# Applications

**Undergraduate applications for Winter 2020 open on December 2nd! Please see below for important dates and information.**

**November 19th: **Graduate student mentor applications open.

**November 27th: **Graduate student mentor applications due at 5:00 pm.

**December 2nd: **Undergraduate student applications open (Apply here).

**December 13th: **Undergraduate student applications due at 5:00 pm.

**December 27th: **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**

**What is Mathematics?
**

*Prerequisites:*None

*Mentor: *Raghavendra Tripathi

*Description: *Hardy’s answer to the question in the title was “mathematics is what mathematicians do.” In this project, we will cover materials from the two books:

a) *What is Mathematics?*

b) *Proofs from the Book.*

Both books cover a wealth of material ranging from algebra, analysis, graph theory, beautiful inequalities and more. The project does not aim to dive deep into a particular area of mathematics, but rather it provides an introduction to “all of mathematics.” This project will allow the student to develop an appreciation for the mathematical aesthetics.

To a fair degree both books (more than 70% content of both books) is written in a way that is accessible to high-school students. So, the only prerequisite it has is the curiosity to discover the beautiful landscape of mathematics (occasionally there are some cactuses here and there but largely it is beautiful).

**Analytic Number Theory
**

*Prerequisites:*Math 126

*Mentor: *Thomas Carr

*Description: *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**

**“Vector Spaces” over the Integers**

*Prerequisites:* Math 308

*Mentor: *Chengyuan Ma

*Description: *In Math 308, we study vector spaces over the real numbers, a number system where all four basic operations are defined. In the next ten weeks, we will generalize the concept of “vector spaces” to the integers, a number system where division is not always defined. Some possible topics include:

1. modules (vector spaces),

2. submodules (subspaces),

3. module homomorphisms (linear transformations),

4. Gauss elimination,

5. first isomorphism theorem (rank-nullity theorem),

6. minimal polynomial,

7. characteristic polynomial,

8. eigenvalues,

9. Cayley-Hamilton theorem.

If time permits, we will explore some of the following topics:

1. quotient modules,

2. tensor products of modules (high dimension matrices),

3. Smith normal form,

4. classification of finitely generated modules over the integers,

5. rational canonical form.

We will use *Abstract Algebra* by Dummit and Foote, chapter 10, 11, 12.

**Fourier Analysis and Its Applications**

*Prerequisites:* Math 327

*Mentor:* Kuan-Ting Yeh

*Description:* Fourier analysis is a classical tool to solve problems in differential equations, with numerous applications in physics, engineering, and economics. We will start from the very beginning, and build all the necessary tools along the way. In the end, we will apply the results we have developed to solve partial differential equations such as the heat equation and the wave equation.

**Markov Chains**

*Prerequisites:* Math 308 and 394

*Mentor:* Anthony Sanchez

*Description:* Markov chains are a random process that arise in many theoretical contexts and also in real world situations such as biology, economics, and the social sciences. The aim of this project will be to learn about Markov chains, their long term behavior, and first-step analysis. We will read out of the friendly text *Essentials of Stochastic Processes* by Durrett. This project will require some background in probability (such as Math 394) and linear algebra (Math 308).

**Computational Learning Theory**

*Prerequisites:* Math 394 and 395 or equivalents. Some programming experience recommended but not necessary.

*Mentor:* Qian Zhang

*Description:* The title of the project is from one of the chapters of Mitchell’s book *Machine Learning*. While you may or may not have seen some methods of machine learning, you probably know the power of it and there are actually strong mathematical properties behind these methods which allow us to know certain things in a given situation. Then the question naturally arises, what bound is given in what condition of a particular method? How computationally expensive is it to use it? It is possible to optimize a method to a better order? In the project you will be led to read about related and background materials, and try to discuss some current problems.

**Rational Points of Elliptic Curves**

*Prerequisites: *A course in ring theory (Math 402) is recommended.

*Mentor:* Juan Salinas

*Description**:* The problem of finding rational points on an elliptic curve is a difficult one. However, a theorem of Mordell says that, under the right circumstances, there exists a finite number of rational solutions such that one can find all other rational solutions of said elliptic curve using simple geometry. We will explore ideas such as the one above and, if time allows, view applications in elliptic curve cryptography (ECC). We will use Silverman and Tate’s book, *Rational Points on Elliptic Curves.*

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

**Homological Properties of Modules**

*Prerequisites:* 402, 403 (the latter can be taken concurrently)

*Mentor:* Chengyuan Ma

*Description:* Modules over a ring generalize the concept of vector spaces over a field. They can carry valuable information of the ring, hence becoming an important class of objects to be studied in various mathematical fields. In the next ten weeks, we will discuss modules and their homological properties, along with some basic category theory. Some possible topics include: modules, module homomorphisms, submodules, quotient modules, Hom functors, projective modules, injective modules, Ext functors, projective dimension, and global dimension. If time permits, we can explore one of the following topics:

1. Hopf modules over Hopf algebras,

2. 0th, 1st, 2nd Hochschild cohomology.

We will use *Abstract Algebra* by Dummit and Foote, chapter 10, 17.

**The numerical pricing of American** **options**

*Prerequisites:* Numerical Analysis, Mathematical Finance

*Mentor:* Guodong Zeng

*Description:* Do you want to be a specialist in derivatives? Here, we can offer you great experience on options. It can be a great point on your resume when you apply for position in some big banks. Join us!

**Topology and Geometry of Surfaces**

*Prerequisites:* Math 324, 327-328, 402. Math 427 could be helpful but is not necessary.

*Mentor:* Nikolas Eptaminitakis

*Description:* We will follow the book *Mostly Surfaces* by Richard Evan Schwartz to explore the topology and geometry of surfaces. We will discuss concepts such as the Euler characteristic of a surface, fundamental groups and covering spaces, and also Euclidean, spherical and hyperbolic geometry. If time permits, and depending on the student’s interests and background, we will discuss Riemann surfaces and translation surfaces. The level of experience needed lies somewhere between intermediate and advanced. Most topics are developed from scratch in the book but a student would get the most out of this project if they feel comfortable with real analysis (Math 327-328) and are to an extent familiar with groups (covered in Math 402). Some exposure to complex analysis (Math 427) would be helpful but is not necessary.

**Complex proofs of real theorems and vice versa**

*Prerequisites:* Complex analysis. A bit of functional/real analysis would be great but is not necessary.

*Mentor: *Raghavendra Tripathi

*Description:* The project will aim at reading parts of the book *Complex proofs of Real Theorems* by Lax and Zalcman. The book contains many beautiful results in real analysis, the proof of which relies on the techniques of complex analysis. For example the slickest proof of the fundamental theorem of algebra employs the complex analysis, so does the proof of uniqueness of Fourier coefficients and so on. The book assumes a good understanding of complex analysis at least a one quarter course in complex analysis. But, it takes one afar into the land of harmonic analysis, PDE, dynamics, functional analysis and shows how simple tools from complex analysis make things simple and elegant. To quote Herman Weyl, “the simplest and shortest path between two real analytic facts often goes via a complex path.”

**Inverse/Ill-posed Problems and Machine Learning**

*Prerequisites:* Analysis

*Mentor*: Kirill Golubnichi

*Description: *An ill-posed problem for the Black–Scholes equation for a profitable forecast of prices of stock options on real market data using deep learning.