Ex Numera

February 27th

Abstract: About 100 years after calculus was conceived in the 17th century, mathematicians around the world began to formulate solutions to many of the famous partial differential equations (PDEs) we know of today. Describing every kind of phenomena in the natural world, from light and sound to blood flow and earthquakes, the beautiful relations we have unearthed speak a great deal about the process of solving PDEs as well as the nature of mathematical progress today. The strategies we have created for solvable PDEs led to insight in understanding differential equations we cannot find a closed form or even general solution for. This talk will discuss methods for solving some higher order PDEs, the nature of those solutions, and how these relate to PDEs we are tackling in the present.
Prerequisites: Calculus of a single variable, ordinary differential equations, multivariable calculus would also be helpful

September 26th

Abstract: This talk will introduce the theory of compositional distributional semantics, which hopes to use linguistic theory to combine the meaning of individual words into the meaning of full sentences in a computational setting. We will begin by introducing a simple theory of syntax using pregroups, and then, in steps, upgrade it to a strictly monodical category equipped with a Frobenius algebra. The category theory will be approached through string diagrams. Moving into a categorical setting allows us to construct the Transfer functor that carries syntactic data into the category of real vector spaces, so that syntactic derivations of a sentence become semantic computations. I will try to limit my examples to the English language. No knowledge of linguistics, natural language processing, or category theory is required, although knowledge of tensors, or at least vector spaces, will be helpful.
Prerequisites: some basic knowledge of vector spaces and tensors would be helpful.

October 3rd

Abstract: Partial Differential Equations (PDEs) are very important in explaining and modeling natural phenomena, but it's often difficult to produce analytical solutions for these problems. Thus, we are often forced to rely on numerical approximations to solutions of PDEs. The two most common numerical methods, Finite Element and Finite Difference, can be ineffective on irregular domains which are common in biological problems. The goal of this work is to implement an alternative numerical method for solving PDEs that overcomes the difficulty introduced by irregular boundaries by involving stochastic processes. In addition, we have involved machine learning methods, like temporal difference learning, to increase the performance of our code. No prior experience in PDEs/ML needed! Joint work with: Cameron Martin (Univ of Toronto), Hongyuan Zhang (Grinnell College) Advisors: Adam Stinchcombe and Mihai Nica (Univ of Toronto)

October 10th

Abstract: Mathematical graphs, made up of vertices connected by edges, can be used to model networks in many disciplines. Eulers paper on the Seven Bridges of Königsberg in 1736 is considered the first graph theory paper, and in the last century the subject has expanded in many different directions. This talk will focus on graph colorings, a way of labeling vertices and edges according to a variety of rules. In applications, colorings can help solve scheduling problems, register allocation, and even Sudoku puzzles. Starting from the basics of graph theory, this talk will introduce several notions of graph coloring. We will begin by defining a proper coloring of vertices and edges, and introduce some interesting graphs such as the Kneser Graph and the Mycielski Construction. In addition, we will explore several different graph colorings, including list coloring and fractional coloring. We will develop enough tools to prove the Five-Color Theorem, the precursor to the more famous Four-Color Theorem of Planar Graphs. No prior knowledge of graph theory is necessary.
Prerequisites: None

October 24th

Abstract: Philosophy of mathematics is a branch of philosophy which poses the basic questions about mathematics of which no mathematicians consider in their day-to-day research: What are numbers? How are mathematical truths necessarily true? What is the relationship between numbers and sentences (propositions)? It is the study of implications, foundations, and assumptions we make about mathematics when doing mathematics. There are lots of philosophers who contributed to this topic, but we will cover some of the most crucial figures in philosophy of mathematics and their key ideas in philosophy of mathematics. Figures we will cover include Plato, Aristotle, Immanuel Kant, Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein. To make the talk less technical and more engaging, I will omit discussing figures such as Gödel, as his theorems cannot be discussed without the discussion of mathematical logic. The talks aim is to give a historical survey of the topic matter and basic familiarity with philosophical theory regarding the nature of mathematics.
Prerequisites: No prior knowledge in philosophy is necessary. Some basic understanding and familiarity with high school mathematics should suffice. Some familiarity in logic will be beneficial.

October 31st

Abstract: Quantum computers represent an existential threat to current techniques in cryptography. Methods relying on supersingular isogenies are potentially resistant to quantum attacks. The difficulty of this isogeny problem is analogous to finding a path between two vertices of an isogeny graph. The graph over $\overline{\mathbb{F}_p}$ is a directed multi-graph, with vertices as isomorphism classes of elliptic curves and edges as isogenies with a fixed degree. Using properties about isogenies, we prove relations between the adjacency matrices corresponding to prime degree graphs and composite degree graphs. Composite degree isogenies are significantly harder to compute. So, this provides a useful method for finding isogenies. Moreover, we use a result by C. Delfs and S. Galbraith to determine the cycle types within supersingular isogeny graphs over $\mathbb{F}_p$. Specifically, we show that isogeny graph has exactly the same shape for curves with endomorphism ring $\mathbb{Z}[\sqrt{-p}]$ and $\mathbb{Z}[\frac{1+\sqrt{-p}}{2}]$. Using this result, we also prove that the adjacency matrices corresponding to a 2-degree and odd prime degree graph are commutative.

September 20th

Abstract: Fermat's last theorem (abbreviated FLT), first conjectured by Pierre de Fermat in 1637, had been one of the most famous unsolved problems in mathematics before finally being proved by Andrew Wiles nearly 358 years later. This talk will focus on the attempts of mathematicians, especially Kummer, in attempts to prove FLT during the 19th century.
More specifically, we first observe that proving the theorem is equivalent to showing that $a^{n}+b^{n}=c^{n}$ has no integer solutions for any odd prime $p$, since having a solution for a given $n$ is equivalent to having a solution to any factor of $n$. Then using the fact that $t^{p-1}= (t-1)(t-w)(t-w^{2})(t-w^{p-1})$, $w$ being a primitive $p$th root of unity and substituting $t$ for $\frac{a}{b}$, we obtain the nice equation $(a+b)(a+bw)(a+bw^{2})(a+bw^{p-1})=c^{p}$ from the original equation $a^{p}+b^{p}=c^{p}$. Thus we can consider solutions to Fermats last Theorem by working in the ring extension $\mathbb{Z}[w]$ over $\mathbb{Z}$. It turns out that if $\mathbb{Z}[w]$ is a UFD, then it is not difficult to show that FLT holds true. Unfortunately, the nice proper ty that $\mathbb{Z}[w]$ possesses unique factorization doesnt hold for all $w$ (For example, when w is a primitive 23rd root unity). However, the remarkable discovery was made that $\mathbb{Z}[w ]$ always uniquely factorize as a product of prime ideals! Using this, Kummer was able to show that FLT holds true for certain primes called regular primes. Unfortunately, while irregular primes exist, Kummers work was still an important stepping stone in the proof of FLT. Moreover, unique factorization plays a huge role in nu mber theory, and Kummers work along with other mathematicians in the 19th century heavily influenced the development of algebraic number theory. Ultimately, this talk will sketch the results of Kummers proof of FLT for regular primes, and will also illustrate the relevance of these results in number theory today.
Prerequisites: some basic knowledge of abstract algebra

September 27th

Abstract: Sphere packing - the problem of placing spheres in n-dimensional space such that their density is maximized - is notoriously difficult. It is trivial in one dimension, tricky in two dimensions, and took nearly 400 years to resolve in three dimensions. In higher dimensions, the optimal packing was not known until 2016, when Maryna Viazovska proved the optimal packings in both 8 and 24 dimensions. While other sphere packing results have required detailed casework, becoming complicated enough to demand automated proof-checking, these new results involve a clean and elegant combination of harmonic analysis and the theory of modular forms. We will discuss basic results in these fields, with a particular focus on modular forms. Topics in harmonic analysis will include the Fourier inversion theorem and the Poisson summation formula. Topics under the umbrella of modular forms will include lattices, elliptic curves, and L-functions. By the end, we will have a sketch of the proof of the optimal sphere packing in eight dimensions.
Prerequisites: Linear Algebra, basic familiarity with complex functions

October 4th

Abstract: A Game Tree is a directed graph whose nodes are positions in a game and whose branches are moves. Many zero-sum two-play er games could be represented by a game tree(e.g. Tic-tac-toe, chess, checkers, and nim). If one knows all the values of the end node, then one could figure out the strategy using the minimax algorithm. However, games like chess have too many leaves to use the minimax algorithm. Because a typical small piece of a ridiculo usly enormous game looks random, assume the end nodes to either be one or zero with probability p. This assumption revea ls interesting natures of a k,l tree where one player chooses from k options and other chooses from l options. For examp le, in the k,l tree there exists a single fixed probability of fairness. Also, the relationship of it with that of the k ,l is exponential. Furthermore, we will prove that if the large game of k,l tree is in ones favor, one only need to kn ow value of log(n) nodes (n being the total number of nodes) to win the game. We will also prove that if the end nodes f ollow a random distribution of a function from 0 to 1, similar results hold as when end nodes are either one or zero. We will end with a brief discussion on randomized algorithm, and calculation of its upper bound.
Prerequisites High School Calculus

October 11th

Abstract: Brief Introduction to Mathematical Control Theory (and why you should care) Abstract: Control theory is a subset of mathematics that deals with augmenting a dynamical system by introducing a controller into the system. It has been applied across all kinds of fields, from self-driving cars to the stock market to drug delivery. In this talk we will look at the strong mathematical foundation behind control theory, specifically the concept of controllability as it applies to linear autonomous systems. We will work through a few proofs relating to controllability, leading up to the proof of the controllability matrix, which allows us to tell which systems are totally controllable from their mathematical makeup. We will frame our investigation through the example of the rocket car, tying the concepts we think about to a concrete example. Adaptation of DRP project completed with Patrick Martin
Prerequisites: Calculus

October 18th - No Talk (Fall Break)

October 25th

Abstract: Quantum computers store data in probabilistic quantum states, and use quantum mechanical phenomena like entanglement to perform computations on it. Quantum computing is good for two things: helping you work as many buzz words into a sentence as possible, and solving certain problems that are computationally difficult on classical computers, like the prime factorization of integers, in polynomial time. However, quantum computers are plagued by $\textit{decoherence}$ when the quantum states lose their crucial probabilistic nature before the computation is completed. Topological quantum computing provides one potential solution to the decoherence problem by storing data in $\textit{topological}$, or globally protected, quantum states of 2-dimensional quasi-particles (yes, fake particles) called $\textit{anyons}$. This talk will introduce the braiding of anyons as a method of computation, and modular categories as models for anyonic systems. It will provide intuition behind a proof that certain anyons can't be used to build a topological quantum computer because they have Property $\textbf{F}$, and discuss the strange physical consequences of this proof. Finally, it will demonstrate an application (who knew) of knot theory.
Prerequisities No physics! Basics of abstract algebra helpful, but not necessary.

November 1st

Abstract: This talk will give an introduction to an alternative formulation of analysis known as Infinitesimal Analysis or Synthetic Differential Geometry (SDG). The idea is to introduce infinitesimal objectsobjects that aligns with intuitive notions of the infinitely smallinto the real line in such a way that, under constructivist logic, standard operations and constructions such that differentiation, integration, and Taylor series can be defined (with their usual properties) without δ formulations or limit computations. The talk will first motivate the Kock-Lawvere Axiom that allows the existence of infinitesimal objects, and then reconstruct roughly two semesters of calculus (hopefully in under 40 minutes) within this new framework. We will also provide a brief description of constructivist logic as an alternative mathematical paradigm. We will end by discussing manifolds in this framework and topos theory as the correct universe of discourse for models of this theory.
Prerequisities Two semesters of calculus. Knowledge of analysis, logic, and algebraic geometry is beneficial but by no means necessary.

November 8th

Abstract: The current national energy consumption is dominated by fossil fuels, which are processed by generators. As we seek to include more renewable energy resources into our grid, the stability of the grid is no longer guaranteed. This lack of equilibrium is due to differences in the inertia generators being high inertia systems and renewables low to no inertia systems. This discrepancy causes frequency fluctuations that can lead to poor operations and power outages. My research project focuses on designing battery controlled inverters to be integrated into the grid to ensure stability. To show this, I begin by setting up the differential equations that govern the relationship between the output power of the generators and the frequency of the grid. I will examine the current solution in place (Droop Control) and show how its transient response can be improved. This is done by considering the proposal of two novel controllers and evaluating them in terms of efficiency and efficacy. To conclude, I will demonstrate how the chosen controller impacts the overall performance of the power grid and discuss further use of renewables in a large scale grid system.
Prerequisities High School Calculus

November 15th

Abstract: Category Theory is the unifying language of mathematics. It can be used to turn a topological problem into an algebraic one, or it can be used to understand the analogy between linear transformations between vector spaces and functions between sets. Well start off with some definitions: categories have objects which can be seen as the nouns of the category, and morphisms between the categories which are like verbs. Well then give some useful examples before quickly moving towards concepts such as isomorphisms, monomorphisms, epimorphisms, universal properties, functors, and natural transformations. Well also explore opposite categories and how every theorem in category have a duel theorem. Then well talk about (co)products, where well see a surprising relationship between the GCD of two numbers and the cartesian product of two sets. Other topics may be explored as time permits.
Prerequisities Knowledge of Abstract Algebra and Topology would be helpful, but not necessary

November 29th

Abstract: In the year 1770, Lagrange proved the well-known four-square theorem, which states that each positive integer can be written as the sum of four squares. That same year, the British mathematician Edward Waring proposed a generalization of Lagrange's result, which states that for all natural numbers k, there is a corresponding positive integer g(k) such that every positive integer can be written as the sum of at most g(k) kth powers of natural numbers. Various cases for k were solved in the next century, and in 1909, Hilbert proved Waring's problem for all natural numbers k. Decades later, Hardy and Littlewood established bounds on g(k) and G(k), defined as the smallest positive integer such that every sufficiently large integer can be written as a sum of at most G(k) kth powers of natural numbers. This talk will be largely based on W.J. Ellison's paper on this topic, and will outline various approaches to the proof of Waring's problem, as well as similar problems in number theory.
Prerequisities Integral calculus, linear algebra and elementary number theory.