**Colloquium Fest 2024**

Please join us for the 2024 Colloquium Fest hosted by the department of Mathematics & Statistics. Please find below the schedule:

9:30 – 10:20 am Bronfman Auditorium Ammar Eltigani (Thesis Defense)

10:30 – 11:10 am Bronfman Auditorium Elijah Washington

10:30 – 11:10 am NSB 017 Jon Carl

10:30 – 11:10 am NSB 326 Cooper Brill

11:15 -11:55 am Bronfman Auditorium Nick Hollings

11:15 -11:55 am NSB 017 Bemnet Getachew Mengistu

11:15 -11:55 am NSB 326 Natalie Jean-Michel

1:00 – 1:40 pm Bronfman Auditorium Tim Gore

1:00 – 1:40 pm NSB 015 Victor Kilel

1:00 – 1:40 pm NSB 326 Daniela Galvez – Cepeda

1:45 – 2:25 pm Bronfman Auditorium Shaurya Taxali

1:45 – 2:25 pm NSB 015 Christina Halloran

2:30 – 3:20 pm Bronfman Auditorium Sarah Lyell (Thesis Defense)

Here are the titles & abstracts:

Ammar Eltigani (Thesis Defense)

The thesis title: “Khovanskii’s Theorem on the Integer Lattice”

The abstract: “In this talk, we introduce Khovanskii’s theorem, a statement about the growth of iterated sums of (finite) subsets of the d-dimensional integer lattice. We sketch a new proof for the complete characterization of the Khovanskii polynomial for sets of size d+2 and discuss recent investigations in search of an effective version of Khovanskii’s theorem. We also briefly mention some computational evidence that points to another potential method for deducing tighter effective bounds.”

Elijah Washington

Title: Architecture and the Wallpaper Groups

Abstract: Wallpaper patterns are two-dimensional infinitely repeating patterns that fill the whole plane without gaps. Finite versions of these patterns appear frequently in art and architecture. A wallpaper group, or plane crystallographic group, is the set of symmetries of a wallpaper pattern. In this talk, I will discuss some architectural examples, the crystallographic restriction, and the remarkable fact that there are only 17 distinct wallpaper groups.

Jon Carl

Title It’s Not Magic—It’s Math: The Banach-Tarski Paradox

Abstract. A single ball is replicated into two balls without stretching the original ball. The only things required are to cut the ball into pieces, shift, and rotate those pieces. It seems like a magic trick; we’ve doubled the volume of the original ball without introducing a different ball during the construction. The axiom of choice plays a crucial role in the Banach-Tarski paradox. Is the axiom of choice responsible for this bizarre result–or is it the fault of infinity?

Cooper Brill

TBA

Nick Hollings

Title: The Sylvester-Gallai Theorem and the Number of Ordinary Lines

Abstract: Does every finite set of points in the Euclidean plane contain a line that passes through exactly two points? The Sylvester-Gallai theorem demonstrates that this is the case as long as the points are not collinear. In addition to proving the theorem, this talk will address its generalizations as well as the number of such lines in various finite sets.

Bemnet Getachew Mengistu

Title: Patterns of Partitions

Abstract: How many ways can you write the number 3 using the addition of numbers greater than or equal to 1? Ignoring the order of summation, we can easily see that 3 is given by 1+1+1, 2+1, or 3 itself. These different ways to write non-negative integers are called partitions. If you have a piece of paper, enough dedication, and hopefully no homework, you might decide to find all the partitions of the integers up to 20. Many mathematicians have been computing the partitions of non-negative integers into the billions since the 1740s. The main focus for a long time, however, has been finding patterns in the number of partitions of non-negative integers. For example, the number of partitions of ‘n’ into distinct parts equals the number of partitions of ‘n’ into odd parts. How can we show this? In this colloquium, we will explore Euler’s generating functions to study these patterns between odd and distinct partitions of a non-negative number.

Natalie Jean-Michel

Title: The Sylow Theorems

Tim Gore

Title: Dirichlet? More Like Dirich-SLAY!

Abstract: One of the most fundamental questions in math concerns the distribution of prime numbers. In 1837, Dirichlet proved that for any number N, each coprime residue class mod N is equally dense with primes. In this talk, we’ll go through the proof of Dirichlet’s Theorem, and look at some beautiful (perhaps even…”slay”) applications of its components.

Victor Kilel

Title: Expressing Real Numbers as Continued Fractions

Abstract: Continued fractions provide a simple yet informative way to express real numbers. This talk presents an exploration of continued fractions, demonstrating that every real number can be expressed in this format. We begin by introducing continued fractions and their role in approximating real numbers. The main focus is a direct proof showing that all real numbers, whether rational or irrational, can be uniquely represented as infinite continued fractions. We then briefly touch on the practical applications of this theorem in mathematical fields like number theory and its relevance in approximating irrational numbers.

Daniela Galvez-Cepeda

Title: Take It or Leave It: the Ultimatum Game

Abstract: Do you ever feel like your niceness affects your negotiation skills? Turns out it does, and we can use game theory to calculate exactly how much it impacts your payoffs. In this talk you will be introduced to the Ultimatum Game, the different tactics different people use to play it, and how evolutionary dynamics dominate it in the long run.

Shaurya Taxali

Title: A Galois Approach to the Fundamental Theorem of Algebra

Abstract: The Fundamental Theorem of Algebra (first proved by Gauss in 1799) states that every single-variable polynomial with complex coefficients has roots within the complex numbers. This revolutionary result helped pave the way for many other discoveries in the world of Algebra. In this colloquium talk I will prove the fundamental theorem of Algebra using methods and concepts in Galois Theory.

Christina Halloran

Title: Dance of the Coprimes: The probability that two random numbers are relatively prime to one another

Abstract: Most people can name the first handful of prime numbers: 1, 3, 5, 7, 11, etc. When given a number at random and asked to determine if it is prime or not, with some work most people can reach an answer. Yet when asked if two numbers are relatively prime to each other (i.e. the only number that divides both of them is 1), it may take a bit longer to determine. For example, clearly 5 and 7 are relatively prime to each other since they are both prime numbers, but what about 376 and 277? To figure this out, we could write out the factors of each and see if any of them match. If the only factor that matches is 1, then they are relatively prime to each other (i.e., they are *coprime*). This leads us to the following question: if you pick two numbers at random, what are the chances that the only number that divides both of them is 1? In this talk, I will prove the precise probability that two random numbers are coprime. In addition to limits, two key functions used in the proof will be the möbius function, μ(n), from number theory and the Riemann Zeta function, (s), from complex analysis.

Sarah Lyell (Thesis Defense)

Title: The Twelvefold Way for Continuous Functions

Abstract: The Twelvefold Way is a classification scheme for twelve related problems in enumerative combinatorics originally proposed by Gian Carlo-Rota. The original scheme was proposed as a way to count the number of functions between two finite sets subject to restrictions on the functions themselves and when those functions are considered equivalent. As this has been a useful scheme for classification of finite combinatorial problems, we wondered if it might be expanded to continuous functions. To narrow this, we restrict ourselves to polynomials, particularly polynomials of lower degree on a set interval. We begin with a thorough review of the twelvefold way in the finite case before moving on to counting polynomial functions in degrees one, two, and three on a restricted (but continuous interval).