In this talk we will discuss extensions of the 4 fundamental products of graphs (cartesian, categorical, lexicographical, and strong products) to quantum graphs, and provide bounds on the resulting graphs akin to those for products of classical graphs. We will pay particular attention to the lexicographical product, discussing our notion of a quantum b-fold chromatic number as a tool for computing the quantum chromatic number of the lexicographical products.

This is joint work with A. Meenakshi McNamara.

We make explicit the relationship between Hilbert modular forms and orthogonal modular forms arising from positive definite quaternary lattices over the ring of integers of a totally real number field.  Our work uses the Clifford algebra, and it generalizes that of Ponomarev, Bocherer--Schulze-Pillot, and others by allowing for general discriminant, weight, and class group of the base ring.  This is joint work with Eran Assaf, Dan Fretwell, Colin Ingalls, Adam Logan, and Spencer Secord.

[pre-talk at 3:00PM]

Epigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multi-cellular organisms. It was previously found via simulations of stochastic models that the time scale separation between establishment (fast) and erasure (slow) of chromatin modifications (such as DNA methylation and histone modifications) extends the duration of cell memory, and that different asymmetries between erasure rates of chromatin modifications can lead to different gene expression patterns. We provide a mathematical framework to rigorously validate these computational findings using stochastic models of chemical reaction networks. For our study of epigenetic cell memory, these are singularly perturbed, finite state, continuous time Markov chains. We exploit special structure in our models and extend beyond existing theory to study these singularly perturbed Markov chains when the perturbation parameter is small. We also develop comparison theorems to study how different erasure rates affect the behavior of our chromatin modification circuit. The theoretical tools developed in our work not only allow us to set a rigorous mathematical basis for highlighting the effect of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other singularly perturbed Markov chains beyond the applications in this work, especially those associated with chemical reaction networks.

The widely-used k-means procedure returns k clusters that have arbitrary convex shapes. In high dimension, such a clustering might not be easy to understand. A more interpretable alternative is to constraint the clusters to be the leaves of a decision tree with axis-parallel splits; then each cluster is a hyperrectangle given by a small number of features.

Is it always possible to find clusterings that are intepretable in this sense and yet have k-means cost that is close to the unconstrained optimum? A recent line of work has answered this in the affirmative and moreover shown that these interpretable clusterings are easy to construct.

I will give a survey of these results: algorithms, methods of analysis, and open problems.

Is every finite group isomorphic to the Galois group of some Galois extension of the rational numbers? Although this question remains open in general, powerful methods have led to an affirmative answer in some cases, including that of solvable groups, symmetric and alternating groups, and most of the sporadic groups. In this talk, we call upon seemingly disconnected areas of algebra, topology, and complex analysis to describe the rigidity method of inverse Galois theory.