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The Undergraduate Friday Seminar (UFS) is a weekly event aimed at showing undergraduate mathematics students a taste of what they can experience from the professional world of mathematics research. Every Friday, we sit down with a different speaker as they tell us all about the interesting work they do and some of the neat results from their field!

Unlike many seminars, we make it a point to have material presented at a level that even the youngest of mathematicians can grasp. We also include undergraduates as our presenters; all of which is done in an effort to cultivate a generation of mathematicians who can accurately and engagingly present their work and interests to others.

### Upcoming Events

*More information coming soon on the following:*

- March 12, Professor Olivia Beckwith
- March 19, Professor AJ Hildebrand
- March 26, Eliot Kaplan
- April 2, Professor Rich Sowers
- April 9, Professor Reuven Hodges
- April 16, Dr. Joey Palmer
- April 23, Movie:
*Secrets of the Surface*

#### Friday, February 5, 2021, 4 p.m. CST

**Speaker: **Vaibhav Karve, UIUC Math

**Title: **Satisfiability: Why some problems are easy, while others are hard

**Abstract:** I will introduce a 50-year-old problem called Boolean Satisfiability (or simply SAT) and I will explain why we should care about it. We will explore how an abstract-looking problem can end up being connected to circuits, airline scheduling, Rubiks cubes, chess, video games, and travelling salesmen. I will explain how small SAT are easy and how big SAT can be hard -- and how we quantify the hardness of a problem. By the end of the talk, we will have learned some computer science as well.

**Zoom Link: **Please email Derek Thomas

#### Friday, January 29, 2021, 4 p.m. CST

**Speaker: **Jared Bronski, UIUC Math

**Title: **The mathematics behind two puzzles

**Abstract:** I plan to talk about two puzzles with very elegant mathematical solutions. I would encourage you to think about (but not Google!) these puzzles, particularly the easier version of the first puzzle, before the talk. All of them can be found in the attachment below. We'll also need volunteers for a demonstration on Friday, so if you'd like to help please email Derek Thomas (at the email address below).

**Puzzle Info: **Download here.

**Zoom Link: **Please email Derek Thomas

**Contact Info: Interested in Presenting?**

If you have any questions about these weekly seminars, or wish to give a presentation during the semester, feel free to email us at undergradseminar@math.illinois.edu. (See below for a list of previous topics and presentations.)

**Previous Seminars (Fall 2020)**

**Speaker: **Alexi Block Gorman, UIUC Math

**Title: **Opportunities in the Illinois Geometry Lab

**Abstract:** Ever wonder what it means to do research in mathematics? Or are you interested in mathematical puzzles and pedagogy? Come learn about research and outreach opportunities that the Illinois Geometry Lab (IGL) provides for undergraduate students. The research students do in the IGL is designed to be understood--and performed--by undergraduate participants. The topics for these projects range from very applied mathematics to highly theoretical. I will explain the IGL's application process, and offer some tips on how to write a successful application. I will also mention some of the community engagement opportunities we offer to get involved with math outreach through the IGL. All are welcome!

**Speaker: **Eric Samperton, UIUC Math

**Title:** Computing with braids

**Abstract:** This talk will be a mash-up of some basic ideas in topology and computer science. The braid group on n strands is a mathematical abstraction of the collection of “braid patterns” one can form by braiding n ropes around each other. It plays an important role in many parts of topology and group theory. I’ll begin by introducing the braid group, it’s relation to knot theory, and the important concept of a "group action.” We’ll then see how various actions of the braid group can be used to simulate computers—both quantum and classical. We’ll learn some basic theoretical computer science along the way.

**Previous Seminars (Spring 2020)**

**Speaker: **Patrick Allen, assistant professor, member of the Number Theory Group @ Illinois

**Title: **The Arithmetic of Quadratic Fields

**Abstract: **It is often said that the prime numbers are the building blocks of the integers, the precise statement of which is the fundamental theorem of arithmetic: any integer greater than one can be factored uniquely as a product of prime numbers. What if we move beyond the integers? The simplest cases to consider are the analogues of the integers in what are called quadratic fields, which are number systems obtained from adding to the rational numbers the square root of some fixed integer. Whether or not these quadratic integers satisfy the analogue of the fundamental theorem of arithmetic turns out to be very subtle and both what is known and what is not known are rather surprising.

**Speaker: **Kyle Begovich, Mathematics and Computer Science Senior @ Illinois

**Title: **Python Workshop

**Abstract: **Kyle will be working at Google as a software engineer at the end of this semester. He'll be running a workshop on Python through Project Euler, a great way to get a jumpstart on learning some of the basics of solving problems through coding. No prior coding experience is needed for this workshop! Project Euler is an online platform for students in any discipline to work on “challenging mathematical/computer programming problems that will require more than just mathematical insights to solve”. In this time of remote work, a platform that is online and well-suited to developing problem-solving techniques can provide a good outlet for learning new skills, developing your analytic senses, and interacting with a community of mathematical thinkers. This seminar will help you set up an environment, discuss common approaches to work on these problems, and walk through some early problems to get you started.

**Speaker: **Sean English, J.L. Doob Research Assistant Professor

**Title: **How to Tile Your Bathroom: An Extremely Impractical Guide from a Mathematician

**Abstract: **Tilings have been considered by mathematicians for centuries and by artists for millennia. The main question for tiling problems involves asking if a small number of shapes can be used to cover an entire geometric region without gaps or overlaps. We will briefly talk about some of the history behind tilings, then we will explore many interesting different directions these sorts of problems can take. We will explore some questions as simple as "which regular polygons can tile the plane?" to questions as obscure as "do chickens give rise to a periodic tiling?" Disclaimer**:** Unless your bathroom is infinite in size, follows spherical or hyperbolic geometry, or has a floor that is more than two dimensional, this talk may not actually be helpful for tiling your bathroom

**Title: **Julia Robinson and Hilbert's Tenth Problem (film)

**Abstract: **Julia Robinson was the first woman elected to the mathematical section of the National Academy of Sciences, and the first woman to become president of the American Mathematical Society. While tracing Robinson's contribution to the solution of Hilbert's tenth problem, the film illuminates how her work led to an unusual friendship between Russian and American colleagues at the height of the Cold War.

**Speaker: **Raghavendra Bhat, Freshman Mathematics Major, UIUC

**Title: **Prime Number Conjectures

**Abstract: **Freshman math major and author (*Math -- A Subtle Language of the Universe*) Raghavendra Bhat will present some of his prime number conjectures, which he has presented on many platforms across the world. His talk will focus on his recent conjectures and thoughts on number theory research and math in general.

**Speaker: **David Altizio, University of Illinois at Urbana-Champaign

**Title: **A Brief Introduction to Problem Writing

**Abstract: **Many of the best mathematics competition problems push the boundaries of pre-calculus math in unexpected ways. While these questions fuel the popularity of contests among middle and high school students, they also make competitions seem inherently unsustainable; constructing these questions appears to be a Herculean task. In this talk, I will shed some insight into how problems are made by exploring my eight-year-long journey through problem writing. In particular, I will discuss common writing philosophies, sources of inspiration, and the stories behind some of my favorite creations.

**Speaker:** Robert Joseph Rennie, University of Illinois at Urbana-Champaign

**Title: **What Is A Mathematics?

**Abstract: **In this talk, I will begin with a mathematization of the process of mathematization. We will then see how category theory and type theory provide a nice general framework for constructing and comparing systems of math. This discussion will motivate, without ever mentioning topological spaces, the study of higher toposes to anyone who cares about theoretical physics (not necessarily just those who study it). This talk requires only an interest in thinking about how math works.** **