Geometry and physics have been, in recent times, a great inspiration for mathematical problems. It is therefore useful to consider numerical methods that pay special attention to various invariant geometric structures, and minimize dependence on choices of coordinate systems. Attention to this aspect can lead to better stability and qualitative behavior. One important tool we have in capturing geometric structure is differential forms; many common differential equations find their most natural expressions in terms of forms. The Finite Element Exterior Calculus (FEEC) provides a framework for discretizing differential forms as finite elements. We present examples of how FEEC recasts problems into a more geometric form, and describe generalization to hyperbolic problems, by, specifically, application of FEEC to solving Maxwell's equations. We describe a choice of discretization (Whitney Forms) and possible generalizations and their issues.

We consider the problem of finding a hidden clique in a random graph. This problem was studied by Alon, Krivelevich and Sudakov in 1998. Using Matlab, we wrote two algorithms that are designed to find a hidden clique. One of the algorithms was suggested by the work of Alon, Krivelevich and Sudakov. The other algorithm is a slight variation that seems to perform better in our experiments.

For a graph $G$, the bipartition number $\tau(G)$ is the minimum number of complete bipartite subgraphs whose edge sets partition the edge set of $G$. In 1971, Graham and Pollak proved that $\tau(K_n)=n-1$. Since then, there have been a number of short proofs for Graham-Pollak theorem by using linear algebra or by using matrix enumeration. We present a purely combinatorial proof for Graham-Pollak theorem. For a graph $G$ with $n$ vertices, one can show $\tau(G) \leq n- \alpha(G)$ easily, where $\alpha(G)$ is the independence number of $G$. Erd\H{o}s conjectured that almost all graphs $G$ satisfy $\tau(G)=n-\alpha(G)$. In this talk, we prove upper and lower bounds for $G(n,p)$ which gives support for Erd\H{o}s' conjecture. Joint work with Fan Chung.

Cryptography has a long history, dating back to the first century BC.
The twentieth century saw huge advances in cryptography as systems for
rapid long-distance communications developed, to the point where now
cryptographic practices are used on a daily basis for commonplace
activities such as checking email. In this talk I will give a brief
survey of mathematics that go into the modern practice of cryptography,
in both designing and attacking cryptographic systems.

Understanding solid liquid phase transition is of great importance in many applications starting from growth of nano-crystals in solutions for solar cells to making the perfect ice cream/slushy. This forms the basis for understanding more complex phenomena such as solvation and self assembly in active matter systems.The element of interest in this talk is to understand the role and the effect of the fluid flow on the phase transition. The same effect that prevents the ice cream or slushy from freezing into a solid block.This talk will consider a system of particles that interact through a pair potential and choose a revised Enskog kinetic theory to describe the time evolution. Using a generalized Chapman-Enskog procedure non-local hydrodynamics that take the form of a density functional theory will be derived. A numerical study on the effect of melt flow on the freezing transition will be discussed in detail. Some potential applications to self assembly in active matter systems and solvation will also be outlined.
This work was done in collaboration with John Lowengrub and Aparna Baskaran.