# AMS Student Chapter Seminar: Difference between revisions

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Title: Induced and Restricted Representations of a Sequence of Groups | Title: Induced and Restricted Representations of a Sequence of Groups | ||

Abstract: Let <math> G </math> be a finite group. Given a sequence of groups <math> G = G_n \supset G_{n-1} \supset \cdots \supset G_2 \supset G_1 = \{1\} </math>, we can construct a formal ring of induction-restriction operators on <math> G </math>, which we denote by <math> \mathbb{Z} \langle Ind(Res), Ind^2(Res^2), \cdots, Ind^n(Res^n) \rangle </math>. Using Frobenius reciprocity, we will show that the formal ring | Abstract: Let <math> G </math> be a finite group. Given a sequence of groups <math> G = G_n \supset G_{n-1} \supset \cdots \supset G_2 \supset G_1 = \{1\} </math>, we can construct a formal ring of induction-restriction operators on <math> G </math>, which we denote by <math> IR_G := \mathbb{Z} \langle Ind(Res), Ind^2(Res^2), \cdots, Ind^n(Res^n) \rangle </math>. Using Frobenius reciprocity, we will show that the formal ring <math> IR_G </math> is in fact a commutative polynomial ring of 1 variable. If time allows, we will also show that for a sequence of symmetric groups <math> S_n \supset S_{n-1} \supset \cdots \supset S_2 \supset S_1 = \{1\} </math>, the formal ring <math> IR_{S_n} </math> is isomorphic to a polynomial ring <math> \mathbb{Z}[x]/(f(x)) </math>, where <math>f(x)</math> is a polynomial of degree <math> n </math>. | ||

=== November 7, TBD === | === November 7, TBD === |

## Revision as of 22:32, 21 October 2018

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.

**When:**Wednesdays, 3:20 PM – 3:50 PM**Where:**Van Vleck, 9th floor lounge (unless otherwise announced)**Organizers:**Michel Alexis, David Wagner, Patrick Nicodemus, Son Tu

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: TBD

Abstract: TBD

### October 31, Sun Woo Park

Title: Induced and Restricted Representations of a Sequence of Groups

Abstract: Let [math]\displaystyle{ G }[/math] be a finite group. Given a sequence of groups [math]\displaystyle{ G = G_n \supset G_{n-1} \supset \cdots \supset G_2 \supset G_1 = \{1\} }[/math], we can construct a formal ring of induction-restriction operators on [math]\displaystyle{ G }[/math], which we denote by [math]\displaystyle{ IR_G := \mathbb{Z} \langle Ind(Res), Ind^2(Res^2), \cdots, Ind^n(Res^n) \rangle }[/math]. Using Frobenius reciprocity, we will show that the formal ring [math]\displaystyle{ IR_G }[/math] is in fact a commutative polynomial ring of 1 variable. If time allows, we will also show that for a sequence of symmetric groups [math]\displaystyle{ S_n \supset S_{n-1} \supset \cdots \supset S_2 \supset S_1 = \{1\} }[/math], the formal ring [math]\displaystyle{ IR_{S_n} }[/math] is isomorphic to a polynomial ring [math]\displaystyle{ \mathbb{Z}[x]/(f(x)) }[/math], where [math]\displaystyle{ f(x) }[/math] is a polynomial of degree [math]\displaystyle{ n }[/math].

### November 7, TBD

Title: TBD

Abstract: TBD

### November 14, Soumya Sankar

Title: TBD

Abstract: TBD

### November 21, Cancelled due to Thanksgiving

Title: TBD

Abstract: TBD

### November 28, Niudun Wang

Title: TBD

Abstract: TBD

### 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, TBD

Title: TBD

Abstract: TBD