Abstract: This talk will give a brief introduction to the basic case of rhombic tiling, and establish the bijection between tiling and plane partition. After introducing the Macmahon‘s formula, the presentation will discuss the case of a punctured hexagon. 

Delivery: PowerPoints/Slides

Abstract: We study the concept of Brownian local time, a measure introduced by P. Lévy to quantify the time that Brownian motion spends near a specific point in space. Traditional measures fail to capture this duration due to the zero Lebesgue measure of level sets in Brownian paths, motivating the introduction of local time. We discuss the relationship between the existence of local time and the differentiability of Brownian motion. A convenient representation of Brownian local time is then provided using the Tanaka Formula. Finally, we present the Trotter Existence Theorem, which proves the existence of local time. 

Delivery: PowerPoints/Slides

Abstract: The sufficiency principle in statistics provides a conceptual framework for thinking about the information retained after a data reduction procedure. We consider the interplay between sufficiency and the data processing inequality in information theory to explore the mathematical foundations of reducing data while retaining all relevant information. 

Delivery: PowerPoints/Slides

Abstract: In this presentation, we will discus the mathematics of knots. First, we will define what a knot is, and then discuss the surprisingly difficult task of determining whether two knots are equivalent. This will be done through knot invariants such as tri-color ability, the Alexander Polynomial, and analyzing the complement of a knot with Algebraic Topology. 

Delivery: PowerPoints/Slides

Abstract: We begin by covering some fundamental properties and results related to Markov chains. This culminates in a brief discussion on spectral methods utilized to analyze convergence of chains and lower bounding mixing. 

Delivery: PowerPoints/Slides

Abstract: Our presentation will introduce the measure spaces and construct the Lebesgue Measure 

Delivery: Paper/Handwritten

Abstract: This presentation explores key concepts of Albert-László Barabási's Network Science textbook, providing a foundation for understanding the structure and dynamics of complex networks. Topics include the principles of network representation, the small-world phenomenon, scale-free networks, network growth mechanisms, and power-law distributions. This topic is highly applicable to various areas of study, such as social networks, biological systems, and telecommunications. The presentation will highlight an example of network science applications to the agriculture industry of Sub-Saharan Africa. 

Delivery: Powerpoint/slides