# Math+ 2023

### Applications for Math+ 2023 are now open!

Math+ is a full-time 8 week program for collaborative student summer research in all areas of mathematics. Math+ 2023 is open to all current undergraduate students at Duke University and North Carolina Central University. We particularly encourage women and underrepresented minorities to apply.

Math+ 2023 runs from May 22 until July 14, 2023. The application deadline is February 15, 2023. The program consists of groups of typically 3-4 undergraduate students working together on a single project. Each team will be led by a faculty mentor assisted by a graduate student. Participants will receive a $4,000 stipend and may not accept other employment or take classes during the program. There are 5 teams planned for Math+ 2023. As part of the application, you will list the number(s) of the projects that you would like to apply for. Information about each of the projects is given below. If you have any questions about Math+ 2023, please email the program co-directors, Professors Lenny Ng (ng@math.duke.edu) and Heekyoung Hahn (hahn@math.duke.edu). ### Projects for Math+ 2023 1. Non-convex optimization using variants of stochastic gradient algorithms, led by Professor Jing An Stochastic gradient descent is widely used in training machine learning models due to its efficiency and simplicity. In recent years, many variants of stochastic gradient algorithms keep developing for practical improvements. This project aims to understand stochastic gradient algorithms’ behavior from a mathematical perspective. We will focus on two types of training algorithms, reparameterized and preconditioned gradient descents, which are not only popular for training neural networks in practice, but also have fruitful theoretical results obtained recently. In particular, the project plan is: ( 1) deriving evolution dynamics of deterministic reparameterized and preconditioned gradient descents in the continuous time limit. With those equipped, we study the convergence of gradient flow. (2) considering the stochastic version of reparametrized and preconditioned gradient descents. We will look at asymptotic and non-asymptotic properties of those stochastic algorithms. Moreover, can we say something about behaviors of stochastic algorithms in non-convex landscapes? Compared to standard stochastic gradient descent, reparameterization or preconditioning may change the training trajectories and result in different minimum selections. Participants should preferably have working knowledge of differential equations, numerical methods, probability and stochastic calculus. It would be helpful to have some programming experience, for example, in Python or MATLAB. 2. Moduli of weighted hyperplane arrangements and degeneration, led by Professor Haohua Deng This project aims at studying the moduli space parametrizing$n$hyperplanes in the$r$-dimensional complex projective space. For example, when$r=1$, this means the space of complex projective line$\mathbb{P}^1$with$n\$ marked points. Specifically, we are interested in some nice-behaved low dimensional families and their boundaries. These boundary strata can be determined by numerical or combinatorial data and have interesting geometric intepretation.

This problem has deep background in algebraic geometry, but the basic knowledge we need for this project are mostly linear/abstract algebra and combinatorics. We will learn more advanced concepts together if needed, and skills in mathematical programming may be helpful.

3. Computing hyperbolic structures from link diagrams, led by Professor Calvin McPhail-Snyder

Many (in some sense, most) interesting 3-manifolds admit hyperbolic structures, which means they admit metrics of constant negative curvature. Studying these structures is an important and well-established technique in 3-dimensional topology. To compute these in practice, one describes the manifolds using ideal triangulations, which determine a set of equations that can be solved for hyperbolic structures.

In this project, we will study a different (but related) method to compute these structures using link diagrams, which record how a collection of circles are knotted in ordinary 3-dimensional space. This method has a few advantages over the traditional techniques and is related to current research in low-dimensional and quantum topology. Our goal is to use it to work out some more families of examples of hyperbolic structures on link diagrams. Time permitting, we might reach other goals, like:
Implementing our method in the existing SnapPy software used for computational hyperbolic geometry.
• Using our method to compute other properties of the links, like hyperbolic volumes or A-polynomials.
• Applying our methods to tangles, which are links with open components.

Participants should have a good knowledge of linear algebra (Math 218/221) and be familiar with basic group theory (included in Math 401). Taking Math 401 is not necessarily required, but students who have not taken it might need to do some background reading. It would be helpful (but not necessary) to have some knowledge of topology and/or programming experience, especially with Python.

4. Automated theorem proving and proof verification, led by Professor Colleen Robles

In the last 20 years there has been a great deal of progress formalizing mathematics, including proof-verification by computer.   In this project we will work with the interactive theorem prover “Lean”.  The goal is to submit “theorems” to the Lean Mathematical library (which is under development).  After learning how to program in Lean, we will identify a set of theorems to formalize in Lean, outline a strategy to do so, and begin the process of formalizing the theorems.

No programming experience is necessary.  Participants should have working knowledge of linear algebra and single-variable calculus.

5. Applications of nonlocal operators, led by Professor Logan Stokols

Differential equations are at the center of modern physics, being an incredibly powerful tool for modeling physical phenomena. These models are fundamentally local: two pieces of a physical system can only interact when they are in the same location, and the derivative of a function at a point only depends on the function values infinitesimally close to said point.  However, in complex systems, sometimes it is helpful to think about nonlocal interactions. This requires us to look beyond differential equations into the world of nonlocal “integro-differential” equations. “Integro-differential” refers to the fact that nonlocal operators behave like derivatives, but are defined using weighted integrals.

We will study some simple nonlocal operators and how they apply to real-world phenomena. For example, the flocking behavior of birds, or the deformation of solid objects as they break and snap (peridynamics). We will begin by developing an analytic understanding of the operators involved, and then try to understand how they can be computed numerically. Since every single point in a nonlocal system interacts with every other point, these operators can be very inefficient to compute. This makes it both challenging and rewarding to implement them efficiently in computer models.

Students should be comfortable with multivariable calculus at a minimum. Coding experience (preferably Python) is also desirable. Additional experience which would be helpful includes prior exposure to real analysis, differential equations, computer modeling, and/or algorithm design.