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Abstract:  Liquid crystals are soft materials that flow like liquids while retaining directional molecular order. In cholesteric liquid crystals, this directional order rotates through space. The Landau-de Gennes (LdG) model attempts to capture this information in a 3x3 matrix called the Q-tensor.

This summer, students learned about the LdG model by first reviewing the background mathematics, then by implementing their own code to simulate it in two dimensions using the finite element software Firedrake. Students then learned to use an already existing code to perform simulations in three dimensions. They then explored the relationship between the chirality (propensity for the directional order to rotate) of the liquid crystal system and the boundary anchoring constants in the LdG model. The students observed that for lower chirality, rotation of the liquid crystals can be eliminated by imposing sufficiently large anchoring values, but for higher chirality, apparently no such values exist. Future work would therefore be to develop a framework to rigorously prove this relationship.

Abstract:  For a graph $G = (V(G),E(G))$, an edge coloring of $G$ is a surjective function $\gamma: E(G) \rightarrow [\ell],\;\ell\in\mathbb{N}$ that assigns a color to each edge of $G$. A wild edge belongs to every color class. Given a graph $G$ with edge coloring $\gamma$, our goal is to determine the corresponding wild number, or the minimum number of edges which must be colored wild so that every color class contains a spanning tree. We denote this by $\mathrm{wld}_\gamma(G)$. One motivation comes from network design---viewing edge-disjoint spanning trees as independent pathways that enhance network reliability, and edge colorings as allocations of pathways across distinct operations. In this setting, wild edges represent pathways that must be upgraded to support multiple operations with the wild number detailing the minimum number of upgrades that must be made to allow for spanning pathways for each operation. We study the wild number of complete graphs and complete bipartite graphs under two common edge colorings, equitable $2$-colorings and minimum proper colorings.

For an equitable $2$-coloring, the ranges of possible wild numbers of a complete graph and a complete bipartite graph are determined, and constructions for a complete graph show that every wild number within the range is attainable by some equitable 2-coloring. For minimum proper colorings, the lower bound of the wild number of a complete graph is established, and we conjecture that the upper bound corresponds, while the wild number of a complete bipartite graph is determined.

We also investigated how this parameter behaves for subgraphs and for Cartesian products. Considering a $2$-coloring, we find that removing a vertex can reduce the wild number by at most $2$, in which case the removed vertex must not have been a cut vertex. Considering a proper coloring, the wild number of the Cartesian product of complete graphs can be bounded below by the sum of the wild numbers of the factors, while the wild number of the Cartesian product of complete bipartite graphs can be bounded below by the product of the wild numbers of the factors.

Abstract:  Maximum mean discrepancy (MMD) is a sample-based way to compare probability distributions, with applications in statistics and generative modeling. One can try to match a model distribution to a target by evolving a cloud of interacting particles along a Wasserstein gradient flow that decreases MMD. For the nonsmooth energy kernels considered in this project, however, purely local transport can become trapped away from the target or converge very slowly when the two distributions are far apart.

We introduced a jump-transport dynamics that augments this continuous motion with occasional nonlocal jumps toward energetically favorable locations. We developed its continuum equation and stochastic finite-particle model. Under suitable assumptions on the data and jump mechanism, we proved global existence and uniqueness, showed that the target is the only stationary state, and established convergence of the continuum dynamics to the target in a weak probabilistic sense. We also studied particle and time-discretized approximations, providing a mathematical foundation for algorithms that combine local transport with jumps.

Abstract:  A very general question one can ask in topology is: given an $n-$dimensional manifold, what $(n+1)-$dimensional manifolds can $M$ be the boundary of? For example, the unit circle (which is $1-$dimensional) is the boundary of the unit disk (which is $2-$dimensional). A particular class of $3-$dimensional manifolds that receives much attention is the class of rational homology 3-spheres; these manifolds have particularly simple topology. A natural question many topologists like to ask is whether a given rational homology 3-sphere can bound a particularly simple $4-$dimensional manifold called a rational homology 4-ball. This question has received considerable attention in the past two decades.

The goal of this project was to answer this question for a particular infinite family of rational homology 3-spheres arising from a well-known topological construction called double branched covers of links. In particular, we considered the double branched covers of an infinite family of links known as weaving links and classified which of these bound rational homology 4-balls and which do not. First, we proved that about half of these rational homology 3-spheres do not bound rational homology 4-balls using lattice theoretic techniques. We then proved that the remaining rational homology 3-spheres do bound rational homology 4-balls by explicitly constructing them using handlebody diagrams.