The Undergraduate Seminar in Mathematics is an hour-long talk, held every week. There you can hear about some of the many deep and interesting areas of mathematics beyond what you would see in the classroom of most math classes. The talks are designed to be accessible to any college student; little to no math background is required. The seminar occurs most weeks on **Tuesdays from 5:30-6:30 pm, in Stevenson 1432**. Come by and enjoy some pizza before the talk, and hear about something new. We hope to see you there!

**Spring 2024’s talks:**** **

**January 23: **Denali Relles

**Title:** Manifolds: Many Shapes

**Abstract. **A manifold is an important concept in many areas of math, from the most abstract to even some applications. Also, manifolds live in a wonderful space right on the edge of what we can visualize conceptually. Luckily, one does not need to know particularly high-level math to understand what manifolds are and why we care about them. In this talk, I will explain what a manifold is, and why we might care about them.

**February 6: **Dan Margalit

**Title:** Fixed points, knots, and matrices.

**Abstract. **We will introduce several fixed-point theorems in mathematics, such as the Brouwer fixed point theorem, the Borsuk-Ulam theorem, the Smith fixed point theorem, and the contraction mapping theorem. We will discuss applications of these theorems to matrix theory, game theory, and topology. We will give conceptual proofs of the first two theorems using knot theory. The talk will be accessible to any undergraduates with an interest in mathematics.

**February 13: **Ekaterina Rybak

**Title:** Undecidable problems in groups and cryptography

**Abstract. **Is math that powerful? Do there exist problems in math that are impossible to solve? Surprisingly, yes. One of the most famous examples is the halting problem. The halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. It is undecidable, meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. We will talk a little about groups and group presentations. Then we will look at some algorithmically undecidable problems in group theory. Also, we will discuss how undecidable problems in groups can be used in post-quantum cryptography. The talk will be accessible to all undergraduate students interested in math.

**February 27: **Junhwi Lim

**Title:** Geometric Interpretation of Elementary Complex Analysis

**Abstract. **Complex analysis is the complex number version of calculus. It allows one to compute complicated integrals and prove many useful inequalities in analysis. In this talk, first, we explain how to add and multiply complex numbers diagrammatically. Then, we introduce a way to visualize the complex version of differentiability by using circles and annuli. Lastly, as an application of this idea, we prove the uniqueness of analytic continuation and the maximum modulus principle.

**March 5: **Spencer Dowdall

**Title:** The Period Puzzle

**Abstract. **Consider a continuous function f from the unit interval [0,1] to itself. Since the source and target are the same, we can iterate the map and look at f^2(x) = f(f(x)) and f^3(x) = f(f(f(x))) and so on. This leads to some interesting dynamical questions. Firstly, a fixed point of the function is a point that stays fixed when you apply the function! That is, f(x) = x. More generally, a periodic point is a point x so that f^k(x) = x for some power k; in this case, the smallest power that works is called the period of the point. So a fixed point is just a periodic point of period 1. Now here is the basic question: For any given map, what periods are possible? Can you have a map with no periodic points? How about a map with just one fixed point but no other periodic point? Is there a map with points of periods 1, 2, and 3 but nothing else? What about periods exactly 2, 5, 11, 17? The answers to these questions are all surprising and reveal a mysterious order amongst the periods. In this talk we’ll use basic calculus to get the heart of the matter and solve the puzzle of the periods.

**March 26:** Sean McAfee

**Title:** Voting Power and the Shapley-Shubik Power Index

**Abstract. **In many voting situations (such as among a board of directors at a company), each voter may have a different number of votes than the others. In this case, how do we determine the power dynamics of the situation? What is gained or lost by members forming coalitions to vote in unison? We will look at some examples and introduce one way to measure voting power: the Shapley-Shubik Power Index. This will allow us to analyze some real-world power dynamics, such as the UN Security Council, the electoral college, and voting law changes in New York’s Nassau County.

**April 2:** Kai Toyosawa

**Title:** Knots and polynomials.

**Abstract. **We say two knots are equivalent if one can distort one knot into the other without breaking it. A fundamental but long-standing problem in knot theory is to classify all knots up to this equivalence. In this talk, we will discuss two polynomial invariants for knots, the Alexander polynomial and the Jones polynomial, and compute an example of knots that is not equivalent to its mirror image.

**April 9:**

**Title:** TBA

**Abstract. **TBA

All current Vanderbilt students and staff are welcome to attend, but pizza will be provided only to undergraduate students and speakers.

Check out all of the awesome talks we have had in the past!