Quantum cellular automata (QCA) are models of discrete-time unitary dynamics of quantum spin systems. They can be characterized algebraically as certain automorphisms of the associative algebra generated by local observables of a spin system. We will give a gentle introduction to this topic, and explain some of our recent contributions to the problem of classification…

Permutations $w$ in $S_n$ for which the (type-A) Schubert variety $\Omega_w$ is smooth are characterized by avoidance of the patterns 3412 and 4231.  The smaller family of codominant permutations, those avoiding the pattern 312, seems to explain a lot about character evaluations at Kazhdan-Lusztig basis elements $C'_w(q)$ of the (type-A) Hecke algebra. In particular, for…

Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost-minimization problem over the factorizations of a permutation into transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection t…

In 1995, Stanley introduced the chromatic symmetric function of a graph, a symmetric function analog of the classical chromatic polynomial of a graph. The Stanley-Stembridge e-positivity conjecture is a long-standing conjecture that states that the chromatic symmetric function of a certain class of graphs, called incomparability graphs of (3+1)-free posets, has nonnegative coefficients when expanded…