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We propose Gamma conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class to a Fano manifold . We say that satisfies Gamma conjecture I if equals the Gamma class . When the quantum cohomology of is semisimple, we say that satisfies Gamma conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by for an exceptional collection in the derived category of coherent sheaves . Gamma conjecture II refines a part of a conjecture by Dubrovin. We prove Gamma conjectures for projective spaces and Grassmannians.
We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order , with . We consider the class of nonlocal operators , which consists of infinitesimal generators of stable Lévy processes belonging to the class of Caffarelli–Silvestre. For fully nonlinear operators elliptic with respect to , we prove that solutions to in , in , satisfy , where is the distance to and . We expect the class to be the largest scale-invariant subclass of for which this result is true. In this direction, we show that the class is too large for all solutions to behave as . The constants in all the estimates in this article remain bounded as the order of the equation approaches . Thus, in the limit , we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.
We classify all locally finite joinings of a horospherical subgroup action on when is a Zariski-dense geometrically finite subgroup of or . This generalizes Ratner’s 1983joining theorem for the case when is a lattice in . One of the main ingredients is equidistribution of nonclosed horospherical orbits with respect to the Burger–Roblin measure, which we prove in a greater generality where is any Zariski-dense geometrically finite subgroup of , .