Let $G=D_p$ be the dihedral group of order $2p$, where $p$ is an odd prime. Let $k$ be an algebraically closed field of characteristic $p$. We show that any action of $G$ on the ring $k[[y]]$ can be lifted to an action on $R[[y]]$, where $R$ is some complete discrete valuation ring with residue field $k$ and fraction field of characteristic $0$

We define an invariant of oriented links in $S^3$ using the symplectic geometry of certain spaces that arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid that, in turn, we view as a loop in configuration space. Fix an affine subspace ${\mathcal{S}}_m$ of the Lie algebra ${\mathfrak{sl}}_{2m}(\mathbb{C})$ which is a transverse slice to the adjoint action at a nilpotent matrix with two equal Jordan blocks. The adjoint quotient map restricted to ${\mathcal{S}}_m$ gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submanifold $L_{\wp {}_{\pm }}$ of a fibre ${\mathcal{Y}}_{m,t_0}$ of this fibre bundle; we regard the braid $\beta$ as a symplectic automorphism of the fibre and apply Lagrangian Floer cohomology to $L_{\wp {}_{\pm }}$ and $\beta (L_{\wp {}_{\pm }})$ inside ${\mathcal{Y}}_{m,t_0}$. The main theorem asserts that this group is invariant under the Markov moves and hence defines an oriented link invariant. We conjecture that this invariant coincides with Khovanov's combinatorially defined link homology theory, after collapsing the bigrading of the latter to a single grading

We show that difference Painlevé equations can be interpreted as isomorphisms of moduli spaces of difference connections (d-connections) on ${\mathbb{P}}^1$ with given singularity structure. In particular, we derive a difference equation that lifts to an isomorphism between $A_2^{(1)*}$-surfaces in Sakai's classification (see [29]); it degenerates to both difference Painlevé V and classical (differential) Painlevé VI equations. This difference equation has been known before under the name of asymmetric discrete Painlevé IV equation

We prove the level one case of Serre's conjecture. Namely, we prove that any continuous, odd, irreducible representation $\stackrel{̲}{\rho }:G_{\mathbb{Q}}\to {\mathrm{GL}}_2(\stackrel{̲}{{\mathbb{F}}_p})$ which is unramified outside $p$ arises from a cuspidal eigenform in $S_k({\mathrm{SL}}_2(\mathbb{Z}))$ for some integer $k\ge 2$. The proof relies on the methods introduced in an earlier joint work with J.-P. Wintenberger [31] together with a new method of weight reduction

We introduce a notion of a piecewise automatic group. Among these groups we describe a new class of groups of intermediate growth. We show that for any function $f:\mathbb{N}\to \mathbb{N}$, there exists a finitely generated torsion group of intermediate growth $G$ for which the Følner function satisfies ${\mathrm{Føl}}_{G,S}(n)\ge f(n)$ for some generating set $S$ and all sufficiently large $n$. As a corollary we see that the asymptotic entropy of simple random walks on these groups could be arbitrarily close to being linear, while the Poisson boundary is trivial