2023 U-M REU Seminars

Ilir Ziba: Post's Lattice and Attack of the Clones

Mentor: Ronnie Chen

Given a set of cardinals $N$ and an underlying set $X$, a function clone is a set of functions $f: X^n \rightarrow X$ for $n \in N$, which contains all projection functions and is closed under composition.

In 1941, Emil Post fully characterized all clones of finite functions $f : 2^n \rightarrow 2$, where $n \in N$ in a lattice famously titled Post's Lattice, ordered by inclusion. We seek to extend the view of clones to include countably-infinite Borel functions and characterize this extended notion of Post's Lattice. This talk will focus on providing some of the general theory behind clones and polymorphisms, and painting a picture of some of the landscape of the extended lattice.

Audy Lebovitz: Minimal fibering degrees of toric varieties

Mentor: David Stapleton

The minimal fibering degree of an algebraic variety is the smallest d > 0 such that there exists a family of degree d curves sweeping out the variety. This invariant is complicated in general to understand, so a natural starting point is computing it for toric varieties. Toric varieties are a special class of algebraic varieties which can be constructed combinatorially from polytopes. So, one expects a combinatorial interpretation of the minimal fibering degree. We discuss an answer to this question in the case of toric surfaces, and some progress on the higher dimensional case.

Zuyuan Han and Haisu Ding: Signature Method in Variance Swap Pricing

Mentors: Qi Feng & Bingyan Han

Variance swaps are financial instruments which offers investors straightforward and direct exposure to the volatility of an underlying asset such as a stock or index. They are swap contracts where the parties agree to exchange a pre-agreed variance level for the actual amount of variance realized over a period. Our project is aimed at applying signature-based model in the pricing problem of the variance swap. We consider asset price models whose dynamics are described by linear functions of the time-extended signature of a primary underlying process, which in our case is a multidimensional continuous semi-martingale. The linearity of the signature model is a crucial tractability feature which allows to derive fast and accurate calibration results.

Andy Chen: A superparticle method for systems of coupled oscillators

Mentor: Danny Forger

Many biological behaviors with limit cycles, such as the Hodgkin-Huxley model of neuron firing, can be represented by coupled oscillators associated to a differential equation. The asymmetric particle population density method takes the limit as the number of oscillators goes to infinity, modeling the evolution of a continuous population distributed in phase space. The method represents the population as a sum of Gaussian particles, which must be split and combined over time. We describe some phenomena about the alignment of particles' major axes with their instantaneous velocity, and outline a new numerical method which simplifies the representation of particles on a limit cycle. 

Cruise Song: Optimization of Model Predictive Control using ilqr and Neural Net Work: an application on Mobile Robot System

Mentor: Dao Nguyen

Finding the best control for a dynamical system is a critical problem in many fields and its solutions are often bottlenecked by computational requirements. In particular, MPC is a very popular method for real-life process control in chemical, aerospace and biomedical applications. We specifically focus on MPC with iterative Linear Quadratic Regulator (iLQR). We are implementing an iLQR controller on a self-defined mobile robot system. If time permitting, we will then explore the combination of iLQR with Neural Network. 

Horacio Moreno-Montanes: Modeling 1-D Cold Electrostatic Plasma with a \\ Lagrangian Particle Method

Mentor: Robert Krasny

The conventional approach to plasma simulations utilizes the particle-in-cell (PIC) method, but PIC simulations often lose resolution as complex features develop in the plasma. This project develops an alternative Lagrangian particle method for a one-dimensional cold, electrostatic plasma with periodic boundary conditions. The primary objective is to develop an efficient method that preserves accuracy as the plasma evolves in time, in contrast to existing methods that . The plasma is described by the Vlasov-Poisson equations for the plasma distribution in phase space and the self-consistent electric field in physical space. The plasma is represented by discrete charged particles (electrons) with a neutralizing constant background distribution of positively charged ions. Two integration techniques, Euler's method and fourth order Runge-Kutta, are used to evolve the electrons, and regularization is applied to ensure continuity of the electric field. We investigate the effects of the numerical parameters, such as time-step size, number of particles, and the regularization parameter. Initial results indicate that the lack of continuity in the non-regularized problem contributes to error growth for both integration schemes. Future extensions of this work will (1) implement a particle insertion scheme to preserve the resolution of the plasma, and (2) apply the method to study the two-stream instability.

Eissa Haydar: An Introduction to Scott Complexity

Mentor: Matthew Harrison-Trainor

This talk will be an introduction to the ideas of Scott sentences and Scott complexity, where structures are studied through automorphism orbits and isomorphism with other structures via sentences in logic. I will give examples in the case of countable structures. The talk will end with the general ideas of my work for this REU, namely extending the concepts to topological, continuous structures, such as Polish spaces.

Liyang Shao: Weighted Diophantine approximation and Schmidt's games---from homogeneous to inhomogeneous

Mentor: Shreyasi Datta

It is well-known that the set of badly approximable numbers in one-dimensional Euclidean space is of Lebesgue measure zero, but of full Hausdorff dimension. In the 1960s, Wolfgang. M. Schmidt introduced some winning properties stronger than being of full Hausdorff dimension, via the notion of games which now bear his names. Recently, it is proven that the set of weighted badly approximable vectors has these winning properties in both R^n and the non-degenerate curves. The relevant techniques to approach these results, including dangerous intervals,  games that are equivalent or similar to Schimdt's games, and quantitative non-divergence estimates, will be discussed in the talk. Meanwhile, I will introduce how these results could potentially be extended from homogeneous case to inhomogeneous, which is exactly the key idea of the project by my mentor and me.

Nianchen Liu: The Ekedahl–Oort type of Artin-Schreier curves

Mentor: Patrick Daniels

It is of great interest to study Ekedahl–Oort (E-O) type of a curve C, which is the isomorphism type of the p-kernel group scheme J[p]. We can understand the E-O type by computing the Dieudonné module of J[p], which are in a bijective correspondence to an isomorphism class of Hasse-Witt triples. We will then compute the Ekedahl–Oort of Artin-Schreier curves by computing the Hasse-Witt triple and proceed to find other equivalent representations. 

Yueyang Ding: Mesh Generation for Solving Ion Channel Modeling using finite element method.

Mentor: Zhen Chao

To solve an ion channel dielectric continuum model using the Poisson-Boltzmann Equation (PBE), a mesh is required for the finite element method. In this research project, I discuss an approach to generate tetragonal meshes based on a given ion channel protein and a surface box. Our goal is to create software that can generate these meshes, consisting of tetragons classified into three regions: membrane, protein, and solvent.

Several processes are involved, such as converting the original pdb file to a pqr file, generating the surface mesh of boxes and the protein, using mesh generation software to generate tetragonal meshes, and using algorithms to extract the membrane and solvent areas and label the tetragons. In this research project, I focused on self-innovated algorithms for box surface generation and solvent extraction, while for other tasks, I used existing software packages like Tetgen and TMSMesh.

Ashvin Pai: Investigating Interpretability of Gender Classification Models 

Mentor: Martin Strauss

Convolutional Neural Networks are a type of deep learning algorithm which are often used for image classification tasks. One such image classification task is the classification of a person's gender based upon an image. These gender classification models pose many social problems, often posing a bias towards and misclassifying transgender and non-white individuals. These social problems are compounded by the fact that large neural networks function as a black-box and lack interpretability; it is difficult to explain why a model classifies an image in a particular way. We are investigating Gradient-weighted Class Activation Mapping (Grad-CAM) as a visualization tool which can explain how gender classification models make their decisions. 

Hao Pan: Small x asymptotics for special function solutions of Painlevé  III

Mentor: Andrei Prokhorov

Painlevé   equations are list of six \textbf{nonlinear second-order ordinary differential equations} in the complex plane with the \textbf{Painlevé property} (the only movable singularities are poles). Up to certain transformations, they can be written in the form of $$y^{{\prime \prime }}=R(y^{{\prime }},y,t)$$ (with $R$ a rational function)\\Their singularities that are $log(z)$ don't depend on the initial conditions, but the singularities that are $1/z$ do depend. Most of them are \textbf{irreducible}. That means most of them can't be reduced to other equations and their solutions can't be reduced to simple special functions. However, there are several exceptions and Painlevé III is one of them. It is easy to study asymptotic behavior of its solutions and that is why we are interested in it.

Helena Heinonen: Connectivity of Spheres in the Curve Graph

Mentor: Alex Wright

I will introduce the curve graph and define the "complexity" of a surface. Then consider spheres on the curve graph for a fixed center point c and arbitrary radius r. I will prove that any spheres of radius r about c are connected for surfaces of sufficiently large "complexity." In doing so I will introduce different techniques used in our proof, such as the Bounded Geodesic Image Theorem. 

Jad Damaj: Degree Spectra

Mentor: Mathew Harrison-Trainor

Given a relation (or function) on a mathematical structure, the computational properties of the relation might depend on how the structure is presented. For example, in some ways of presenting a vector space, independence might be computable, while in other presentations, it might not be. The degree spectrum of the relation measures this. In this talk we will introduce degree spectra and focus on relations on the natural numbers as a linear order. We attempt to answer the “on-a-cone” version of a question posed by Wright about which degree spectra are possible on $(\mathbb{N}, <)$ by showing the existence of many previously unknown degree spectra.

Neil Patram and Ben Scott: Numerical Investigation of Singularities of Optimal Transport Between Triangular Domains

Mentor(s): Mattias Jonsson, Nicholas McCleerey

Following work done by Hultgren et. al., 2023, we investigate the singularities of optimal transport maps between the surface of a tetrahedron and its dual. This problem can be reduced to studying the singularities of transport between triangular domains in the plane. We discuss results of numerical simulations of this planar problem, obtained through entropic regularization and the assignment problem.

Roshan Klein-Seetharaman: Bounded Geodesic Image Theorem

Mentor: Alex Wright

I will present the statement of the theorem and how it was relevant to our project on connectivity of spheres on the curve graph.

Minghan Sun: Basic properties of the curve graph

Mentor: Alex Wright

I will define the curve graph associated to a closed 2 manifold and to such a manifold with finitely many punctures.I will give some idea of why the curve graph is always connected and has infinite diameter. 

Syed Akbari: Computable Machine Learning

Mentor: Matthew Harrison-Trainor

In machine learning theory, Probably Approximately Correct (PAC) learning provides a foundational framework for studying when a (binary classification) problem is learnable, or when it is not learnable. Intuitively, it is PAC learnable if you have enough information such that you can "learn" how to almost-certainly approximate the ground truth, given any finite sample.

Recent work proves that there exist problems that are PAC learnable, but where there cannot exist any computable learner. Our work is set out to prove, or disprove, the idea that every "natural" PAC learnable problem will also in fact have a computable learner.

Ruohan Hu: Homogeneous Median Algebra

Mentor: Ronnie Chen

This talk will be an introduction to homogeneous structures and my work on an instance of these structures, namely the homogeneous median algebra. Homogeneous structures are known to exist in various settings and have many interesting properties. I will define homogeneous structures and give examples from graph theory and algebra. Then, I will define the median algebra and explain how a homogeneous median algebra can be constructed.

Jake Hofgard: Convergence of the Deep Galerkin Method for the Mean Field Control Problem

Mentor: Asaf Cohen

We consider the convergence of the deep Galerkin method (DGM), a deep learning-based scheme for solving high-dimensional nonlinear PDEs, for Hamilton-Jacobi-Bellman (HJB) equations that arise from the study of mean field control problems (MFCP). First, we show that the loss functional of the DGM can be made arbitrarily small given that the value function of the MFCP possesses sufficient regularity. Then, we show that if the loss functional of the DGM converges to zero, the corresponding neural network approximators must converge uniformly to the true value function on the simplex. We also provide numerical experiments demonstrating the DGM's ability to generalize to high-dimensional HJB equations.

Hahn Lheem and Nir Elber: How to Build a Gamma Factor in Three Easy Steps

Mentors: Jialiang Zou and Elad Zeligher

Finite fields and the groups attached to them are objects of classical interest in number theory. One good way to study such groups is by examining how they interact with a vector space via a representation. In this talk, we study representations of the general linear groups GL1(Fq) and GL2(Fq) using an invariant called the ``gamma-factor.'' This gamma-factor is a finite-field analogue of a gamma-factor appearing in the local Langlands correspondence, and it has historically been defined for certain representations or pairs of representations. In this talk, we review what is known, and we define and study gamma-factors attached to triples of representations.

Abhi Shukul: Gromov Boundary of Hyperbolic Groups and Finite State Automata

Mentor: Teddy Weisman

Hyperbolic groups are a class of finitely presentable groups to which we can assign a metric with ""coarse negative curvature"". It turns out that these groups give rise to an automatic structure, which means there exist finite state automata that solve certain computational problems about the group. The automatic structure is powerful in that, among other things, it can give us a solution to the word problem, a problem that's undecidable for arbitrary finitely generated groups. 

In this talk, I will define and give examples of hyperbolic groups and describe how the automatic structure can give information about the ""boundary at infinity"" of its corresponding hyperbolic group. 

Haoyan Shi: Soliton Solutions to the Korteweg-de Vries Equation

Mentor: Elliot Blackstone

The Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation, models waves in shallow water. My study explores soliton solutions to the KdV equation using the scattering transform. Example initial value problems are given to illustrate this process.