On February 15th, from 4 to 7:35PM, in Matherly 0015, UMS will be hosting our second Undergraduate Talk Extravaganza! The event will consist of eight short undergraduate talks, and a break for pizza. The schedule for the event, including the talks and their abstracts, are listed below.
4:00 – 4:20: Omar Elbasri, “The First Computer Scientists were just Mathematicians (In Disguise…)”
Abstract: My talk is intended to be a brief overview of the intersection between math and computer science and how the first computer scientists, such as Alan Turing, were just mathematicians. I’ll probably briefly touch on turing machines and type systems, maybe the Busy Beaver numbers, but it’ll be more about the history of the field’s connections to mathematics.
4:25 – 4:45: Sarah Bell, “What is a Topology, Really?”
Abstract: The notion of a topological space lacks apparent meaningful substance in of itself as it is more of a logical than mathematical one. The study of topology is about gauging various layers of ‘spatialness’ and what particular property they yield, so that spatial reasoning can cover the most broad fields. Topology share a strong analytic connection, and its pathological nature is tied to the same reason real functions ‘as a rule’ can’t be uniformly defined. To understand the reality of spaces and of functions, topology is essential, as function spaces, like the space of continuously infinitely differentiable functions, are modeled not as a metric space but as a topological space.
4:50 – 5:10: Andersen Wall, “Number Theory in Encryption”
Abstract: This talk aims to briefly introduce the audience to some of the applications of number theory to encryption. The main focus of the talk will be the math behind RSA public key cryptography. Properties of both prime numbers and modular arithmetic will be discussed.
5:15 – 5:35: Sierra Edelstein, “The Mathematics of Crochet: An Expository Presentation on Crochet Modeling of Mathematical Objects”
Abstract: Mathematicians from around the world have developed methods of transforming complex and abstract mathematical concepts into figures that can be realized through methods of crochet and closely related yarn techniques, like knitting. These models can be found in a variety of disciplines within math, including, but not limited to, Hyperbolic Geometry, Abstract Algebra, and Topology. I highlight various applications of crochet to represent certain mathematical concepts in a geometric manner. In particular, I focus on developing and understanding models of Hyperbolic Planes developed by Daina Taimina in her book “Crocheting Adventures with Hyperbolic Planes,” representations of a specific type of fractal, the space-filling fractal, with crochet and techniques of knitting torus knots and links developed by Kyle Calderhead and Sarah-Marie Belcastro, respectively, in the book “Figuring Fibers,” and a representation of the group of symmetries of a regular hexagon developed by Andrea Heald.
5:40 – 6:00: Break for Pizza
6:00 – 6:20: Dylan Burke, “Unraveling High-Dimensional Data with Manifold Learning Techniques”
Abstract: Manifold learning techniques offer powerful tools for understanding the underlying structure of high-dimensional data, facilitating visualization, dimensionality reduction, and pattern recognition tasks. This talk delves into the mathematical principles behind manifold learning methods, with a focus on techniques such as t-Distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP). Through intuitive explanations and visual demonstrations, attendees will gain insights into how manifold learning uncovers hidden patterns and relationships in complex datasets, enabling more effective data analysis and interpretation in a variety of fields.
6:25 – 6:45: Austin Lam, “A Full Description of Electric Fields in Space and Time”
Abstract: Coulomb’s law, although powerful, is sufficient only for static charges. A general method of calculating electric fields given an initial time-dependent charge and current is derived and used for a simple example. Prerequisite: Vector calculus, delta functions. step functions. Recommended: Green functions, electromagnetism
6:50 – 7:10: P Beall, “An Equivalent Form of Choice in Linear Algebra”
Abstract: The Axiom of Choice is equivalent to the assertion that every vector space has a basis. First, we review transfinite recursion. Next, we use transfinite recursion and the Axiom of Choice to construct a basis for an arbitrary vector space. Conversely, we follow a non-obvious proof that the Axiom of Choice follows from the assertion that every vector space has a basis.
7:15 – 7:35: Sai Sivakumar, “Circulant Matrices and Convolutions”
Abstract: We will define circulant matrices and summarize basic properties they satisfy, then discuss what the circular convolution of $n$-tuples is and reveal how they are implemented as linear transformations. Lastly, we peek at the effect the discrete Fourier transform has on the circular convolution.