Presentations will take place from 11:00AM to 2:00PM on Saturday, April 25 in Beering 1248. Snacks will be provided!
11:00-11:10 - Caleb Dai (mentor: General Ozochiawaeze)
Title: A Note on Markov Chains
Abstract: In my presentation, I’m going to introduce a basic overview of what a Markov Chain is. Then I’m going to go over a very basic example of a Markov Chain. After that, I will introduce several key properties of a Markov Chain, but not all, because that would be difficult to explain in 10 minutes without going into too much detail. Then explain one of the properties, specifically the Chapman-Kolmogorov Equation, with an example before going onto the last subject, which is a brief mention of real world application.
11:15-11:25 - Runguang (Kevin) Li (mentor: Zhilin Luo)
Title: The Riemann Hypothesis for Curves over Finite Fields
Abstract: This presentation explores the theoretical bridge between classical number theory and the geometry of algebraic curves. Beginning with the foundational concepts of prime numbers, the Prime Counting Function, and Euler’s zeta function, the discussion contextualizes the classical Riemann Hypothesis as a pivotal, unsolved challenge regarding the distribution of prime numbers. The presentation emphasizes the complexity of this classical problem, suggesting that valuable insights can be gained through the study of analogous mathematical structures.
11:15-11:25 - Nicolas Jagelka (mentor: Jelena Mojsilovic)
Title: Nakayama’s Lemma and its Applications in Commutative Algebra
Abstract: Nakayama’s Lemma is a fundamental result in Commutative Algebra, giving rise to many key properties of finitely generated modules. In this presentation, I will introduce the lemma and highlight several consequences.
11:15-11:25 - Matthew Young (mentor: Esen Aksoy)
Title: Kakeya Set Problem in Finite Vector Spaces
Abstract: Let Fq be a finite field with q elements, where q = p^m, p a prime. A Kakeya set in finite fields is a set K ⊂ Fq^n such that ∀y ∈ Fq^n there exists x ∈ Fq^n such that the line Lx,y = {x+a· y|a ∈ Fq} is contained in K. The main result on the size of these sets is from Zeev Dvir, Noga Alon, and Terence Tao (2008), which states that if K is a Kakeya set in Fq^n then |K| ≥ Cn · q^n. In this talk, we will go over this result and a brief sketch of its proof.
11:15-11:25 - Jose Gutierrez-Perez (virtual) (mentor: Matthew Wackerfuss)
Title: The Strong Law of Large Numbers
Abstract: This presentation introduces the Strong Law of Large Numbers in the independent and identically distributed case, with an emphasis on the structure of its proof rather than full technical detail. After stating the theorem and interpreting it as a result about almost sure convergence, we outline the main ideas behind the proof, including truncation to control large values, reduction to bounded random variables, and the use of probabilistic tools such as Chebyshev’s inequality and the Borel–Cantelli lemma to establish convergence. A simple coin flip simulation is used to illustrate how sample averages stabilize over time. Overall, the talk highlights how measure theory concepts and probabilistic reasoning combine to show that random behavior becomes predictable in the long run.
11:15-11:25 - Riley Hall (mentor: Yusra Qasem)
Title: Introduction to Steiner triple system
Abstract: A Steiner triple system of order v, denoted STS(v), is a collection of triples from a v-element set such that every unordered pair lies in exactly one triple. First posed by Woolhouse in 1844 and resolved by Kirkman in 1847, the existence question has a remarkably clean answer: an STS(v) exists if and only if v≡1 or 3(mod6).
This talk introduces Steiner triple systems as a special case of balanced incomplete block designs, derives the necessary conditions on v, and presents two explicit constructions: the method of differences and Skolem’s quasigroup construction. These constructions are illustrated through examples of STS(7) and STS(9). The talk concludes with applications on STS.
BREAK
11:15-11:25 - Advait Panicker, Dhruv Upreti (mentor: Kale Stahl)
Title: A Brief Introduction to Brownian Motion
Abstract: In this talk we will discuss the basic underpinnings of Brownian motion; including Brownian motion as a Gaussian variable and analyzing several constructions. We also look at Brownian motion as a martingale to use stopping times and Wald's identity, finishing with the strong Markov property of Brownian motion.
11:15-11:25 - Dhruv Jain (mentor: John Sterling)
Title: An introduction to Topological Spaces and Manifolds
Abstract: In this talk, we introduce the definition of a topological space, together with the notions of continuous maps, homeomorphisms, and homotopies. Building on these concepts, we define manifolds and discuss a simple example.
11:15-11:25 - Alperen Uraz Tasci (mentor: John Sterling)
Title: The Schwarz-Christoffel Mapping
Abstract: The Schwarz–Christoffel mapping is a conformal map that transforms the upper half-plane onto the interior of a polygon, sending points on the real line to the boundary of the target polygon. Its inverse is useful for simplifying boundary value problems posed on polygonal domains by mapping them to the upper half-plane, whose boundary is the more manageable real line. In this talk, we introduce the basic idea of conformal mapping and then discuss the Schwarz–Christoffel transformation. We also outline some of its limitations, particularly the difficulties that arise for polygons with more than three vertices and the crowding phenomenon. Finally, we present a simple toy contour example and describe an application to steady-state heat distribution.