Mixed-Precision Numerical Linear Algebra
Faculty Mentor: Zhaojun Bai
Description: Mixed-precision, in particular, using low-precision hardware, for solving problems in numerical linear algebra.
Requirements: MAT128ABC, MAT167, MATLAB, Python, Julia.
Enrollment: 2 Openings
Numerical Simulation of Adiabatic Control in Open Quantum Systems
Faculty Mentor: Martin Fraas
Description: The goal of the project would be to numerically simulate an adiabatic control equation in open quantum system. This equation has a Lindblad form. We will start with some toy problems in numerical simmulation of Lindblad equation and slowly work towards the actual equation that we aim to simulate.
Requirements: MAT 119A/B and programming skills. Familarity with quantum mechanics is helpful.
Enrollment: 2 Openings
Exploration of Chiral Central Charge
Faculty Mentor: Martin Fraas
Description: We are exploring how to microscopically define chiral central charge. The exploration follows a proposal described by Kitaev.
Requirements: Quantum mechanics and functional analysis.
Enrollment: 1 Opening
Markoff Triples
Faculty Mentor: Elena Fuchs
Description: The Markoff equation $x^2+y^2+z^2=3xyz$ has a graph associated to its solutions which is known to be connected over the positive integers. One natural question is whether or not the graph is still connected when we take the solutions to the equations modulo a prime. This was recently shown to be true for large enough primes $p$. Our goal is to program an algorithm based off of Daniel Martin's recent paper on connectivity of Markoff triple graphs modulo $p$ to either finally show that they are connected for all $p$ or get very close.
Requirements: MAT 108, MAT 150A, some programming (MATLAB/Python/Sage) and 115A/B is helpful.
Enrollment: 1 Opening
A Dynamical Systems Analysis of Vertical Transmission in Host–Microbe Models
Faculty Mentor: Tim Lewis
Description: Microbes that live in or on a host can strongly affect its health and survival. Dynamical systems are a valuable tool in understanding the complex interactions between hosts and their microbial communities. A recent study used a system of ordinary differential equations to model interactions between a host, pathogenic (harmful) microbes, and mutualist (helpful) microbes. One limitation of the original model is that it assumes microbes are only acquired from the environment (horizontal transmission). In some organisms, microbes can also be passed from parent to offspring (vertical transmission). Vertical transmission can be beneficial when helpful microbes are inherited, but harmful when it spreads pathogens, creating a tradeoff between vertical and horizontal transmission. This project will (1) extend the dynamical systems model to include vertical transmission, (2) analyze the extended model using bifurcation analysis and simulation experiments and (3) assess the optimal vertical transmission rate under different scenarios.
Requirements: MAT 119A, ECS 32, R/Python
Enrollment: 2 Openings
Distribution of Error Terms in Lattice Points Counting
Faculty Mentor: Junxian Li
Description: Given a nice convex region \(\Omega\) and a large number \(x\), we know that the number of lattice points in \(x\Omega\) is asymptotically the volume of the region \(x\Omega\). How does the error term in the approximation behave as \(x\) gets larger? What is the distribution of the error term as we vary \(x\) in a large interval? If the region is a disk, then this related to the Gauss circle problem, which gives an upper bound for the error term. The normalized error term is known to have a non-gaussian distribution. In this project, we will explore the distribution of the error term in these lattice point counting problems for general ellipsoids in higher dimensions.
Requirements: MAT 127A, MAT 185A or programming skills.
Enrollment: 3 Openings
$D$-modules and Habiro rings
Faculty Mentor: Motohico Mulase
Description: The project is aimed at exploring the recently conjectured relationship between $D$-modules and the Habiro rings. One of the original motivations for introducing Habiro rings was to prove number theoretic properties of quantum topological invariants of $3$-manifolds. Since then this area of research has attracted attentions from algebraic geometry, $D$-module theory, number theory, quantum field theory, and complex analysis. An important open problem is to construct categorification of Habiro universal invariants of knots and links. Our goal of this quarter is a modest one: concrete construction of examples of Habiro-type universal knot/link invariants, identification of their number theoretic properties, and discovering the conjectured passage to $D$-modules.
Requirements: MAT 127AB, MAT 150AB, MAT 185A
Enrollment: 1 Opening
The Plactic and Hypoplactic Monoid
Faculty Mentors: Anne Schilling and Chenchen Zhao
Description: Consider a finite alphabet $A$. The set of all words with letters in the alphabet $A$ is the free monoid $A^*$, where multiplication is just concatenation of words. The plactic and hypoplactic monoid are obtained from the free monoid by identifying certain words which differ by Knuth and/or quartic relations. These monoids play a central role in combinatorics and representation theory, in particular in the combinatorics of Young tableaux and ribbon tableaux. The aim of this project is to program the plactic and hypoplactic monoid in SageMath and to discover properties of these monoids. Ideally by the end of the project, the code will be merged into SageMath through the github interface.
Requirements: MAT 150A, Python or SageMath.
Enrollment: 2 Openings