The Summer 2024 DRP at UIC Presentation Session was held on September 9, 2024. There were four presentations:
Fantastic Fractals
Janien Hammonds, mentored by Lisa Cenek
Abstract: Fantastic Fractals explores the paradoxical nature of fractals by building on the concept of the Cantor set. I
use the Koch curve and Koch island to exemplify this characteristic.
Pólya's Recurrence Theorem & Electric Networks
Marcelo Lozano, mentored by Karoline Dubin
Abstract: "A drunk man will return home, but a drunk bird may get lost forever." Will a point wandering randomly on an n-dimensional grid return to its initial position? George Pólya's Recurrence Theorem tells us that simple random walks are recurrent on one and two dimensional lattices and are transient in higher dimensions. This talk will apply the language of electric networks to develop intuition for this famous result (and outline one of its proofs).
Wait, There is More than One Geometry?
Juan Jose Rosendo, mentored by Amelia Pompilio
Abstract: When taking a high school level geometry class, we are exposed to the reality of shapes. It is all we know, until one decides to learn more about a "geometry." Therefore, we will explore the differences between various geometries, including Euclidean, Hilbert, Spherical, Hyperbolic, and Projective geometry. A brief introduction will be given for each as well as a fun fact of why each one differs from the other.
Why Lebesgue? Understanding the Limitations of Riemann Integration
Sebastian Tous, mentored by Nick Christo
Abstract: While Riemann integration provides a straightforward and geometrically intuitive approach to finding the area under a curve, it falls short in cases involving discontinuous functions and is insufficient for nice general convergence theorems to hold. In this presentation, I will discuss the motivation behind Lebesgue integration, its relation to the Riemann integral, and some of its limitations.