Transitive factorizations of permutations and geometry

We give an account of our work on transitive factorizations of permutations. The work has had impact upon other areas of mathematics such as the enumeration of graph embeddings, random matrices, branched covers, and the moduli spaces of curves. Aspects of these seemingly unrelated areas are seen to be related in a unifying view from the perspective of algebraic combinatorics. At several points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th birthda

Topologie (hybrid meeting)

The Oberwolfach conference "Topologie" is one of only a few opportunities for researchers from many different areas in algebraic and geometric topology to meet and exchange ideas. On this occasion, because of the Corona pandemic, only about 20 participants attended in person, but another $\sim$ 25 attended online. Speakers were selected from both groups. A topic of special interest emphasized at the workshop was the rational homotopy theory of embedding spaces and relations to graph complexes and formality. Two 50 minute lectures on this theme were given by Thomas Willwacher, and one by Victor Turchin. The rest of the program covered a wide range of topics, among them: homotopy properties of diffeomorphism groups of high dimensional manifolds, advances in the classification of high-dimensional highly connected smooth manifolds, parametrized algebraic surgery in relation to hermitian algebraic K-theory, other advances in and geometric applications of algebraic K-theory, stable homotopy interpretation of link invariants, geometry of surface bundles and cohomology of mapping class groups, boundary concepts in geometric group theory, and Koszul duality for operads

KMS states on Quantum Grammars

We consider quantum (unitary) continuous time evolution of spins on a lattice together with quantum evolution of the lattice itself. In physics such evolution was discussed in connection with quantum gravity. It is also related to what is called quantum circuits, one of the incarnations of a quantum computer. We consider simpler models for which one can obtain exact mathematical results. We prove existence of the dynamics in both Schroedinger and Heisenberg pictures, construct KMS states on appropriate C*-algebras. We show (for high temperatures) that for each system where the lattice undergoes quantum evolution, there is a natural scaling leading to a quantum spin system on a fixed lattice, defined by a renormalized Hamiltonian.Comment: 22 page

Arithmetic lattices and weak spectral geometry

This note is an expansion of three lectures given at the workshop "Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces" held at Kyoto University in December of 2006 and will appear in the proceedings for this workshop.Comment: To appear in workshop proceedings for "Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces". Comments welcom