# Fall 2018

In Fall 2018, important dates and times for WDRP are as follows:

**Monday Oct 1st, 5:00 pm: **Start-of-quarter kick-off event.

**Monday Dec 3rd, 5:00 pm: **End-of-quarter presentations.

More details about these events will be provided here as they are available.

Outside of these times, undergraduate students and graduate students are expected to meet for 1 hour per week at a time of their choosing. In between meetings, undergraduate students are also expected to spend 3-4 hours per week reading and working on their own. Graduate students are expected to take the requisite time prepare for meetings on their own time as well.

**Fall 2018 Projects
**

**Generating Functions**

*Mentor: Peter Lin*

*Mentee: Zhongyan Wang
*

Generating functions are a powerful tool to compute and approximate combinatorial sequences. For example, the theory easily gives an explicit expression for the number of binary trees with n nodes. In this project we would try to understand how to use the theory to get explicit expressions for other counting problems in math and computer science. If time permits, we could also look into the connections with complex analysis. A possible reference is the book “Generatingfunctionology” by Herbert Wilf.

**Computability Theory**

*Mentor: Nico Courts
*

*Mentee: Jordan Brown*

Can a computer program be written that analyzes computer programs and determines whether they always terminate (vs getting caught in an endless loop)? This is an example of a question that investigates the limits of computability. We will begin by learning some fundamental mathematics and the core definitions before diving into what makes some problems computable and others not! The majority of the time will be spent exploring the nexus of mathematics and theoretical computer science. We will be following the book by Rebecca Weber. Depending on the interests of the student we can incorporate a small programming component, but this is not at all necessary.

**Combinatorial Games**

*Mentor: Lucas Van Meter
*

*Mentee: Madelynne Zornes*

Explore the math behind two player games like Nim and Hackenbush. The book “Winning Ways for Your Mathematical Plays” offers a wealth of different topics to think about. Along the way we might even discuss the surreal numbers.

**Representation theory**

*Mentor: Graham Gordon*

*Mentee: Thalya Paleologu
*

Like linear algebra? Like permutations? You might like representation theory! It is a way to study abstract algebraic structures called “groups” using only tools from linear algebra. Bruce Sagan’s book, “The Symmetric Group,” introduces representation theory in general and then discusses how it applies to the structure of the symmetric group. We could try reading some of this book, computing some examples, and doing some math together.

**Understanding Markov Chains**

*Mentor: Anthony Sanchez
*

*Mentee: Katie Gao*

The aim of this project would be to learn about Markov chains. Markov chains are a random process that arise in many theoretical contexts and real world situations. We will read out of the friendly text “Understanding Markov Chains”. As such, this directed reading would be perfect for any student who has some background in probability (Math 394 would be ideal) and wishes to further their knowledge in probability theory.

**Linear Algebra (Done Right!)**

*Mentor: David Simmons
*

*Mentee: Jean Galleon*

Interested in learning linear algebra beyond Math 308? Then this project is for you. Here you will have a chance to solidify your knowledge of vector spaces and linear transformations as we take a proof-based approach and work our way through the first few chapters of Linear Algebra Done Right by Sheldon Axler. Depending on student interest we will cover related topics as well.

**Mathematics of Medical Imaging**

*Mentor: Nikolas Eptaminitakis
*

*Mentee: Jesse Loi*

We will use the book “Introduction to the Mathematics of Medical Imaging” by Charles Epstein to discover some of the mathematical ideas underlying widely used medical imaging techniques such as CT tomography. This project will be an introduction to Inverse Problems, a very active and exciting research area in Mathematics.

**Optimal Mass Transport**

*Mentor: Mathias Hudoba de Badyn
*

*Mentee: Shannon Mallory*

Optimal mass transport is the rigorous study of moving piles of dirt (probability measures) with the least amount of effort (cost functions). It’s used in everything from training neural networks to controlling swarms of robots. In this project, we will go over the basics of optimization building up to the formulation of OMT, and then explore both theory and applications based on student interest.

**Differential equations. How symmetry properties are related to solutions**

*Mentor: Ravi Shankar
*

*Mentee: Ryan Bushling*

The elementary differential equations course seems like a “bag of tricks” with no discernible pattern, but in fact, all these methods for solving equations relate to the symmetry properties of the equation itself. Separation of variables comes from spatial shifts, homogeneous equations come from a dilation symmetry, the superposition principle for linear equations is itself a symmetry, etc. We will follow “Solving Differential Equations by Symmetry Groups” by Starrett.

**Symmetries**

*Mentor: Stark Ledbetter
*

*Mentee: Vicente Velasco*

Following “The Symmetry of Things” we will explore and classify different types of symmetry on the plane, the sphere, the torus and more. This is a beautiful synthesis of both algebra and geometry.

**Group Actions: Orbits, stabilizers and all that, plus applications to number theory**

*Mentor: Sam Roven
*

*Mentee: Blanca Viña Patiño*

Throughout many areas of math lie the idea of group actions. Anyone who is interested in learning any advanced mathematics can benefit from having the machinery of groups actions in their toolbox. What may seem like a relatively tame idea at first, will quickly reveal itself to be a powerful problem solving tool. Given an action of a group on a set, the notion of an orbit and stabilizer, related to the given action, will lead to many applications and settings in which group actions apply. After getting familiar with the basics, diving into some number theoretic applications will be next on the list, but the ultimate direction that the project takes can be flexible based on the interests of the students involved.