Mailing List: https://forms.gle/f4XdQH874yM9CkFo7
When: Tuesday 2:30 - 3:30 pm
Where: Vincent 207
Organizers: Sam (coyle158 AT umn.edu), Amin (shari179 AT umn.edu), Tel (fahre062 AT umn.edu)
The Undergraduate Math Research Seminar (UMRS) is a seminar aimed to create a welcoming undergraduate research community at UMN. We will have opportunities for students to give low stakes talks on the research and reading they’ve done as well as hold sessions devoted to learning what research is like, the different areas of math research and how to create research presentations. If you’d like to be updated on the seminar, see the mailing list above. See here for the 2024-2025 schedule.
Sep 16: First Meeting
Sep 23: Samuel Coyle "Finite Spaces and Finite Cobordism Theory"
Abstract: At first glance, finite topological spaces may appear too simple to be of significant interest; however, they can encode a substantial amount of topological information. In this talk, we will first define finite spaces and see some of this rich topological knowledge encoded by looking at the difference between simple homotopy equivalences and weak homotopy equivalences with respect to Whitehead torsion. From here, we will take a brief digression to look at smooth manifolds and a surprising relation between Whitehead torsion and cobordism theory. Motivated by the case in the smooth manifold setting, we will be able to develop an appropriate setting to discuss cobordism in finite spaces. Time permitting, we will conclude with some recent results and open conjectures concerning finite cobordism, specifically focusing on the gap between simple homotopy equivalences and weak homotopy equivalences.
No prerequisite knowledge will be assumed.
Sept 30: Board Games and Social Hour
Join us for some fun and games with people who are also interested in Math!
Oct 7: Tel Fahrendholz "Dynamical Systems and The Lorenz Attractor"
The Lorenz Attractor is one of the most iconic examples of chaos in mathematics and science. This talk will begin with an introduction to dynamical systems, with a focus on the tools used to classify behavior in 2D systems. We will then see how these tools fail to describe the 3D Lorenz equations, and walk through some of the same steps Edward Lorenz himself followed as he explored their surprising outcomes. The journey will culminate in the emergence of a new concept made possible by the addition of an extra dimension: deterministic chaos.
Oct 14: Serena Hall "Groups and Symmetry"
As an introduction to group theory, we discuss the elementary elements of this area as they are presented in the text Groups and Symmetry by M.A. Armstrong. We define the group axioms and see how, from this definition, we derive the notion of cosets, normal subgroups, and quotient groups. At the point of realizing these objects, we have access to language to introduce and unravel Lagrange’s theorem about the order of finite groups.
Oct 20: Amin Shariff "Making Counting Easy: An Expository Look into Polya Theory"
Enumerative combinatorics is an area of math concerned with counting the number of distinct configurations for a set of discrete objects. Some examples of this are poker hands, configurations of chess pieces, isomers of a chemical compound, and the arrangement of atoms into a crystal lattice. Polya’s Enumeration Theorem (also known as Polya counting) is a powerful combinatorial method that counts the number of distinct arrangements, while considering geometric symmetry. This talk focuses on using symmetry-based counting techniques to better categorize structures under rotation and reflection. We will also discuss some applications of Polya counting, including isomer identification and grid enumeration.
Oct 28: Two Proofs and a Lie
Nov 4: Nathan Elango "Mathematics of Quantum 2D Materials"
Theoretical predictions from mathematics and experimental results from physics motivate each other. In this talk, I will briefly introduce some ways in which these two fields interact, before describing and motivating my research and its applications, which are examples of such interactions. I will only assume the mathematical and scientific background of a Freshman in a STEM-related major. I will introduce the basics of quantum mechanics, and of AB-stacked bilayer graphene. I will then discuss quantum geometry, particularly the quantum geometric tensor and my successful simulation of this quantity. Lastly, I will introduce slightly more advanced quantum mechanics, circular-dichroic angle resolved photoemission spectroscopy, and my challenges in simulating it.
Nov 11: Andrew Gill "Category Theory and Universal Properties"
Category theory is a beautiful and widely useful area of mathematics which has been immensely successful since its inception less than a century ago. Unfortunately, the field is not always accessible; categories are highly general and abstract objects, and many motivating examples of categories arise in algebraic geometry and homological algebra, which are by no means simple subjects. In this talk, we will first provide a simple motivation for categories with a question about the Cartesian product. We will then define categories and some related notions, emphasizing basic examples. Finally, we will return to our question, which can be answered by describing the universal property of the Cartesian product.
No knowledge beyond basic set theory is assumed.
Nov 18: Joint REU Panel with Math Club
We will be hosting a joint event with Math Club to talk about REU and research opportunities available to undergraduate students over the summer. The event will be hosted in Vincent 207 at UMRS usual time from 2:30 to 3:30.
Nov 25: No Meeting (Enjoy Thanksgiving's Break!)
Dec 2: TBD
Dec 9: TBD