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: MAT 128ABC, MAT 167, MATLAB, Python, Julia.
Enrollment: 2 Openings
Algorithms to find ham-sandwich cuts of convex bodies
Faculty Mentor: Jesús De Loera
Description: Mass equipartition problems in geometry determine when a collection of continuous measures (informally they should be thought as convex sets) can be simultaneously divided into equal fractions by geometric objects. The most celebrated result of this type is the Ham-Sandwich theorem, which states that any $d$ finite, continuous Borel measures in $\mathbb{R}^d$ can be simultaneously bisected by a single affine hyperplane. Our project aims to investigate (1) algorithms for doing equipartitions (not just existence!!) and (2) Investigate what happens with more than one hyperplane, what equipartitions are possible. More concretely, the Grünbaum--Hadwiger--Ramos problem generalizes this setup to multiple cutting hyperplanes. Let $\mu_1, \mu_2, \dots, \mu_j$ be $j$ finite, continuous Borel measures (mass distributions) on $\mathbb{R}^d$. We seek $k$ affine hyperplanes $H_1, H_2, \dots, H_k$ that simultaneously partition each of the $j$ measures into $m$ parts, with $1 \leq m \leq 2^k$. First natural question: Can we partition into $m$ parts for any value of $m$? It will depend on $k$ the number of hyperplanes of course. More strongly, the version we really want to answer is can we partition with $m$ equal parts? Most important case is when $m=2^k$ because then these parts correspond to the $2^k$ orthant-like chambers created by the arrangement of the hyperplanes. We hope to at least completely answer the question for dimension 2.
Requirements: MAT 108, MAT 114
Enrollment: No New Openings
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) required, MAT 115A/B recommended.
Enrollment: No New Opening
Neural networks for function approximation
Faculty Mentor: Abi Gopal
Description: A recurring task in computational mathematics is to fit a function to some given data points. Neural networks provide a flexible class of models for this purpose, particularly when the underlying relationship is nonlinear or the data are noisy. In this project, we will implement simple neural networks from scratch and train them to approximate functions from sampled data. We will investigate how choices such as network architecture, activation function, and training procedure affect the quality of the approximation.
Requirements: MAT 108 and Python
Enrollment: 2 Openings
Optimization, Polyhedral/Algebraic Geometry, Satisfiability Modulo Theories in passagemath
Faculty Mentor: Matthias Koeppe
Description: passagemath is a modernized version of Sage, the comprehensive open source mathematics software system. Its modular design aligns well with the contemporary Scientific Python ecosystem. passagemath is trivial to install either locally or in Google Colab or other generic cloud computing systems. We will develop new functionality for passagemath in Mixed Integer Optimization, Polyhedral/Algebraic Geometry, and Satisfiability Modulo (Arithmetic) Theories.
Requirements: MAT 168 or MAT 170 or MAT 167 or MAT 141 or MAT 114, Python, Git.
Enrollment: 3 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: No New 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: No New Openings
Decomposition of Polyhedra
Faculty Mentor: Maria Trnkova
Description: In this project we will study decomposition of polyhedra into smaller pieces which satisfy a special geometric property. Each piece has a face serving as its base, and all other faces project onto this base. In other words, there are no overhangs: every point of the piece projects vertically onto the base. In this case we say that the piece is a "hight function" over its base. For example, a tetrahedron with two obtuse dihedral angles cannot be placed on a face without creating overhangs. This problem is motivated by 3D printing process. Overhanging parts require additional support during manufacturing and may negatively affect the quality of the final object. The goal of this project is to study the minimal number of hight-function pieces required to decompose a polyhedra with n vertices. We will explore both convex and non-convex polyhedra with small numbers of vertices and will 3D print them.
Requirements: MAT 108 required, MAT 114 recommended.
Enrollment: 3 Openings