Mathematical Sciences

► Bounding Lifts of Markoff Triples mod $p$
Brenda (Siran) Chen

Advisor: Elisa Bellah
Abstract: In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds for the Markoff surface in most cases. That is, the modulo $p$ solutions to the equation $x^2+y^2+z^2=3xyz$ are covered by the integer points for most primes $p$. To prove this result, Bourgain, Gamburd, and Sarnak constructed a path in the associated Markoff mod $p$ graphs. In this project, we implemented this path finding algorithm in order to obtain upper bounds on lifts of Markoff triples modulo $p$. We then collected numerical evidence that these bounds can be improved on average and with high probability.

► Two-Phase Free Boundary Problems
Thomas Horstkamp

Advisor: Giovanni Leoni
Abstract: This project aims to investigate two-phase free-boundary problems motivated by the flow of two liquids in models of jets and cavities. Specifically, the project considers the minimization of functionals involving a two-phase function and its derivative, subject to boundary conditions.

The existence and regularity of solutions are of interest. Since the study of two-phase free-boundary problems is still in its early stages, this project seeks to contribute to its development by providing more elementary proofs of previously established results and studying novel variations of previous problems. The methodology involves using techniques from the Calculus of Variations, Functional Analysis, and Sobolev Spaces to parameterize critical points of the functionals and reduce to standard minimization problems. The results of this research may have useful real-world applications in modeling physical phenomena and industry-related problems.

► Capillary droplets inside a container
Iris Jiang

Advisor: Robin Neumayer
Abstract: The shape of the region occupied by a droplet of liquid inside a container is modeled by an isoperimetric problem, which consists of minimizing Gauss’ free energy among shapes of a fixed volume. This project focuses on a two-dimensional analogue of this model and investigates how the container’s geometry impacts the shape and energy of energy-minimizing liquid droplets. We show that in any non-circular container, when the mass of the droplet is sufficiently small, the minimizing droplet’s energy is strictly smaller than the energy of a droplet of the same mass in a circular container of the same size.

► Turbulence and self-homeomorphisms of the Sierpinski carpet
Dhruv Kulshreshtha

Advisor: Aristotelis Panagiotopoulos
Abstract: Let $K$ be a compact metric space and consider the homeomorphism group $H(K)$ of all symmetries of $K$. How difficult is it to classify those symmetries up to conjugacy? If $K$ is the interval $[0,1]$ then $H(K)$ can be easily classified using certain countable structures as invariants: these countable structures have to simply keep track of the maximal open intervals on which each homeomorphism is increasing, decreasing, or constant. The problem becomes much more complex if one is to replace the interval $[0,1]$ with the square $[0,1]^2$. Indeed it is a theorem of Hjorth that there exists no definable way to classify self-homeomorphisms of the square $[0,1]^2$ by using countable structures as invariants. The proof of this result goes through his celebrated theory of turbulence.

The main goal of this project is to consider the problem of classifying $H(\mathbb{S})$ where $\mathbb{S}$ is the classical one-dimensional Sierpinski Carpet:

► A generalization of the regulated integral
Subhasish Mukherjee

Advisor: Ian Tice
Abstract: We investigate set-algebra pairs and prove properties about the regulated functions, the uniform closure of the set of step functions in the uniformity of uniform convergence for functions taking values in uniform spaces. In a very elementary way, with respect to finitely additive measures we define the Cauchy integral on the regulated functions taking values in locally convex spaces.

We prove the Cauchy integral satisfies several properties, such as linearity, versions of monotonicity, a Riesz representation theorem, an elementary Fubini theorem, the fundamental theorem of calculus and its standard consequences, and various continuity properties. We apply these properties for regulated functions from $\mathbb{R}^n$ to locally convex spaces and show that this is a robust theory of integration and an alternative to the Riemann or Bochner integrals as a first pass at integration.

► CSPs and Compactness Principles
Anthony Li, Ishin Shah, Matthew Snodgrass, Rui Zhou

Advisor: Riley Thornton
Abstract:  Constraint Satisfaction Problems (CSPs) are a well studied class of problems from computer science which come with a rich notion of complexity. Each CSP can be associated with a set theoretic compactness principle (a weak form of the axiom of choice). By studying the relative strengths of these compactness principles, one can obtain a partial (quasi) order on the difficulty of corresponding CSPs with some interesting connections to the algebraic and computational properties of CSPs.

In this project we characterized those CSPs whose compactness principles are at the bottom of this order (those provable from ZF). We found an infinite sequence of compactness principles of increasing strength, corresponding to axioms of choice for larger and larger finite sets.

We found an infinite sequence of incomparable compactness principles. And, we separated out two intermediate classes of compactness principles for 2-valued CSPs, answering a question of Katya, Toth, and Vidnyanszky.

► Embedding dimension gaps in sparse codes
Henry Siegel, David Staudinger, Yiqing Wang

Advisor:  Amzi Jeffs
Abstract:  We discuss the open and closed embedding dimensions of a convex code FP, derived from the Fano plane. We are motivated by the following question: do there exist convex codes whose open embedding dimension is large, but which have codewords of bounded size? The code FP has codewords of size at most three, and may be the first example of such a code whose embedding dimension is larger than 5. We explain that the closed dimension of FP is equal to 3, while the open dimension is either 4, 5, or 6. In particular, FP is the first example of a code whose codewords have size at most three, whose closed embedding dimension is three, and whose open dimension is not equal to closed dimension.