Math+ 2023 ran from **May 22** until **July 14, 2023**. The 2023 program featured 5 projects and 22 undergraduate student researchers.

This page will be updated soon with project reports.

### Projects for Math+ 2023

**Analysis of preconditioned stochastic gradient descent with non-convex loss**

**Project leader:** Professor **Jing An****Project manager:** Victor Amaya**Team members: **Will Chen, Ethan Ouellette, Nick Sortisio, Shuhuai Yu

Stochastic gradient descent (SGD) is a popular optimization algorithm used to train large scale machine learning models. Despite its widespread use, there are a lot of unanswered questions about the theory behind its empirical behaviors, especially in so-called “deep” neural networks. Over time people have made improvements on SGD, leading to a variety of SGD variants with different predictive performance and efficiency. In order to understand the performance of SGD variants, we derived mathematical models of these SGD algorithms as stochastic differential equations (SDEs) to study how certain SGD algorithms systematically lead to particular performance characteristics. For example, a wide class of SGD algorithms known as preconditioned SGD are shown to lead to models that do very well on data they have seen before, but perform worse when exposed to a new setting. We also studied the adequacy of different SDE representations of SGD and their implications for the theoretical performance guarantees we can derive, an area of active debate in the field. Our work contributes to our understanding of how SGD and its variants behave theoretically, in order to allow machine learning practitioners to make informed decisions in the process of building models. **Read more**

**Moduli spaces of stable weighted hyperplane arrangements**

**Project leader:** Professor **Haohua Deng****Project manager:** Chongyao Chen**Team members: **Ronan Hallinan, Erick Jiang, Santino Panzica, Felicia Wang

Our project aimed to investigate the properties of a particular algebraic concept called moduli spaces. Intuitively, moduli spaces "parametrize" certain types of mathematical objects—in our case they parametrize "weighted stable hyperplane arrangements." Moduli spaces have many connections to deep topics within algebraic geometry, but we attempted to examine them using largely undergraduate-accessible techniques. We were particularly interested in the boundaries of these spaces, which is where their "nice" properties start to degenerate, and how various functions behaved on these boundaries. **Read more**

**Computing hyperbolic structures from link diagrams**

**Project leader:** Professor **Calvin McPhail-Snyder****Project manager: **Jason Ma**Team members: **Rebecca Lan, Khiyali Pillalamarri, Lorenzo Valerio, Wendy Wang

Take a string and tie it into a knot, then put the knot in hyperbolic space. The knot complement is all of the space around the knot, excluding the knot itself. Our project aims to understand the knot complement by dividing it up into tetrahedra. We make this problem algebraic by drawing a diagram of the knot, then assigning three numbers *(a, b, m)* to each segment of the diagram. For the tetrahedra to glue together correctly the various *(a, b, m)* must satisfy certain equations. Our project explores these equations, particularly their application to links. Links are like knots, except that they allow for multiple strings to be tied together. This makes the equations more difficult to solve, because for a knot *m* is the same everywhere, but for a link there is a different *m* for every string. *Read more*

**Automated theorem proving and proof verification**

**Project leader:** Professor **Colleen Robles****Project manager:** Chun-Hsien Hsu**Team members: **Yannan Bai, Annapurna Bhattacharya, Stavan Jain, Kurt Ma, Ricardo Prado Cunha, Anoushka Sinha

A mathematical proof is a logical argument that aims to establish the validity/soundness of a mathematical claim. Traditionally, mathematical proofs have been done using intuitive reasoning and natural (i.e. spoken) language. Such informal proofs rely on the reader’s interpretation and cannot be verified algorithmically. Formal proofs, on the other hand, are written using precise rules and syntax which allows every assumption and claim made within a proof to be checked by a computer, and demands that all such assumptions and claims be justified using predefined rules. Proofs written formally are thus much more rigorous and certain. The verification of a formal proof is done using a proof assistant: software that helps in developing and checking mathematical proofs. For our project we worked with one such proof assistant: Lean. Lean is open source software which allows encoding formal proofs in its own functional programming language. Using Lean we have been working on developing a teaching aid for Duke’s introductory linear algebra course—Math 221. It plays out like a game and allows students to write formal proofs in linear algebra. **Read more**

**Applications of nonlocal operators**

**Project leader:** Professor **Logan Stokols****Project manager: **Karim Shikh Khalil**Team members: **Billy Cao, Alvaro Gonzalez, Erik Novak, Ra Pryor

In our project, we explored the applications of nonlocal operators in peridynamics. Peridynamics gives a non-local description of the behavior of particles and the deformation of bonds between particles in a solid. So, to understand the behavior of a particle, our model takes the positions of nearby particles around it into consideration. The goal of our project was to create an equation that can model the behavior of particles and the bonds between them, thought of as springs, when a fixed force is applied at the edge of our object. We also aimed to use numerical processes on our model so that we could corroborate our equation with known real world behavior. In our project we were able to derive a nonlocal operator, define and apply the inclusion of stress forces on the boundary of an object, and create a digital simulation of them for specific peridynamic spring model behavior. For the simulation, we approximate the object and its boundary through an evenly spaced lattice, and computed time-steps of effect on each lattice point over a period of time. **Read more**