We study isolated quotient singularities by finite group schemes in positive characteristic. We compute invariants, study the uniqueness of the quotient presentation, and compute some deformation spaces. A special emphasis is laid on the dichotomy between quotient singularities by linearly reductive group schemes and by group schemes that are not linearly reductive. We essentially classify the linearly reductive ones, give applications, and make some conjectures. This is joint work with Gebhard Martin (Bonn) and Yuya Matsumoto (Tokyo).

We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. This induces a random ``disordered” geometry on the lattice. Our primary interest is in the empirical measures of edge-weights observed along geodesics in this geometry, say from 0 to [n\xi], where \xi is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.

I will present a $C^{1,\alpha}$-regularity theorem for m-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p > m$, in every dimension and codimension.

The space of finitely generated marked groups, denoted by $\mathcal G$, is a locally compact Polish space whose elements are groups with fixed finite generating sets; the topology on $\mathcal G$ is induced by local convergence of the corresponding Caley graphs. I will describe a necessary and sufficient condition for a closed subspace $\mathcal S\subseteq \mathcal G$ to satisfy the following zero-one law: for any sentence $\sigma$ in the infinitary logic $\mathcal L_{\omega_1, \omega}$, the set of all models of $\sigma$ in $\mathcal S$ is either meager or comeager. In particular, the zero-one law holds for certain subspaces associated to hyperbolic groups. This leads to the following (somewhat unexpected) corollary: generic limits of non-cyclic, torsion-free, hyperbolic groups are elementarily equivalent. We will discuss other applications and open problems.

This talk is concerned with the problem of off-policy evaluation in the multi-armed bandit model with bounded rewards. We develop minimax rate-optimal procedures under three different settings. First, when the behavior policy is known, we show that the Switch estimator, a method that alternates between the plug-in and importance sampling estimators, is minimax rate-optimal for all sample sizes. Second, when the behavior policy is unknown, we analyze performance in terms of the competitive ratio, thereby revealing a fundamental gap between the settings of known and unknown behavior policies. When the behavior policy is unknown, any estimator must have mean-squared error larger---relative to the oracle estimator equipped with the knowledge of the behavior policy---by a multiplicative factor proportional to the support size of the target policy. Moreover, we demonstrate that the plug-in approach achieves this worst-case competitive ratio up to a logarithmic factor. Third, we initiate the study of the partial knowledge setting in which it is assumed that the minimum probability taken by the behavior policy is known. We show that the plug-in estimator is optimal for relatively large values of the minimum probability, but is sub-optimal when the minimum probability is low. In order to remedy this gap, we propose a new estimator based on approximation by Chebyshev polynomials that provably achieves the optimal estimation error. This is a joint work with Banghua Zhu, Jiantao Jiao and Martin Wainwright.

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