Based on geometric considerations, J. Roger and T. Yang in 2014 defined a version of the Kauffman bracket skein algebra for punctured surfaces that includes arcs going from puncture to puncture. We'll provide a brief survey of known results about this arc algebra. In particular, I'd like to mention a recent algebraic result whose […]
TOPIC: Graph Theory, Part I On the surface, graphs seem to be some of the simplest objects you might encounter in mathematics. After all, they are made up of just […]
Approximately 500 million tweets are sent everyday. Scientists monitor these tweets to predict the spread of disease, better allocate social welfare services, help first responders during natural disasters, and many other important tasks. […]
Tight frames in Euclidean spaces are widely used convenient generalizations of orthonormal bases. A particularly nice class of such frames is generated as orbits under irreducible actions of finite groups […]
In the 1970s, James O’Keefe and his team observed that certain neurons in the brain, called place cells, spike in their firing rates when the animal is in a particular […]
Online social networks and other networks of interest are known to exhibit community structure, where a community is defined to be a highly interconnected group of nodes with possibly shared […]
Given integers $k,l$ and a graph $G$, how large can be the fraction of $k$-vertex subsets of $G$ which span exactly $l$ edges? The systematic study of this very natural […]
Understanding how fluid flows through heterogeneous materials, and how it can make these materials fail, are among the hardest challenges in materials science. Experiments and simulations show that flow through […]
Candidate for Assistant Professor in Mathematics, Scripps College A surprising application of mathematics: How to name a color Your brain likes patterns and categories; by grouping related ideas together, it […]
An agent comes to a fork in a road. There is a sign that says that one of the two roads leads to prosperity and another to death. The agent must take the fork, but she does not know which road leads where. Does the agent have a strategy to get to prosperity? On one […]
Abstract: A great deal of my research journey has involved the study of m-ary partitions. These are integer partitions wherein each part must be a power of a fixed integer m > 1. Beginning in the late 1960s, numerous mathematicians (including Churchhouse, Andrews, Gupta, and Rodseth) studied divisibility properties of m-ary partitions. In this talk, I will discuss work I completed […]
In 1968, Milnor famously conjectured that the smooth 4-genus of the torus knot T(p,q) is given by (p-1)(q-1)/2. This conjecture was first verified by Kronheimer and Mrowka in 1993 and has received several other proofs since then. In this talk, we discuss a nonorientable analogue of this conjecture, first formulated by Josh Batson. We prove […]
