertion that every odd number greater than 5 can be written as the sum of three primes. This ends up being a (relatively) easier statement to tackle, with much progress towards a proof being made in recent years. This talk will introduce the circle method, a way to transform the problem into the evaluation of an integral. This will lead to the large sieve, a technique used to estimate sums involving trigonometric functions.

## Past Seminars

- April 5, 2024 at 4pm (Building 460, Room 324)
- Speaker:
**Dr. Larissa Richards** - Title:
**The math behind the source apportionment of environmental pollutants** - Abstract: When chemical contaminants are detected in the environment they can come from multiple sources and their distributions can change over space and time. Identifying the contaminant sources and how much each source contributes to the contaminants measured at a given location is a key problem in environmental forensics known as source apportionment. Depending on the amount of chemical information known (e.g., the number of sources, their chemical composition) the methods used for source apportionment differ. In the best-case scenario, the number of sources and their chemical composition are well defined, and a chemical mass balance can be used to determine the contribution of each source at each location. Often, however, this level of information is not available, and a different approach is needed. In these cases, algorithms known as self-training receptor models are used. These models take as their input the chemical data from multiple locations and then ‘unmix’ the chemical data to determine how many sources are present, the chemical composition of each source, and the contribution of each source to each sample. I will discuss the math behind self-training receptor models, illustrating with examples from my PhD research where I apportioned the sources of volatile organic compounds measured in the air in Vancouver Island communities using data collected from a mobile lab.

- Speaker:
- March 8, 2024 at 4pm (Building 460, Room 324)
- Speaker:
**Matthew Debolt** - Title:
**The Slow Localization Game on Graphs** - Abstract: A robber is hiding in the vertices of a graph, and you send out a group of cops to capture them; each cop may probe a vertex to determine how far it is from the robber’s current hiding spot. If you can locate the robber using this information, you win the game. What is your strategy? How many cops do you need to guarantee victory? These questions motivate the study of Localization games on graphs. In this talk, we explore a novel game variant: Slow Localization. To situate ourselves, we introduce graph theory and the basic results of the Localization game literature. We will define the Localization number and Slow Localization number of graphs. To conclude, we will sketch the characterization of all graphs having Slow Localization number 1. This work results from a summer research project supervised by Dr. Melissa Huggan.

- Speaker:
- February 16, 2024 at 4pm (Building 460, Room 324)
- Speaker:
**Dr. Evangelia Aleiferi** - Title:
**A Category-Theoretical Approach to Mathematics** - Abstract: Traditionally, math students are introduced to foundational mathematics through set theory, which studies mathematical structures by considering them as collections of objects that satisfy various properties. The idea of an “element belonging to a set” is fundamental in set theory. On the contrary, category theory aims to study mathematical structures not by looking at their elements, but by observing how these structures interact with one another. That is, by focusing on the relationships between structures rather than what exists within them. In fact, during his undergraduate studies, F. W. Lawvere was convinced that he could do set theory without using the elements of a set - an idea which, at the moment, others thought was impossible. However, Lawvere proved them wrong through his PhD thesis (1963), after describing set theory solely by using functions and their compositions. Essentially, what Lawvere was talking about was the category of sets, one of the archetypical examples of category theory. In this talk, we will introduce the main definitions of category theory and we will explore a variety of examples, drawn from different mathematical areas. We will also talk about universality, and more specifically, the universal properties of categorical products.

- Speaker:
- February 2, 2024 at 4pm (Building 460, Room 324)
- Speaker:
**Dr. Cobus Swarts** - Title:
**The Eternal Chromatic Number of a Graph** - Abstract: Graph colouring problems are one of the central areas of research in graph theory. In this talk, we will first give an introduction to graph colouring and then discuss the Eternal Colouring Game. Given a graph, G, Alice starts the game by giving a proper colouring of the vertices of G with at most k colours. Bob then requests a vertex and Alice must recolour this vertex, maintaining a k-colouring of G. These requests and re-colourings continue on subsequent rounds of the game. Bob wins if after finitely many turns, Alice has no valid moves left. Alice wins otherwise. The
*eternal chromatic*number of G is the smallest k for which Alice can win the game on G. The aim of the talk is give an introduction to the ideas above and to compare the eternal chromatic number to the regular chromatic number of a graph.

- Speaker:
- December 6, 2023 at 4pm (Building 460, Room 324)
- Speaker:
**Dr. Melissa Huggan** - Title:
**The Damage Number of a Graph** - Abstract: In adversarial situations on networks, we often concern ourselves with minimizing resources required for neutralizing a threat. Here we consider a different parameter which addresses the situation where the aggressor is damaging each unique location they visit. Framed within the context of the game of Cops and Robbers on graphs, the robber tries to maximize the number of unique vertices they visit in order to maximize the damage to the graph, while the cops aim to minimize the damage by limiting the robber territory. This model was first introduced in 2019 by Cox and Sanaei. We build on their results. This is joint work with Margaret-Ellen Messinger and Amanda Porter.

- Speaker:
- October 25 at 4pm (Building 210, Room 270)
- Speaker:
**Dr. Heather Wiebe** - Title:
**The Mathematics of Computational Chemistry** - Abstract: With the increase in computer power over the past 50 years, simulation has become an increasingly important tool in a chemist’s toolbox. Simulations allow us to uncover molecular-level explanations for experimentally observed phenomena as well as make predictions about the properties of molecules that are either too dangerous or too expensive to study in the lab. Even though its application is in the field of chemistry, this incredibly useful tool is based on a solid mathematical foundation. In this seminar, I will discuss the mathematics that drives computational chemistry. This will include an introduction to quantum chemistry and the Schrödinger equation (an eigenvalue problem), as well as a discussion of the computational techniques and approximations that are used to solve it. The utility of these computational techniques will be illustrated by their application to real research questions.

- Speaker:
- September 27 at 4pm (Building 210, Room 270)
- Speaker:
**Matthew Debolt** - Title:
**The Abstract Nonsense of Category Theory** - Abstract: What can be said to exist
In the 19th and 20th centuries, significant mathematical advances were made by taking interest in things and their properties. Algebraic topologists Eilenberg and Mac Lane shifted their attention to the relationships between things in the 1940s, and the result was category theory. Owing to its extensive use of diagrammatic argument, it is endearingly referred to by practitioners as “abstract nonsense.”

We will tour the main ideas of category theory and consider their applications. A category is a thing which collects things and their relationships—the category’s objects and morphisms. It follows naturally to consider morphisms between categories, known as functors. Next, we consider morphisms between functors; these are natural transformations. Using this language, we can characterize objects without reference to their internal properties. Instead, we find universal properties of how certain objects relate to different objects. Time permitting, we may discuss the Yoneda Lemma and the dimension of categories.

- Speaker:
- October 4 at 4pm (Building 210, Room 270)
- Speaker:
**Dr. Dave Smith (Yale-NUS College, Singapore)** - Title:
**Linear algebra in initial boundary value problems** - Abstract: Initial boundary value problems and partial differential equations model many phenomena in physics, from conduction of heat to waves on the ocean, but present challenging mathematical problems. We will show how they can often be encoded as eigenvalue problems which are then much easier to solve. We will explore how derivatives can be seen as linear operators in infinite dimensional vector spaces, how they can be diagonalized, and how all this relates to solving initial boundary value problems. This will lead us to some recent research in solving initial boundary value problems with applications to water waves and diagonalizing differential operators.

- Speaker:
- April 14, 2023 at 3:30pm (Building 460, Room 324)
- Speaker:
**Dr. Gara Pruesse** - Title:
**Partially Ordered Sets and their hard, easy, and approximate algorithmic problems** - Abstract: We are frequently called upon to compare objects according to some metric, with one found to be greater and one lesser. Comparisons of the elements of a set result is a list of the elements in sorted order, each element less than the next, like the numbers 1; 2; 3; etc. Or so you would think. Some sets of objects, such as tennis players, may comprise a partial order: perhaps it is quite clear that Novak is a better player than Gara, but maybe Gara and Bette are “incomparable” in that one is not clearly better than the other. Partial orders arise naturally in many areas, in knowledge representation, preference rankings, operations research, and in scheduling, such as task scheduling for computer processors. There are journals dedicated just to partial order research, and other journals that publish only articles on scheduling.

This talk introduces partial orders and some of the interesting problems that arise from them. Some of the optimization problems on partial orders are computationally “hard” (NP-hard), but are nevertheless so useful that researchers focus on finding approximate solutions. The jump number problem is such a problem. Gara will present a classic result on jump number, with a new, elegant proof, as well as some other recent results that she will be presenting at the upcoming SIAM Conference on Applied and Computational Discrete Algorithms (ACDA ’23).

- Speaker:
- March 24, 2023 at 3:30pm (Building 460, Room 324)
- Speaker:
**Dr. Alex Fok** - Title:
**Fun with spheres** - Abstract: Among all closed and bounded surfaces, spheres are the simplest, yet the most symmetrical and occur most frequently in the nature. Since the time of Archimedes, they have been objects of great interest in mathematics. In this talk, we will take a stroll through some fascinating properties of spheres and use them as toy models to get a taste of some far-reaching results in topology, an area of mathematics dubbed the rubber-sheet geometry.

- Speaker:
March 17, 2023 at 3:30pm (Building 460, Room 324)

- Speaker:
**Dr. Melissa Huggan** - Title:
**Pursuit-Evasion Games: An Introduction** - Abstract: Pursuit-evasion games have been vastly studied over several decades by dozens of researchers worldwide. A central focus of this research has been on the game of Cops and Robbers. Player 1 controls a set of k cops, while Player 2 controls a robber. Player 1 chooses k (or fewer) vertices of a graph for the cops to occupy, then Player 2 chooses a different vertex for the robber to occupy. Players then alternate turns moving along edges of the graph. The goal of the cop player is to capture the robber by occupying the same vertex. The goal of the robber is to avoid capture indefinitely. This talk will take participants on a journey through key results from the literature of Cops and Robbers. We will conclude with a discussion of current research directions within the area of pursuit-evasion games.

- Speaker:
- February 17, 2023 at 3:30pm (Building 460, Room 324)
- Speaker:
**Dr. Cobus Swarts** - Title:
**The Complexity of (Di)graph Homomorphisms** - Abstract: In this talk we will be discussing the computational complexity of (di)graph homomorphisms. This refers to the ease/difficulty with which this problem can be solved on a computer. The talk will begin by discussing what graphs are and what it means to colour a graph. At this point we will discuss the computational complexity of graph colouring. We then show that graph colouring is a special case of graph homomorphisms and that there is really more to the colouring problem. We will then introduce graph homomorphisms and talk about their computational complexity. If there is time left, we will discuss digraph homomorphisms and their computational complexity.

- Speaker: