We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for nonamenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on nonamenable graphs. While we rely on the equivalence of the LRRW to a mixture of Markov chains, the proof does not use the so-called magic formula which is central to most work on this model. We also derive analogous results for the vertex reinforced jump process.

In a previous work, we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen’s slice genus bound $s$ for each stable cohomology operation. We show that in the case of the Steenrod square ${Sq}^2$ our refinement is strictly stronger than $s$.

We study the Chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties (ppav). We describe the subring generated by divisor classes, and we compute the class of the partial compactification of the universal zero section, which turns out to lie in this subring. Our formula extends the results for the zero section of the universal uncompactified family.

The partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of ${\mathcal{A}}_g$. Our formula provides a first step in a program to understand the Chow groups of ${\overline{\mathcal{A}}}_g$, especially of the perfect cone compactification, by induction on genus. By restricting to the image of ${\mathcal{M}}_g$ under the Torelli map, our results extend the results of Hain on the double ramification cycle, answering Eliashberg’s question.

Let $(G,K)$ be a symmetric pair over an algebraically closed field of characteristic different from $2$, and let $\sigma$ be an automorphism with square $1$ of $G$ preserving $K$. In this paper we consider the set of pairs $(\mathcal{O},\mathcal{L})$ where $\mathcal{O}$ is a $\sigma$-stable $K$-orbit on the flag manifold of $G$ and $\mathcal{L}$ is an irreducible $K$-equivariant local system on $\mathcal{O}$ which is “fixed” by $\sigma$. Given two such pairs $(\mathcal{O},\mathcal{L})$, $(\mathcal{O}\text{'},\mathcal{L}\text{'})$, with $\mathcal{O}\text{'}$ in the closure $\overline{\mathcal{O}}$ of $\mathcal{O}$, the multiplicity space of $\mathcal{L}\text{'}$ in a cohomology sheaf of the intersection cohomology of $\overline{\mathcal{O}}$ with coefficients in $\mathcal{L}$ (restricted to $\mathcal{O}\text{'}$) carries an involution induced by $\sigma$, and we are interested in computing the dimensions of its $+1$ and $-1$ eigenspaces. We show that this computation can be done in terms of a certain module structure over a quasisplit Hecke algebra on a space spanned by the pairs $(\mathcal{O},\mathcal{L})$ as above.

We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modeling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a “curved orbit decomposition.” The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal, and CR geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third-order differential equation that governs the existence of a compatible Einstein metric, and in conformal geometry we discuss almost-Einstein scales. Finally, we discuss analogues of the two latter equations in CR geometry, which leads to invariant equations that govern the existence of a compatible Kähler–Einstein metric.