# AMS Student Chapter Seminar

The AMS Student Chapter Seminar is an informal, graduate student-run seminar on a wide range of mathematical topics. Pastries (usually donuts) will be provided.

Everyone is welcome to give a talk. To sign up, please contact one of the organizers with a title and abstract. Talks are 30 minutes long and should avoid assuming significant mathematical background beyond first-year graduate courses.

The schedule of talks from past semesters can be found here.

## Fall 2018

### September 26, Vladimir Sotirov

Title: Geometric Algebra

Abstract: Geometric algebra, developed at the end of the 19th century by Grassman, Clifford, and Lipschitz, is the forgotten progenitor of the linear algebra we use to this day developed by Gibbs and Heaviside. In this short introduction, I will use geometric algebra to do two things. First, I will construct the field of complex numbers and the division algebra of the quaternions in a coordinate-free way. Second, I will derive the geometric interpretation of complex numbers and quaternions as representations of rotations in 2- and 3-dimensional space.

### October 3, Juliette Bruce

Title: Kissing Conics

Abstract: Have you every wondered how you can easily tell when two plane conics kiss (i.e. are tangent to each other at a point)? If so this talk is for you, if not, well there will be donuts.

### October 10, Kurt Ehlert

Title: How to bet when gambling

Abstract: When gambling, typically casinos have the edge. But sometimes we can gain an edge by counting cards or other means. And sometimes we have an edge in the biggest casino of all: the financial markets. When we do have an advantage, then we still need to decide how much to bet. Bet too little, and we leave money on the table. Bet too much, and we risk financial ruin. We will discuss the "Kelly criterion", which is a betting strategy that is optimal in many senses.

### October 17, Bryan Oakley

Title: Mixing rates

Abstract: Mixing is a necessary step in many areas from biology and atmospheric sciences to smoothies. Because we are impatient, the goal is usually to improve the rate at which a substance homogenizes. In this talk we define and quantify mixing and rates of mixing. We present some history of the field as well as current research and open questions.

### October 24, Micky Soule Steinberg

Title: What does a group look like?

Abstract: In geometric group theory, we often try to understand groups by understanding the metric spaces on which the groups act geometrically. For example, Z^2 acts on R^2 in a nice way, so we can think of the group Z^2 instead as the metric space R^2.

We will try to find (and draw) such a metric space for the solvable Baumslag-Solitar groups BS(1,n). Then we will briefly discuss what this geometric picture tells us about the groups.

### October 31, Sun Woo Park

Title: Induction-Restriction Operators

Abstract: Given a "nice enough" finite descending sequence of groups $\displaystyle{ G_n \supsetneq G_{n-1} \supsetneq \cdots \supsetneq G_1 \supsetneq \{e\} }$, we can play around with the relations between induced and restricted representations. We will construct a formal $\displaystyle{ \mathbb{Z} }$-module of induction-restriction operators on a finite descending sequence of groups $\displaystyle{ \{G_i\} }$, written as $\displaystyle{ IR_{\{G_i\}} }$. The goal of the talk is to show that the formal ring $\displaystyle{ IR_{\{G_i\}} }$ is a commutative polynomial ring over $\displaystyle{ \mathbb{Z} }$. We will also compute the formal ring $\displaystyle{ IR_{\{S_n\}} }$ for a finite descending sequence of symmetric groups $\displaystyle{ S_n \supset S_{n-1} \supset \cdots \supset S_1 }$. (Apart from the talk, I'll also prepare some treats in celebration of Halloween.)

### November 7, Polly Yu

Title: Positive solutions to polynomial systems using a (mostly linear) algorithm

Abstract: "Wait, did I read the title correctly? Solving non-linear systems using linear methods?” Yes you did. I will present a linear feasibility problem for your favourite polynomial system; if the algorithm returns an answer, you’ve gotten yourself a positive solution to your system, and more than that, the solution set admits a monomial parametrization.

### November 14, Soumya Sankar

Title: The worlds of math and dance

Abstract: Are math and dance related? Can we use one to motivate problems in the other? Should we all learn how to dance? I will answer these questions and then we will have some fun with counting problems motivated by dance.

### November 28, Niudun Wang

Title: Continued fraction's bizarre adventure

Abstract: When using fractions to approximate a real number, continued fraction is known to be one of the fastest ways. For instance, 3 is close to pi (somehow), 22/7 was the best estimate for centuries, 333/106 is better than 3.1415 and so on. Beyond this, I am going to show how continued fraction can also help us with finding the unit group of some real quadratic fields. In particular, how to solve the notorious Pell's equation.

### December 5, Patrick Nicodemus

Title: Applications of Algorithmic Randomness and Complexity

Abstract: I will introduce the fascinating field of Kolmogorov Complexity and point out its applications in such varied areas as combinatorics, statistical inference and mathematical logic. In fact the Prime Number theorem, machine learning and Godel's Incompleteness theorem can all be investigated fruitfully through a wonderful common lens.

### December 12, Wanlin Li

Title: Torsors

Abstract: I will talk about the notion of torsor based on John Baez's article 'Torsors made easy' and I will give a lot of examples. This will be a short and light talk to end the semester.

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