We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic zero point induced from the automorphic representation is formally smooth of the correct dimension. We do this by employing the Taylor–Wiles–Kisin patching method together with Kisin’s technique of analyzing the generic fiber of universal deformation rings. Along the way we give a characterization of smooth closed points on the generic fiber of Kisin’s potentially semistable local deformation rings in terms of their Weil–Deligne representations.

We provide a mirror symmetry theorem in a range of cases where state-of-the-art techniques that rely on concavity or convexity do not apply. More specifically, we work on a family of FJRW potentials (named for the Fan, Jarvis, Ruan, and Witten quantum singularity theory) which is viewed as the counterpart of a nonconvex Gromov–Witten potential via the physical Landau–Ginzburg/Calabi–Yau (LG/CY) correspondence. The main result provides an explicit formula for Polishchuk and Vaintrob’s virtual cycle in genus zero. In the nonconcave case of the so-called chain invertible polynomials, it yields a compatibility theorem with the FJRW virtual cycle and a proof of mirror symmetry for FJRW theory.

Let $W$ be a smooth complex variety with the action of a connected reductive group $G$. Adapting Teleman’s stratification approach to a microlocal context, we prove a vanishing theorem for the functor of $G$-invariant sections—that is, of quantum Hamiltonian reduction—for $G$-equivariant twisted $\mathcal{D}$-modules on $W$. As a consequence, when $W$ is affine we establish a sufficient combinatorial condition for exactness of the global sections functors of microlocalization theory. When combined with the derived equivalence results of our recent work, this gives precise criteria for “microlocalization of representation categories” in the spirit of Kashiwara–Rouquier and many other authors.

In this paper we prove a functional transcendence statement for the $j$-function which is an analogue of the Ax–Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions with the $j$-function are governed by modular relations.