# Applications

**Undergraduate applications for Autumn 2020 open on September 18th! Please see below for important dates and information.**

**September 18th: **Undergraduate student applications open (Link here).

**September 27th: **Undergraduate student applications due at 5:00 pm.

**October 2nd: **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.

**Explorer Level: Entry level undergraduates with some math experience**

There are no explorer level projects this quarter, but check back next quarter!

**Beginner Level: May require some calculus**

**Probability Theory IRL**

*Prerequisites:* Math 126

*Mentor:* Madeline Brown

*Description:* Using the book “Probability Tales” by Grinstead, Peterson, and Snell, along with some supplemental material, we will explore the probability theory behind real-life occurrences in sports, the stock market, and lotteries. These specific examples will serve as a simple and fun introduction to a variety of probability distributions and stochastic (random) processes. Any students who have finished the calculus sequence and are interested in learning about probability are encouraged to apply. This project would be useful when taken concurrently with Math/Stat 394, but no background in probability or statistics is necessary!

**Elliptic Curves**

*Prerequisites:* Math 126

*Mentor:* Thomas Carr

*Description:* Elliptic curves are notoriously complicated. Recently, these curves were the key to the proof of “Fermat’s Last Theorem”, a legendarily difficult math problem which had gone unsolved for over 350 years! In another recent use, the mind-boggling complexity of these curves turned out to be an advantage that can be exploited to make Bitcoin transactions virtually impossible to intercept. We’ll start to unravel the mysteries of these fascinating mathematical objects by following the book “Rational Points on Elliptic Curves” by Silverman and Tate. Depending on your interests, this project can also involve a programming aspect.

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

**Chip Firing; Algebra and Graph Theory Exploration**

*Prerequisites:* Math 300, Math 308, some exposure to Combinatorics

*Mentor:* Tafari Clark-James

*Description:* I would like to delve into some of my undergraduate research thesis work through some of the papers I cited in my thesis. We will also be taking a close look at my own thesis work, scouring for open problems. I hope to show the student some of the friendlier parts of abstract algebra, and how it can streamline the process of doing combinatorics. See https://www.msri.org/web/msri/education/for-undergraduates/msri-up/2016/research at the bottom of the page for more rough details.

**Dimensionality reduction of the Hodgkin-Huxley model**

*Prerequisites:* AMath 351 or Math 307; AMath 301 or coding experience; AMath 352 or Math 308

*Mentor:* Megan Morrison

*Description:* The Hodgkin-Huxley model is a four variable nonlinear dynamical system that describes the dynamics of action potentials in neurons. The FitzHugh–Nagumo model also describes action potentials in neurons, however it only uses two variables to describe the dynamics. The appeal of the Hodgkin-Huxley model is that each variable describes an individual entity in the cell (voltage or ion channel), while the appeal of the FitzHugh–Nagumo model is that it describes the dynamics with two variables instead of four. The FitzHugh–Nagumo model was derived from the Hodgkin-Huxley model using expert knowledge and a clever choice for variables. However, could a low-dimensional representation for this system be found without expert knowledge, and is the FitzHugh–Nagumo model in fact the best low-dimensional representation of this system? In this project we will use machine learning, as opposed to expert knowledge, to find a low-dimensional model for the Hodgkin-Huxley dynamical system and compare our resulting system to the FitzHugh–Nagumo model. Do we recover the FitzHugh–Nagumo model? We will explore the advantages of analyzing systems in a low-dimensional space and relate our findings back to the original system.

**Learning the intrinsic dimension of dynamical systems with machine learning and data-driven discovery **

*Prerequisites:* AMath 351 or Math 307; AMath 301 or coding experience; AMath 352 or Math 308

*Mentor:* Megan Morrison

*Description:* Many high-dimensional dynamical systems have dynamics that exist on a lower-dimensional manifold that is often not obvious. For example the Hodgkin-Huxley model is a four dimensional dynamical system that has intrinsic dynamics that exist in a two dimensional subspace — the FitzHugh-Nagumo model. Finding low-dimensional models is important because it makes systems easier to analyze and visualize and it reduces the computational expense of processes involving these systems. Deep neural network autoencoders can compress dynamics to a low-dimensional subspace. For example, previous studies have shown that two-dimensional linear embeddings can be found for some systems, while nonlinear systems can be found with a combination of autoencoders and SINDy. In this project we will use deep neural network autoencoders to successively compress high-dimensional dynamical systems to lower dimensional systems and measure the extent to which we can compress each system. We will test this process on randomly generated dynamical systems and canonical systems found in the systems biology literature. We will develop a process for finding the minimum dimension of the system while minimizing computational resources.

**Category Theory**

*Prerequisites: *Math 300, Math 308, some exposure to groups and rings would be helpful, but not necessary

*Mentor:* Eric Zhang

*Description:* Category theory is about structures and structure-preserving functions. For instance, group is an algebraic structure and group homomorphism is the corresponding structure-preserving function. In another context, continuous maps are structure-preserving functions for topological spaces. Category theory also finds its application in computer science. Don’t be intimidated by the word “structure.” I will give an intuitive example: in an Euclidean topological space (a plane), a line is transformed to another line by a continuous function. It can’t be transformed to two disconnected pieces of line segments. The abstruse wording “structure” is captured by the visually intuitive concept of connectedness. We will read together on Paolo Aluffi’s Algebra: Chapter 0 and teach each other. It will be an exciting and useful reading. Basic concepts such as groups and homomorphisms will be studied from the beginning so you don’t need a background in algebra. The plan can be adjusted to accommodate personal interest.

**Complex Analysis / Complex Numbers**

*Prerequisites: *Math 126, Math 300, it would be helpful to have taken Math 324

*Mentor:* Eric Zhang

*Description:* The real number field is complete but that is not the whole picture. Complex numbers kick in when we try to solve equations like x^2+1=0. It is a different structure (not ordered for instance) but provides further insights. When learning Calculus or Real analysis, have you wondered the Taylor series of real function exp (-1/x^2) at 0 (after you take care of the removable discontinuity) is identically zero? Thinking about it as a complex function may resolve the mysteries! Complex analysis sounds intimidating but it is not. In fact, many problems and theorems in complex analysis have a nicer structure than in the real case. So don’t be afraid! We will take a gentle touch on elementary complex analysis (requires Math 300), covering topics such as holomorphicity, Cauchy’s integral formula, and Residue theorem. If you don’t feel like doing a lot of theory, we can focus on the geometry and arithmetic property of complex numbers themselves instead (does not require Math 300). We will read Visual Complex Analysis by Tristan Needham and the plan may be adjusted to accommodate personal interest.

**Finite fields and error correcting codes**

*Prerequisites: *Math 308, 340

*Mentor:* Tuomas Tajakka

*Description:* When transmitting data across a computer network or through a difficult environment like deep sea or space, bits can flip and errors often occur. How can the recipient detect or even correct an error in a message sent through a noisy channel? We will explore this question through the lens of linear algebra over finite fields. We can use chapters 1 and 3 of “Finite Fields and Applications” by Mullen & Mummert as our guiding source, supplementing with necessary abstract algebra background. Check out this 3Blue1Brown video for a quick teaser!

**Graph Theory**

*Prerequisites: *Math 300 recommended, Math 308 would be helpful

*Mentor:* Sami Davies

*Description:* 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.

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

**Riemann Surfaces**

*Prerequisites: *Math 428, exposure to some group theory would be helpful

*Mentor:* Albert Artiles

*Description:* We will learn how to do calculus with complex numbers on surfaces. The surfaces where we can do this are called Riemann surfaces and they lie in the intersection of many branches of mathematics such as number theory, dynamics, hyperbolic geometry, etc.

**Abstract Nonsense and You**

*Prerequisites:* Math 404

*Mentor*: Caelan Ritter

*Description: *Sometimes, stuff is like other stuff. But how? Category theory—or, as it is sometimes known (by both its supporters and detractors), “abstract nonsense”—attempts to answer this question. Category theory is the study of objects and arrows between those objects. For example, in one category, the objects are groups and the arrows are group homomorphisms. In another, the objects are topological spaces and the arrows are continuous maps. Nonetheless, both categories have a notion of injections, isomorphisms, products, quotients, etc. In category theory, we ignore the internal structure of particular objects (say, the elements of a group or the points of a topological space) and instead characterize these concepts in terms of which objects have arrows to which other objects. We also explore connections *between* categories. One example: when you stir a cup of coffee, there is a point that doesn’t move. This is a theorem in topology, but one proof involves porting the problem over to the category of groups (using something called a functor) and then realizing that 0 is not equal to 1. Depending on your interest and background, we will likely focus on “The Rising Sea” by Ravi Vakil or “Category Theory in Context” by Emily Riehl. While category theory can be studied independently of any other field with only knowledge of basic set theory, it helps to have working familiarity with a few algebraic or geometric objects: vector spaces, groups, rings, topological spaces, smooth manifolds, etc. That way, when you see an abstract construction or concept, you know *what* is being abstracted.

**Ill-posed Problems/Inverse Problems**

*Prerequisites:* Math 426 (Analysis)

*Mentor*: Kirill Golubnichiy

*Description: *We will study finite difference / finite element methods and machine learning