# Spring 2019

In Spring 2019, important dates and times for WDRP are as follows:

**Wednesday April 3rd, 5:00 pm: **Start-of-quarter kick-off event in PCAR 297.

**Wednesday May 1st, 5:00 pm: **Mid-quarter event (for undergraduate students only) in PCAR 297.

**Wednesday June 5th, 5:00 pm: **End-of-quarter presentations in PCAR 297 and 492.

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 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. To facilitate meeting we have PCAR 297 reserved on Wednesdays from 5:00-6:20 pm throughout the quarter.

Spring 2019 Projects

**How Computers Send Secret Messages
**

*Mentee: Hyesu Lee*

*Mentor: Caleb Geiger*

When sending personal information such as emails or passwords across the internet, how do you stop an eavesdropper from learning your secret information? If you and the party you are talking to are aware of a secret language only you two speak, then it’s reasonable to expect privacy; however, if you have never met the other party, how do you agree on a secret language without the eavesdropper also learning the language? It might seem almost impossible, but mathematics is here to save the day! Two of the most prominent cryptographic algorithms used today (RSA and ECC) rely on some incredibly beautiful and interesting mathematics to send messages securely. In learning how to send secret messages, we will dive into some interesting group theory and number theory, and perhaps as a little bit of computer science to try and implement some of these ideas. It does not matter if you have taken Math 300 before, as we will start from wherever you are at in your learning and work our way up! Anyone is welcome!

**Insights into Game Theory**

*Mentee: Kenny Le
Mentor: Stark Ledbetter*

Mathematical game theory can be applied to medical school matching, social justice models, and strategies for cooperative games. We will read Insights into Game Theory: An Alternative Mathematical Experience, by Ein-Ya Gura and Michael B. Maschler.

**Conservation Laws of Partial Differential Equations
**

*Mentee: Rocky Schaefer*

Mentor: Ravi Shankar

Mentor: Ravi Shankar

In an ordinary differential equation, a “first integral” or “conservation law” is a constant on solutions of the equation, i.e. d I(x)/dx = 0. The conservation of energy is the classical example from Newtonian mechanics. How do first integrals generalize to partial differential equations in multiple independent variables, and to what extent do they influence the behavior of solutions? We will be exploring the 2002 paper by Rosenhaus in Journal of Mathematical Physics, where conservation laws sometimes imply non-existence of solutions!

**Fourier Analysis
**

*Mentee: Derek Tseng*

Mentor: Kevin Chien

Mentor: Kevin Chien

This is an introduction to Fourier series and the Fourier transform. Roughly speaking, we look at how functions decompose into component frequencies. Fourier analysis is not only ubiquitous in engineering and the hard sciences, but also widely used in pure math. After some basic theory, we will be able to explore surprisingly diverse applications, depending on interest and time: possibilities include the Heisenberg uncertainty principle, the wave equation, and Dirichlet’s theorem. We will use Stein and Shakarchi’s book, “Fourier Analysis.”

**Beyond Linear Algebra
**

*Mentee: Yifei Chen*

Mentor: Alessandro Slamitz

Mentor: Alessandro Slamitz

We’ll continue from the topics of linear algebra seen in Math 308 to see more advanced linear algebra. We can both take a very theoretical approach or an applied approach depending on your personal interest.

**Abstract Applications of Linear Algebra
**

*Mentee: Hongda Li*

Mentor: Graham Gordon

Mentor: Graham Gordon

Are you curious about applications of linear algebra to other areas of math, including discrete geometry, graph theory, and algorithms? We’ll explore these by studying some of the examples found in “33 Miniatures” by Jiřì Matoušek.

**Heads or Tails?
**

*Mentee: Joshua Ramirez*

Mentor: Anthony Sanchez

Mentor: Anthony Sanchez

You may recall from your probability class the Law of Large numbers which tells us that the average of several rolls of a fair 6-sided dice is 3.5. Or maybe you remember the Central Limit theorem from a statistics class that tells us that many data sets tend to follow a bell shaped curve. Less known is the Large Deviations estimate that answers the informal question of, “how likely trials of a random process are to deviate from the mean?” These three theorems are concerned with the structure of many trials of a random experiment and are called Limit Theorems. We will explore these theorems and other Limit Laws in the context of coin flips, making this project extremely concrete and hands on! While this setting may seem too restrictive, much of the complexity of probability is already demonstrated through the experiment of flipping coins. We will use the wonderful text, “Heads or Tails” by Lesigne.

**Fractals and Geometry
**

*Mentee: Chen Xu*

Mentor: Max Goering

Mentor: Max Goering

Fractals are a type of sets that exhibit exotic geometric properties and arise in many different ways. Through fractal-like constructions, you could create a continuous function that maps the interval [0,1] onto any $n$-dimensional cube. Alternatively, you can create sets that don’t even have an integer dimension, and still manage to discuss their size! In the project we will explore some parts of fractal geometry.

**The Fourier Transform and Fourier Series**

*Mentee: Ayush Kanthalia
Mentor: Josh Southerland*

Fourier analysis has played a huge role in mathematics over the last 200 years as it slowly evolved into one of the most powerful tools in an analyst’s toolkit. It is a sophisticated topic, but one with many applications, ranging from signal processing in astronomy and geology to music. Whether you are more inclined to study algebra and number theory, or geometry and analysis, there is a project waiting for you!

**Graph Theory**

*Mentee: Youngmin (Janice) Kim
Mentor: Sami Davies*

The “graphs” we’ll be studying are not at all like the ones you might have seen in calculus or pre-calc! Graphs are structures which model relationships between a set of objects. They relay huge amounts of information about the underlying system, making them one of the most fundamental objects of study in math and computer science.

**Mathematics of Medical Imaging**

*Mentee: Jesse Loi
Mentor: Nikolas Eptaminitakis*

We will explore the mathematical ideas underlying widely used medical imaging techniques such as Computed Tomography (CT). We will focus on studying the Fourier Transform and Radon Transform and understanding how they relate to tomography. We will be using the book “Introduction to the Mathematics of Medical Imaging” by Charles Epstein. This project will be an introduction to Inverse Problems, a very active and exciting research area in Mathematics.

**The Geometry of Polynomials**

*Mentee: Weiyi Hu
Mentor: Tejas Devanur*

Explore the inherent geometry of polynomial equations. We will use a computational approach and depending on interest, learn about projective geometry and applications to robotics along the way. Material will come from Ideals, Varieties and Algorithms by Cox, Little and O’Shea.

**Geometric Group Theory
**

*Mentee: Allen Yuan*

Mentor: Yiping Hu

Mentor: Yiping Hu

Typical undergraduate algebra curriculum focuses mainly on finite groups. However, students can graduate without being exposed to the vast and interesting world of infinite groups. This project will be a gentle introduction to this exciting, relatively new field with the book “Office hours with a geometric group theorist.”