# Applications

**Applications for Autumn 2019 have now opened! Please see below for important dates and information.**

**August 28: **Graduate student mentor applications open (Apply Here).

**September 6: **Graduate student mentor applications due at 5:00 pm.

**September 11: **Undergraduate student applications open (Apply Here).

**September 18: **Undergraduate student applications due at 5:00 pm.

**September 23: **Latest possible time that we will announce project pairings.

**Project Proposals
**

Project proposals for the upcoming quarter will be given below, including required courses and brief project descriptions. We expect to run approximately 10 of these projects each quarter, depending on student interest. The descriptions are roughly divided into 3 categories, depending on prerequisites.

**Beginner Level: May require some Calculus**

**Number Theory!
**

*Prerequisites:*Calculus may be useful as far as “mathematical maturity” goes, but anyone who is enthusiastic about math and learning new things can benefit!

*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**

*Prerequisites:* Familiarity with some programming language is preferred.

*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.

**Procedurally Generated Art**

*Prerequisites: *Required: CSE 331 (or equivalent); Preferred: familiarity with vector graphics, image classification.

*Mentor: *Tyler Chen

“Pareidolia is the tendency to interpret a vague stimulus as something known to the observer, such as seeing shapes in clouds, seeing faces in inanimate objects or abstract patterns, or hearing hidden messages in music.” The aim of this project is to generate unique, visually appealing, minimalist depictions of common objects automatically.

**Group Theory and Brain Swapping**

*Prerequisites: *Math 126

*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**

*Prerequisites:* Math 126

*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.

**Intermediate Level: Require Math 300 (proofs) and possibly other 300-level courses**

**Hyperbolic Geometry**

*Prerequisites: *Calculus, and have familiarity with complex numbers

*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?**

*Prerequisites: *Math 327 and 395

*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.

**The two projects below are roughly the same and can run simultaneously. Candidates may rank these projects equally if they would be happy with either mentor.**

**The two projects below are roughly the same and can run simultaneously. Candidates may rank these projects equally if they would be happy with either mentor.**

**Calculus of Variations (I)**

*Prerequisites: *Previous experience with analysis (eg. 327) and comfort with proofs would be extremely helpful.

*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)**

*Prerequisites*: Math 324 is certainly needed

*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.

**Only one of the two following projects will be selected to run. Applications to these two projects will be considered together and only one person will be chosen from among them.**

**Only one of the two following projects will be selected to run. Applications to these two projects will be considered together and only one person will be chosen from among them.**

**Ill-Posed Problems and Machine Learning**

*Prerequisites:* Differential equations, numerical analysis and real (basic level) analysis

*Mentor:* Kirill Golubnichiy

We will learn basic techniques of minimization.

**The Regularization Problem**

*Prerequisites:* Differential equations, numerical analysis and real (basic level) analysis

*Mentor:* Kirill Golubnichiy

We will study the idea of numerical analysis and how it applies to machine (deep) learning.

**Advanced: Require upper-level (400-level) mathematics courses**

**Programming a Quantum Computer**

*Prerequisites: *Math 402+, Math 424+

*Mentor: *Caleb Geiger

We will dive deep into the creation of quantum circuits, how quantum computers break all modern cryptographic schemes, and discuss the algorithms of Shor and Grover.