# Fall 2019

Welcome to the 2019/2020 school year!

This Autumn 2019, the important dates and times for WDRP are as follows:

**Wednesday October 2nd, 5:00 pm: **Start-of-quarter kick-off event in DEN 110.

**Wednesday November 6th, 5:00 pm: **Mid-quarter event (for undergraduate students only) in DEN 110.

**Wednesday December 4th, 5:00 pm: **End-of-quarter presentations in **SAV 138/139.**

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 DEN 110 reserved on Wednesdays from 5:00-6:20 pm throughout the quarter.

**Completed Projects:**

**Number Theory!
**

*Mentee: *Yinxi Pan

*Mentor: *Jessie Loucks

The best numbers are arguably the natural numbers: 1, 2, 3, 4, 5, … (I could go on). Number theory is the study of these very nice numbers. Carl Gauss, a German mathematician, once said: “Mathematics is the queen of the sciences—and number theory is the queen of mathematics.” We will study number theory at an introductory level, with the goal of becoming well-acquainted with the natural numbers and their beautiful properties. We will also explore some of the history of the development of number theory into the fascinating field it is today.

Some topics include: Pythagorean triples, Fermat’s last theorem, factorization and the fundamental theorem of arithmetic, and congruences (with the option of including more or other topics). We will most likely read from Joseph H. Silverman’s very friendly book, *A Friendly Introduction to Number Theory*.

**Numerical Algorithms and Analysis**

*Mentee: *Aeron Langford

*Mentor: *Tyler Chen

Roughly speaking, numerical analysis is about understanding the effects of rounding errors on algorithms. When you add floating point numbers on a computer, the sum calculated by the computer is not always equal to the exact sum! This means that most algorithms will behave slightly (or even very) differently when implemented using floating point arithmetic than they would in exact arithmetic. Understanding how these errors can effect algorithms is of both practical and theoretical interest. Exact topic is flexible based on your interests and background.

**Group Theory and Brain Swapping**

*Mentee: *Frances Chang

*Mentor:* Caleb Geiger

Say you invent a machine that can swap the minds of any two individuals, but it has the unfortunate drawback that it will allow a swap between any particular pair of bodies only once. After performing a swap, is it possible to put the minds back in the correct bodies by possibly bringing in more people to swap?

This is the question proposed in an episode of *Futurama*, and it leads to some pretty interesting mathematics! In fact, some of the writers of the show, who are mathematicians, proved a nifty theorem on the topic.

In order to answer this, we will dive into some introductory group theory concepts, learn about the representation of groups within the symmetric group, and depending on interest/background perhaps prove up through Cayley’s theorem and the classification of finitely generated Abelian groups!

**Analytic Number Theory**

*Mentee: *Mary Elworth

*Mentor:* Thomas Carr

Is it possible to write down a formula that produces the prime numbers? We will investigate this and many other fascinating questions in Number Theory by (surprisingly!) using techniques from Calculus. We’ll also learn about the famous Riemann Hypothesis, an important unsolved math problem with a bounty of $1,000,000 for a correct solution! A possible text we could follow is “Introduction to Analytic Number Theory” by Apostol.

**Hyperbolic Geometry**

*Mentee: *Shine Sun

*Mentor: *Albert Artiles

Hyperbolic geometry has a rich history and beauty to it. It explores a world different from our day to day perception (at least at first glance). This course will bring together ideas from calculus and group theory to understand space, although very little knowledge is required. We will start from the ground up.

**Heads or Tails?**

*Mentee: *Joshua Ramirez

*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.

**Calculus of Variations (I)**

*Mentee: *Ruimin Zhang

*Mentor:* Kevin Chien

What do protein folding structures, Schwarzschild black holes, and the (ideally) shortest path you take as you’re running late have in common? They can all be found using techniques from calculus of variations! This is an extremely useful generalization of calculus that has applications in engineering, theoretical physics, geometry, and much more. If you’re interested in any of the sciences, this powerful tool is great to know! For this reading project, we’ll be using the classic text of the same name by Gelfand and Fomin. Our goal is to understand the basic theory and then apply it to an interesting problem.

**Calculus of Variations (II)**

*Mentee: *Alexander Waugh

*Mentor*: Haim Grebnev

Suppose that you have two points on a surface, how do you find the shortest path between them lying on the surface? The idea is simple: if you had your hands on the minimizing curve (a.k.a the “shortest path”) and went ahead and deformed it in any way, then the length of the curve must increase since that minimizing curve represents a local length minimizer. In other words, any variation to the curve’s geometry will always increase its length. Applying such an analysis locally to each point of the curve will lead to a differential equation that describes the geometry of the curve, which in turn can be solved (if possible) to actually find an explicit equation for the minimizing curve. This sort of variational technique is the central theme of an extremely important branch of mathematics called the “calculus of variations” which deals with the question of how to find the extrema of functionals (such as the arclength functional in our case) over spaces of functions such as curves, surfaces, hypersurfaces, etc. The differential equation described in our example above is a special case of a general equation called the “Euler-Lagrange equation” that in this particular case also takes the name of “the geodesic equation.” This branch of mathematics has a wide variety of applications such as in optimization, differential geometry, and of course physics. A solid knowledge of multivariable calculus is definitely needed for this material. In this reading project we will most likely use the excellent book “Calculus of Variations” by I. M. Gelfand and S. V. Fomin.

**Ill-Posed Problems and Machine Learning**

*Mentee: *Tianyang Wang

*Mentor:* Kirill Golubnichiy

We will learn basic techniques of minimization.

**Random Walks on Groups**

*Mentee:* Allen Yuan

*Mentor:* Yiping Hu

We will explore this beautiful connection between group theory and probability theory. In particular, we will discuss interesting examples such as random walks on lamplighter groups.