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 $A_F$ to a Fano manifold $F$. We say that $F$ satisfies Gamma conjecture I if $A_F$ equals the Gamma class ${\hat{\Gamma }}_F$. When the quantum cohomology of $F$ is semisimple, we say that $F$ satisfies Gamma conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by ${\hat{\Gamma }}_{F}Ch\left(E_i\right)$ for an exceptional collection $\left\{E_i\right\}$ in the derived category of coherent sheaves ${\mathcal{D}}_{\mathrm{coh}}^b\left(F\right)$. 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 $2s$, with $s\in (0,1)$. We consider the class of nonlocal operators ${\mathcal{L}}_{\ast }\subset {\mathcal{L}}_0$, which consists of infinitesimal generators of stable Lévy processes belonging to the class ${\mathcal{L}}_0$ of Caffarelli–Silvestre. For fully nonlinear operators $\mathrm{I}$ elliptic with respect to ${\mathcal{L}}_{\ast }$, we prove that solutions to $\mathrm{I}u=f$ in $\Omega$, $u=0$ in ${\mathbb{R}}^n\setminus \Omega$, satisfy $u/d^s\in C^{s+\gamma }\left(\overline{\Omega }\right)$, where $d$ is the distance to $\partial \Omega$ and $f\in C^{\gamma }$. We expect the class ${\mathcal{L}}_{\ast }$ to be the largest scale-invariant subclass of ${\mathcal{L}}_0$ for which this result is true. In this direction, we show that the class ${\mathcal{L}}_0$ is too large for all solutions to behave as $d^s$. The constants in all the estimates in this article remain bounded as the order of the equation approaches $2$. Thus, in the limit $s\uparrow 1$, 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 $\Gamma \setminus G$ when $\Gamma$ is a Zariski-dense geometrically finite subgroup of $G={PSL}_2\left(\mathbb{R}\right)$ or ${PSL}_2\left(\mathbb{C}\right)$. This generalizes Ratner’s 1983joining theorem for the case when $\Gamma$ is a lattice in $G$. 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 $\Gamma$ is any Zariski-dense geometrically finite subgroup of $G=SO(n,1{)}^{\circ }$, $n\ge 2$.