# Seminar

This seminar is intended for an undergraduate audience. If you have an interesting talk that would be suitable for undergrads, feel free to contact the organizers and volunteer as a speaker. We especially appreciate talks by undergrads for undergrads!

A large part of mathematical culture happens in seminars, where ideas are discussed informally and with a healthy dose of friendly laughter. There are no grades, evaluations, or judgement. Before the COVID-19 Pandemic, we had pizza!

Seminar runs Wednesdays 14-15:00 (EST) on Zoom. We start seminar at 14:05, to let people arrive and get settled.

After attending seminar, we ask that participants fill out this feedback form. If you are not able to use Google Forms and would still like to give feedback, please e-mail the organizers.

The *prime factorization* of the Meeting ID is: $11 \times 59 \times
124877917$. The passcode for the meeting is: seminar. If you have trouble
accessing the seminar, please contact the co-organizers: Parker Glynn-Adey and
Lisa Jeffery. We will send you a direct
link.

At the bottom of this page, there FAQ for speakers.

Date | Speaker | Title | Notes |
---|---|---|---|

May 26 | Brian Li | Cops and Robbers with Many Variants | video / slides |

June 2 | Parker Glynn-Adey | The Probabilistic Method | video / slides |

June 9 | Ben Chislett | Ray Tracing and the Light Transport Equation | video / slides |

June 16 | Jesse Maltese | An Introduction to Mathematical Logic | video / slides |

June 23 | David Schrittesser | Your life will be better with infinitesimals (Part 1) | video |

June 30 | David Schrittesser | Your life will be better with infinitesimals (Part 2) | Cancelled |

July 7 | Yuveshen Mooroogen | Shining a rainbow-coloured light on the fundamental theorem of algebra | video / slides |

July 14 | Albert Lai | Partial orders and application to semantics of computer programs | video / slides |

July 21 | Kevin Santos | An Introduction to Group Theory through Puzzles | video / slides |

July 28 | Rakan Omar (York) | Influence Centrality | video / slides |

August 4 | Andrew Fallone (York) | The Projected Number of Underreported COVID 19 Cases in Canada | No slides or recording available, due to technical issues |

August 11 | Julie Midroni | Artificial neural networks: The fundamentals | video / slides |

August 18 | Jiawei Chen, Ran Li and Junru Lin | Rapid Testing in COVID and Modified SIR Model | video / slides |

Friday August 20 |
Richard Ye and Xin Ya Xu | Hero Rats - Detecting Tuberculosis | video / slides |

August 25 | Amalrose Vayalinkal | Virtual Ring Routing | video / slides |

We are listed on ResearchSeminars.org as UndergraduateSeminar.

## Speakers

### August 25: Amalrose Vayalinkal: Virtual Ring Routing

*Abstract*:

Virtual Ring Routing (VRR) schemes define a routing algorithm for communication between devices by establishing a virtual network overlay given a physical network of $N$ devices. Using graphs to model the physical network, we introduce the algorithm and explore the pros and cons of VRR. Time permitting, we take a closer look at the simpler case where the physical network is also a ring (circle) and discuss future directions. This work is part of an NSERC USRA this summer under the supervision of Professor Almut Burchard.

**FRIDAY** August 20: Xin Ya Xu and Richard Ye: Hero Rats - Detecting Tuberculosis

*Abstract*:

Tuberculosis has been the leading infectious killer in the world with an infection rate of approximately 10 million people per year. Slow and inaccurate detection methods contribute heavily to making tuberculosis so deadly. In this talk, we will highlight the efforts of APOPO, a non-profit organization that trains scent-detecting African giant pouched rats, to tackle tuberculosis. We will also talk about our approach to improve APOPO’s scent detection technology by using statistical analysis. We will explore the process of applying clustering techniques, such as hierarchical clustering and K-means clustering, to the problem.

Work by Xin Ya Xu and Richard Ye, supervised by Professor Sohee Kang and Professor Marco Pollanen. This is a project where the data was brought to us by Professor Pollanen who works with APOPO, a firm focused on detecting land mines and tuberculosis mainly in sub-saharan Africa.

### August 18: Jiawei Chen, Ran Li and Junru Lin: Rapid Testing in COVID and Modified SIR Model

*Abstract*:

The traditional PCR test for coronavirus typically takes one day to produce results, which cannot satisfy the great volume of demand for testing. However, the recently developed “Rapid Test” (which detects Covid antigen) can help us obtain reasonably accurate results in about 15 minutes. In this talk, we will use a modified SIR model (a classical ODE system for modelling epidemic) to demonstrate how the dynamics of Covid-19 can be affected under the introduction of this new testing technique. We will then discuss how effective the test needs to be in order to control the pandemic.

### August 11: Julie Midroni: Artificial neural networks: The fundamentals

*Abstract*:

Artificial neural networks (ANNs) are a powerful class of machine learning algorithms that can be used for a variety of purposes, including image classification, function approximation, and natural language processing. This talk will present the mathematical basics of ANNs, and briefly explore different ANN algorithms and their uses.

### August 4: Andrew Fallone: The Projected Number of Underreported COVID 19 Cases in Canada

*Abstract*:

As Canada focuses more on mitigation strategies rather than eradication ones, mathematical modelling could play an important role in preserving lives. The model used in this project is a modified SIR model which retains a conservative calculation for COVID 19 cases while giving an insight into how unreported COVID 19 cases are in Canada with the assumption that we have a limited amount of data. This approach seems to be the most effective due to the uncertainty of COVID 19 progression in Canada. Furthermore, this modified SIR model uses a basic reproduction parameter “to estimate the attack rate, epidemic duration”, and critical points of COVID 19 cases in Canada.

### July 28: Rakan Omar: Influence Centrality

*Abstract*:

In graph theory and network analysis, the notion of centrality refers to assigning nodes in a network an index representing the extent to which each node is central - important to the network, or well positioned in it - based on some mathematical property. There are a variety of measures of centrality, each of which measures ‘centrality’ differently, based on (or resulting in) a different definition of ‘importance’ or ‘prominence’.

I will define an original measure of centrality, a variation of pageRank centrality, which I call ‘influence centrality’, that measures the extent to which a node contributes a relation (what is represented by the arcs) to the graph in a directed weighted graph with a finite number of nodes, which are initially labelled with values. I will discuss some properties, applications, and extensions of influence centrality.

I assume some familiarity with graph theory terminology and linear algebra.

### July 21: Kevin Santos: An Introduction to Group Theory through Puzzles

*Abstract*:

The concept of a group is a powerful tool that we can use to understand the structures of mathematical objects. In this talk, we’ll use well-known puzzles such as Peg Solitaire, the 15 puzzle, and Rubik’s cube to motivate a brief introduction to the concepts of group theory. We’ll explore how groups are related to symmetry and give some examples of groups, such as the Klein 4-group and the permutation groups. We’ll then see how these ideas can be applied to understand the puzzles.

### July 14: Albert Lai: Partial Orders and Application to the Semantics of Computer Programs

*Abstract*:

Partial orders generalize total orders by allowing all four possibilities: $x<y$, $x>y$, $x=y$, or none of the above. (Total orders disallow the fourth possibility.) A familiar example is the subset relation over a family of sets. I will show a computer-science application of partial orders to modelling recursive programs. This will be a glimpse of denotational semantics, the study of describing program behaviour by mapping programs to suitable mathematical structures and partial orders.

### July 7: Yuveshen Mooroogen: Shining a rainbow-coloured light on the fundamental theorem of algebra

*Abstract*:

The graphs of real-valued functions on the real line are subsets of a two-dimensional space. As a result, we can sketch them on a piece of paper.

The graphs of complex-valued functions on the complex plane, however, are subsets of a four-dimensional space. Good luck sketching that on a piece of paper.

In this presentation, I will introduce “domain colouring”, which is a technique used to illustrate functions of the complex numbers.

This talk will be in two parts. In the first part, I will explain how to read and construct domain colouring plots, focusing on a number of simple examples. In the second part, I will discuss the Fundamental Theorem of Algebra (which states that every nonconstant polynomial has a complex root) and explain how to visualise one of its proofs using domain colouring. This part of the talk will draw heavily from D. J. Velleman’s beautiful expository article [1].

Prerequisites: Familiarity with the concept of dimension (for a vector space) and with the complex numbers. (Basic principles only. You should know what the notation $x + iy$ means, and how to convert it to modulus-argument/polar form.) Knowledge of multivariable calculus and complex analysis is not expected.

[1] Velleman, D.J. The Fundamental Theorem of Algebra: A Visual Approach. Math Intelligencer 37, 12–21 (2015). https://doi.org/10.1007/s00283-015-9572-7.

### June 23 and 30: David Schrittesser: Your life will be better with infinitesimals

*Abstract*:
When Leibniz, Newton, and others first developed calculus, they used the
metaphor of infinitely small, or infinitesimal, quantities to try to justify
their methods. Later, infinitesimals were expelled from mathematics and
calculus was made rigorous using the familiar notions of limit and
epsilon-delta formulations.

But infinitesimals have been making a come back! Using methods from logic, in particular model theory, they have been restored as respected citizens in rigorous mathematical arguments. This approach, called non-standard analysis has been described as ``the analysis of the future’'. And indeed, it sometimes allows us to do miraculous things. A recent case in point is my joint result with D. Roy and H. Duanmu, with which we solve a long-standing open problem in statistics (namely giving a Bayesian interpretation of admissibility).

In this talk I will give an introduction to the non-standard method and describe some applications. (And if I manage to spark your interest, come to the course I will teach about this topic at U of T in this fall!)

### June 16: Jesse Maltese: An Introduction to Mathematical Logic

*Abstract*:

In this talk, I will give an introduction to the various ideas underlying the foundations of math. I will first talk through some of the history of the field. Then, I will introduce basic concepts in logic including the definition of truth, a model, and notions like completeness and consistency. Then, I will state two important theorems in the field: Gödel’s Completeness Theorem, and the Compactness Theorem. Finally, I will give some interesting applications of compactness, namely the infinite four-colour theorem and the Ax-Grothendieck Theorem.

### June 9: Ben Chislett: Ray Tracing and the Light Transport Equation

*Abstract*:

Ray Tracing is the primary technique for rendering photorealistic images. Advancements in ray tracing techniques have enabled its use in a variety of computer graphics applications, from animated films to real-time video games. In this talk, we explore the Light Transport Equation, Monte Carlo integration, and some of the many optimizations that have led to ray tracing’s rapid rise in popularity.

### June 2: Parker Glynn-Adey: The Probabilistic Method

*Abstract*:

Mathematicians love purity, intuition, and clarity. We like to construct things explicitly with no messiness. But sometimes, mathematical reality is too weird for our explicit methods. Sometimes, we need to look at random examples to find what we’re looking for. In this talk, we use randomness to produce colourings of complete graphs with no monochromatic cliques, a foundational example of the probabilistic method pioneered by Paul Erdős.

Background in counting and graph theory at the MAT A67 / MAT 202 level will help.

### May 26: Brian Li: Cops and Robbers with Many Variants

*Abstract*:

Mathematics and games complement each other in both mathematical research and learning. Cops and Robbers is a game played on graphs between a set of cops and a single robber. The cops begin the game by moving to a set of vertices, with the robber then choosing a vertex to occupy. All players move from vertex-to-vertex along edges. The cops win by successfully occupying the robber’s vertex, hence catching the robber. We will discuss theorems that help us better understand such games and unleash our creativity to explore many different variants. No prior knowledge required.

## FAQ

- How long should my talk last?
- The longest possible talk is 50 minutes, which allows for five minutes for people to arrive and five minutes for questions. Feel free to use up to 50 minutes. Any talk in the range 15 to 50 minutes would be great.

- Do I need to use a particular format for my slides?
- No, feel free to use any software that you like to make your slides. You don’t even need to make slides. Feel free to draw, or use a whiteboard. For math talks, lots of speakers use Beamer.

- Will my talk be recorded and shared online?
- By default, yes and yes. The talks are recorded to Zoom Cloud and
then Parker edits the recording and puts it on YouTube. If you do
**not**want your talk put online, please let Parker know. He will not post your talk online. We want you to feel comfortable speaking and will accodomate privacy concerns.

- By default, yes and yes. The talks are recorded to Zoom Cloud and
then Parker edits the recording and puts it on YouTube. If you do
- Do I need to present original research?
- No! Feel free to present anything. We especially appreciate introductory talks, which introduce the audience to an novel topic. If you have original research that you would like to present, then please do so in a friendly and introductory manner.

- Do you have any suggestions for things to present?
- Jocelyn R. Bell & Frank Wattenberg (2020) The Slippery Duck Theorem, Mathematics Magazine, 93:2, 91-103, DOI: 10.1080/0025570X.2020.1708693
- Sam Chow, Ayla Gafni & Paul Gafni (2021) Connecting the Dots: Maximal Polygons on a Square Grid, Mathematics Magazine, 94:2, 118-124, DOI: 10.1080/0025570X.2021.1869493