We prove that the homology of the mapping class group of any $3$-manifold stabilizes under connected sum and boundary connected sum with an arbitrary $3$-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks and includes as particular cases homological stability for symmetric automorphisms of free groups, automorphisms of certain free products, and handlebody mapping class groups. Our methods also apply to manifolds of other dimensions in the case of stabilization by punctures.

We study the high-energy asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface $M$ of Anosov type. To do this, we look at families of distributions associated to them on the cotangent bundle $T^{*}M$ and we derive entropic properties on their accumulation points in the high-energy limit (the so-called semiclassical measures). We show that the Kolmogorov-Sinai entropy of a semiclassical measure $\mu$ for the geodesic flow $g^t$ is bounded from below by half of the Ruelle upper bound; that is, ${\displaystyle h_{\mathrm{KS}}(\mu ,g)\ge \frac{1}{2}{\int }_{S^{*}M}{\chi }^{+}(\rho ) d\mu (\rho ),}$ where ${\chi }^{+}(\rho )$ is the upper Lyapunov exponent at point $\rho$.

The universal enveloping algebra of any simple Lie algebra $\mathfrak{g}$ contains a family of commutative subalgebras, called the quantum shift of argument subalgebras. We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free $^LG$-opers on ${\mathbb{P}}^1$ with regular singularity at one point and irregular singularity of order $2$ at another point. We also prove a multipoint generalization of this result, describing the spectra of commuting Hamiltonians in Gaudin models with irregular singularity. In addition, we show that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of $\mathfrak{g}$ has a cyclic vector in any irreducible finite-dimensional $\mathfrak{g}$-module. As a by-product, we obtain the structure of a Gorenstein ring on any such module. This fact may have geometric significance related to the intersection cohomology of Schubert varieties in the affine Grassmannian.

We prove a new omega result for extreme values of high-energy Hecke-Maass eigenforms on arithmetic hyperbolic surfaces. In particular we show that they exhibit much stronger fluctuations in the $L^{\infty }$-aspect than what the random wave conjecture would have predicted. We adapt the method of resonators and connect values of eigenfunctions to global geometry of these surfaces by employing the pre-trace formula and twists by Hecke correspondences.

A twist construction for manifolds with torus action is described generalizing certain T-duality examples and constructions in hypercomplex geometry. It is applied to complex, SKT, hypercomplex, and HKT manifolds to construct compact simply connected examples. In particular, we find hypercomplex manifolds that admit no compatible HKT metric, and HKT manifolds whose Obata connection has holonomy contained in $\mathrm{SL}(n,\mathbb{H})$.