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Seminar

May 5, 2023

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!

Seminar runs Wednesdays 13-14:00 (EST) in IC 318 at UTSC and on Zoom. We start seminar at 13:05, 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.

If you want to receive regular announcements about Seminar, please sign up for our ListServ. We’re also listed on ResearchSeminars.org which you can add to your Google Calendar.

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

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Summer 2023 Schedule

May 17th: Alegbraic Complexity Theory by Mohannad Shehata (UTSC)

Abstract: The question of P vs NP is of great interest to computer scientists, but it is hard to tackle given how unstructured the model of Turing machines is. In this talk, we will introduce a model for computing polynomials called arithmetic circuits, and show how its structured nature allows us to derive lower bounds for complexity, a rather difficult task in the general model. Moreover, we will go over some classical results and common proof techniques (combinatorical arguments and defining complexity measures) and transition into more recent breakthroughs. Finally, we conclude with how close we are to resolving P vs NP when restricted to that model. Familiarity with basic complexity theory is an asset.

May 31st: A Hitchhiker’s guide to braids by Ty Ghaswala (Waterloo)

Abstract: Besides being a useful hair taming technique, braids are also a beautiful object in mathematics. In this talk I will give a selective tour of the wonderful world of braids. We will explore some interesting mathematical nooks and crannies in which braids arise, focusing on symmetries of surfaces and whether or not the following question makes any sense at all: Given two braids, which one is bigger?

Winter 2023 Schedule

January 19th: Automatic Sequences by Anatoly Zavyalov (University of Toronto, St. George Campus)

Abstract: Automatic sequences are a class of sequences that are generated by finite automata. Representing these sequences using automata allows us to use tools and ideas from automata theory and formal language theory, creating a fascinating intersection between theoretical computer science and number theory. In this talk, we will introduce automatic sequences and their properties, finite state transducers, as well as showcase intriguing examples and applications. No prior knowledge of automata theory will be assumed.

January 25th: The Toeplitz Conjecture by Maitreyo Bhattacharjee (IACS, Kolkata)

Abstact: The Toeplitz Conjecture The Toeplitz Conjecture, or the Inscribed Square Peg Problem is an open question in geometry with a long history. It asks that, given any Jordan curve in the plane, does there always exist an inscribed square. It was originally posed by Toeplitz in 1911, and remains open as of 2022. In this talk, we would focus on the problem statement, it’s long history of partial resultsThe talk will be accessible to first year students in their first Analysis Course, and no background in Metric Geometry will be assumed.

February 1st: Exploring the QUAKE III Fast Inverse Square Root by Daniel Harrington (University of Toronto, Scarborough Campus)

Abstract: QUAKE III’s fast inverse square root algorithm. The mysterious and ingenious function that made waves on the internet in 2002. Why did this strange piece of game engine code, with comments containing profanity and calling itself evil, make a common operation 4x faster? We will look into how this famous approximation worked, outsmarting the standards it was written in before applying clever bit manipulation and some calculus. Then we will discuss how it can be improved, and where it is today. Basic knowledge of programming is recommended.

February 8th: Making Juggling Mathematical by Erik R. Tou (University of Washington, Tacoma)

Abstract: Juggling has a long history, dating back over 3000 years. During most of this long history, juggling was the avocation of entertainers and artists. Only in the 1980s was a method developed to keep track of different juggling patterns using a numerical code, now known as a siteswap. This numerical shorthand gives rise to a set of integer-like objects that can be analyzed using number-theoretic techniques. We’ll consider how these techniques provide a way to count the number of “primitive” juggling patterns of a certain type.

February 15th: Into the Infinite-Dimensional: An Intro to Functional Analysis by Justin Fus (University of Toronto, Scarborough Campus)

Abstract: We often enjoy living in the comfort of our nice finite vector spaces, but what happens if we strip that away and take the plunge into the infinite world of functional analysis? An introduction to infinite dimensional vector spaces is given and the powerful ability of abstraction that it possesses. We will discuss norms and convergence, operators, continuity, and present Banach spaces all under a visual lens. Suitable for students taking a first year linear algebra course.

March 1st: Proofs = Programs: The Curry-Howard Isomorphism by Andrew Feng (University of Toronto, St. George Campus)

Abstract: We give an introduction to simple type theory in the Church style and minimal propositional logic. The introduction to logic is from a proof-theoretic point of view, emphasizing the notions of deduction. We discuss the striking similarity between typing rules in simple type theory and inference rules in minimal propositional logic, and how that gives an “isomorphism” between proofs and programs. Finally, we give a glimpse of more powerful type theories and logics.

March 9th: Fun Applications of Geometric Algebra! by Logan Lim (University of Toronto, Scarborough Campus)

Abstract: From physics, to computer graphics, to quantum computing and neural networks, geometric algebra is a modern re-imagining of linear algebra to simplify and generalize the role of matrix computation in a wide variety of settings. In this two-part presentation, we give a fun tour of projective geometric algebra (PGA 3d) and its relationship to perspective in art and 3d graphics, quantum computing based on complex Clifford algebras, and a brief discussion on the current progress of neural networks using geometric algebras. Who said math couldn’t be exciting?

March 15th: Abstracting Reality: Symmetry Ideas in Physics by Aditya Chugh (University of Toronto, St. George Campus)

Abstract: Fundamental Physics makes use of a variety of fields of abstract math: algebra, topology, etc. In fact, many important theorems which successfully predict how the universe behaves find origin in beautiful and yet very abstract ideas in Math. In this talk, we analyse some of these ideas and discuss how abstraction of different aspects of physical systems allows us to gain insight that may have been hard to see otherwise. We will take a look at Noether’s Theorem, Ehrenfest’s Theorem, and perhaps particle spin, which make use of group theory and algebraic topology. Suitable for undergraduates in all years- we will try to provide small intros/refreshers as we go along.

March 22nd: Advantages of Working Remotely by Faiza Robbani, Sharon Alex, Yunni Qu, Yushu Zou (University of Toronto, Scarborough Campus)

Abstract: The Pandemic has altered the lives of each and every individual in one way or another. For staff and students, it has taught them to teach and study remotely. As many public health measures have been lifted, classes have resumed in-person, which may have created an inconvenience to both staff and students for multiple reasons. Remote settings have been suggested for the purpose of helping with climate change, personal life and reducing the time spent commuting. This study was designed to estimate the amount of time and energy saved by remote learning, teaching, and conferences. As a result, we divided our research into two parts. Students and staff from UofT were the focus of the first study, and online vs. in-person conferences were the focus of the second. In the first part of the study, data were collected via surveys administered to students and staff at UofT. The second part of our study used data already available in an open-source format. We estimated the time and energy both students and faculty would be able to save if they were to work and study at home. We also looked at how much time and energy could be saved by holding online conferences instead of in-person ones with open attendance data from American Math Society. We predicted the possible commute plan, and estimated the energy cost based on the commuting.

April 5th: Ray Marching and Visualising 3D Fractals by Young Chen

Abstract: When a geometry becomes too complex to efficiently visualise using rasterisation or ray tracing techniques, ray marching can be used instead. We will introduce the basics of how ray marching works and see how to render simple scenes using this technique. We will then look at an area where ray marching has been used extensively: rendering fractal geometry. This part of the presentation will consist of a brief introduction to fractals, followed by an introductory look at the different algorithms used to render fractals using ray marching.

Fall 2022 Schedule

September 14: Periods: from pendula to the present by Brent Pym

Abstract: Many important quantities can be expressed as integrals whose domain and integrand are determined by polynomials with rational coefficients. Such numbers, which include $\sqrt{2}$, $\log(7)$, $\pi$, etc. are called “periods”, because their systematic study was inspired, in part, by the classical calculation of the period of oscillation of a pendulum. Despite the apparent simplicity of the definition, periods have a rich and beautiful structure, and are the subject of some of the deepest open problems in mathematics, connecting geometry, algebra, number theory, physics and more.
I will give an introduction to this circle of ideas and the questions that drive current research in the area, assuming only basic knowledge of multivariable calculus.

September 21: An Introduction to Combinatorial Games by Zack Wolske

Abstract: We’ll introduce a collection of two player games that anyone can play – they’re fun for all ages. Some games have strategy patterns we can find quickly, some have well-hidden patterns that we can uncover with more tools, and others have patterns that no one in the world has found. We will play and analyze those games and share currently open problems.

October 5: Permutations with quipu by Scott Carter

Abstract: According to Wikipedia, a quipu is an accounting system in which knots are tied in a sequence of strings. They were used as a method of storing tax and other financial records. Here we consider cyclic subgroups of groups and catalogue the cosets by means of a string bundle (the English word, not the mathematical word is intended). The quipu are elements in the cyclic subgroups. The traditional methods of multiplying braids by means of vertical juxtaposition is mimicked in the case of permutations-with-quipu. The quipu are allowed to pass upwards through the crossings of transverse strings. So a permutation-with-quipu represents an element in a semi-direct product.

We have developed appealing diagrams that represent the elements in the dihedral groups, and especially in the finite subgroups of SU(2). Have no fear! The groups in question will be described explicitly, and we’ll play with the diagrams in ways that allow easy computations. In fact, the permutations-with-quipu can be thought of as matrices in disguise. I will show how to go between the quipu and the corresponding matrices. Emphasis will be upon examples.

October 19: An Algebraic Proof of the Fundamental Theorem of Algebra by Blake Madill

Abstract: Recall that the Fundamental Theorem of Algebra states that every non-constant polynomial over the complex numbers completely factors as a product of linear terms. In a typical undergraduate experience, students will see proofs of this theorem using topology and/or complex analysis. In this talk, we will explore a completely algebraic (with the exception of some basic calculus) proof of this algebraic theorem. Students will be introduced to aspects of group theory, Sylow theory, field theory, and Galois theory. No prior knowledge of abstract algebra will be assumed.

October 26: Star rosettes in GeoGebra: constructing traditional patterns with contemporary technologies by Sarah Brewer

Abstract: Star rosette patterns are ubiquitous in geometric architectural ornament of the Islamic world. These patterns are traditionally built on a mathematically elegant system of polygons and tangent circles in their underlying Euclidean compass and straightedge constructions. Of particular interest are star rosette patterns built on univalent circle packings whose intersection graph is any k-uniform tiling, where varying the angle of the star rosette pattern lines serves as the transition between a tiling and its dual. In simple terms, you’ll learn how to make some pretty patterns in GeoGebra.

November 2: Spheres, Donuts and Crazy Bottles: An Introduction to The Classification Theorem of Surfaces by Alex Teeter

Abstract: Mathematicians love to classify mathematical structures in order to understand them better. In this presentation, I will take you on a journey through the World of Surfaces and prove their classification. We will see many examples of Surfaces, such as the one-sided Mobius Strip, the Torus, and the Klein Bottle, a bizarre surface that cannot be embedded in 3-dimensional space. We will also cover how to construct surfaces easily and conveniently using surgery, and prove that all surfaces (that fulfill certain conditions) can be constructed from such surgery. I hope you are excited as I am to delve into the wonderful world of Topology!

No prior knowledge of Topology is needed.

November 9: What’s the deal with Homological Algebra? by Ben Briggs

Abstract: Back in the 70’s, David Mumford accused algebraic geometry of “secretly plotting to take over all the rest of mathematics”. While that battle was raging, the topologists attacked from the side and annexed most of algebra (as well as a good deal of number theory, combinatorics, statistics, physics). Nowadays you cannot go outside without stepping in homology or cohomology, or some kind of homotopy. All of this started from homological algebra (the Trojan horse?), which sort of began with David Hilbert and Emmy Noether way back in the 1890’s, but which really got going in the 50’s. I will explain what homological algebra is in very gentle terms, starting with chain complexes, and I’ll give a lot of concrete examples of the cool things you can do with it. I might also give a less concrete idea of how it ended up everywhere (or at least, in math I do).

November 16: What caused Coxeter many restless nights? by Özgür Esentepe

Abstract: “Frieze patterns” kept Coxeter up at night. This talk will introduce these surprisingly ubiquitous grids of integers. We will discuss some basic properties and how they appear in representation theory. We will assume almost zero background.

November 23: A SAT Solver + Computer Algebra Attack on the Minimal Kochen-Specker Problem by Brian Zhengyu Li

Abstract: One of the most fundamental results in the foundations of quantum mechanics is the Kochen-Specker (KS) theorem, a `no-go' theorem that states that contextuality is an essential feature of any hidden-variable theory. The theorem hinges on the existence of a mathematical object called a KS vector system. While the existence of a KS vector system was first established by Kochen and Specker, the problem of the minimum size of such a system has stubbornly remained open for over 50 years. In this paper, we present a new method that is based on a combination of a SAT solver and a computer algebra system (CAS) to address this problem. Using our approach, we improve the lower bound from 22 to 23, with a significant speed-up over the most recent computational methods. Finding the minimum KS system could enable applications in security of quantum cryptographic protocols, zero-error classical communication, and dimension witnessing.

November 30: An Introduction to Modal Logic and Its Applications by Jesse Maltese

Abstract: Modal logic refers to an extension of the language of classical logic wherein two new operators are added.
Referred to as modal operators and denoted $\square$ and $\lozenge$ (‘box’ and ‘diamond’ respectively), these operators change the truth value of a proposition, allowing one to reason about modalities such as necessity, knowledge, and time.
This talk will first present some of the historical development of modal logic, and then introduce the Kripke semantics. We will talk briefly of some alternative semantics, before presenting other modal logics, specifically temporal, epistemic, and doxastic. We will conclude with a discussion of the applications of these systems of logic to computer science. Specifically, how they are used to reason about distributed systems and concurrent programs, and for verifying correctness of software. Throughout, we will make note of the philosophical applications.

This talk will assume an understanding of first-order logic.

TBD: Speedrunning Differential Geometry by Davide Radaelli

Abstract: In this talk, I’ll be speedrunning as much of differential geometry as I can, building everything up from first-principles. I’ll start off with logic, then move onto set theory, topology, manifolds, bundles, linear algebra, tangent spaces, tensors, differential forms, connections and more.

TBD: Dr. Kripke And The Multiverse of Logics by Albert Lai

Abstract: I will introduce and motivate some extensions of basic (classical proposition or first-order) logic and how to formalize their meanings.
Specifically: Intuitionistic logics insist that a proof of a statement should be also a recipe for outputting a witness to that statement, and thus reject that, e.g., “P or not P” can be proved unless tailor-made to a specific P. Modal logics start with basic logic (propositional logic) and adds unary operators intended to mean “necessarily” and “possibly”, to help express “John must be sleeping” and “John may be sleeping”. Temporal logics have a state transition system in mind, and starts with basic logic and adds unary operators intended to mean “for every state in the next step”, “for some state in the next step”, “for every step from now on”, “infinitely often”, etc.
They have evident applications in verification of program correctness. We understand the semantics (meaning) of a logic by imagining a mapping from symbolic names (e.g., the propositional variable “P” above) to concrete things, and asking how the logical operators evaluate the latter.
In the 1960’s, the then open question of giving semantics to modal logics and intuitionistic logics were both solved by Saul Kripke (as a young prodigy) in one stroke by the same idea of: What if there are multiple (but connected) worlds, and different worlds can have different mappings; naturally, the idea carries over to temporal logics (which arose after Kripke’s solution).
I will introduce said logics and explain Kripke’s solution by a multiverse of, uh, truths!

Winter 2022 Seminar

March 16: Modelling Mathematics with Knitting and Crochet by Kevin Santos

Abstract:

When studying subjects like geometry and topology, it can be difficult to visualize or understand certain abstract concepts. Being able to hold and manipulate a physical model of a mathematical object can give deeper intuition into its properties. The process of constructing such an object also offers further insight. In this talk, we will investigate how geometric and topological objects can be constructed using the crafts of knitting and crochet, which offer unique advantages in creating models. We will describe how the hyperbolic plane can be modelled with crochet and we will explore how topological surfaces such as the sphere, the torus, and the Klein bottle can be knitted. No prior knowledge in geometry, topology, knitting, or crochet is required.

March 23: Why Geometric Algebra Should be in the Standard Linear Algebra Curriculum by Logan Lim

Abstract:

Geometric algebra is an extension of $\mathbb{R}^n$ that includes as special cases: The complex numbers, quaternions, exterior algebras, dual numbers, split-complex numbers, dual quaternions and more! When applied to multivariable calculus, it generalizes the fundamental theorem of calculus on manifolds to include the divergence theorem, curl theorem, and gradient theorem, and as a result Green’s and Stoke’s theorem, as special cases of a single statement. It also simplifies many geometric operations in computer graphics by eliminating the need for matrices in projections, rotations, and reflections. Though we can only cover the ‘main idea’ of geometric algebra in the allotted time, this talk will be a buffet of ideas you can explore for this fascinating and deceptively simple algebraic object.

Mathematicians hate him!!!! See how Clifford generalized rotations and orthogonal complements in n-dimensions with this one simple trick! $$ \mathbf{uv} = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \wedge \mathbf{v}. $$

March 30: Seifert Surfaces and Knot Genus by Alex Teeter

Abstract:

Knot Theory, a field of mathematics born from a misguided model for atoms, has since grown to become an important subfield of Topology. Not only does it have much of mathematical interest, but also numerous connections to fields such as graph theory, the study of manifolds, and applications to Biology and Physics. We will analyze the connection between Knot Theory and the Topology of Surfaces. Along the way, we will cover the Euler Characteristic, Genus and the beautiful Classification Theorem of Surfaces. Through this consideration, we develop an algorithm to associate each Knot with a surface, and uncover an important invariant, the Genus of a Knot. This will not only allow us to distinguish between different Knots, but will also be vital in establishing fundamental properties of prime and composite Knots. No prior background in Knot Theory or Topology is assumed.

Fall 2021 Seminar

November 17: Fractional Brownian Motion

Abstract: The classical Brownian motion (or Wiener process) serves as the fundamental object of probability theory with vast theoretical and practical applications in a plethora of fields. In the early 1940’s Kolmogorov sought a natural one parameter extension of the process in aims of modelling turbulence in liquids, the extension consists of retaining the framework of the classical motion by constructing still a continuous centered Gaussian process that retains self similarity (of a now distinct index from that of the classical motion) and stationarity of increments but draws its distinction by parameterizing its specifying covariance structure with what is known as the Hurst index, $H \in (0,1)$, resulting in (for ‘most’ values of $H$) a non-Markov process allowing it to serve as a popular model for dependent phenomena; this extension has since been kept in common parlance as fractional Brownian motion (fBm). This talk aims to provide discussion and (at times demonstration) of the fundamental properties of the fBm as well as investigate sample path properties' dependence on the Hurst parameter.

November 10: Matrix Analysis with a Focus on Inequalities

Abstract: Matrix inequalities are one of the key components in matrix analysis. They have a wide range of applications in statistics, computer science, economics, and physics.

In this presentation, we focus on some classical matrix inequalities such as the Rayleigh-Ritz quotient, the Courant Fischer theorem, Weyl’s inequality, the interlocking eigenvalue lemma, and the Woodbury matrix identity. In particular, some of these show the importance of the maximum and minimum of eigenvalues and singular values.

Recommend Background: Linear Algebra and Calculus. (In the beginning of the presentation, some definitions and key properties such as positive definiteness will be reviewed.)

November 24: Getting Started: Proving with the Lean Interactive Theorem Prover

Abstract: Have you heard of automated and interactive theorem provers? Did you know that American mathematician Alex Kontorovich predicts that the Lean Interactive Theorem Prover will be so widely used that it will be as necessary as LaTex in doing mathematics? This seminar series will take you on a journey, using Lean to give you insights on how to go about proving, which route to choose, how to check for errors, and how to verify a computation or a proof. Particularly, you will be learning to use Lean by playing a number game. Come join us and have some fun!

DRAFT Summer 2021 Seminar

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.
• 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?
  1. Jocelyn R. Bell & Frank Wattenberg (2020) The Slippery Duck Theorem, Mathematics Magazine, 93:2, 91-103, DOI: 10.1080/0025570X.2020.1708693
  2. 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
  3. Bhargava, Manjul. “The factorial function and generalizations.” The American Mathematical Monthly 107.9 (2000): 783-799.pdf
  4. Birman, Joan S. “New points of view in knot theory.” Bulletin of the American Mathematical Society 28.2 (1993): 253-287. pdf
  5. Margalit, Dan. “The Mathematics of Joan Birman.” Notices of the American Mathematical Society (2019). pdf

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