Abstracts for Arbeitstagung 2017 on "Physical Mathematics" in honor of Yuri Manin

Given a symplectic manifold $M$, one can consider its deformation quantization, i.e. an associative multiplication law on functions on $M$ that depends on a formal parameter $h$. When $M$ is the cotangent bundle of a manifold $X$, one essentially recovers the algebra of differential operators on $X$, or rather of $h$-differential operators $P(x, hd/dx)$. Modules over differential operators are well known to have interesting applications to PDE and other topics, so it is natural to hope that modules over deformation quantization would be interesting as well.

In this talk I will explain how to compute,  in terms of linear algebra data, the Fourier-Sato
transform of perverse sheaves over a complex affine space, smooth along a hyperplane arrangement.
As an example a geometric interpretation, and a generalization, of Lusztig's braid group action on
representations of quantum groups will be given. A joint work with M. Finkelberg and M. Kapranov

The little $n$-disks operad $D_n$ is a topological operad of rectilinear embeddings of a number of disjoint "little" disks into the unit disk. Its real homotopy type is known: Due to work of Kontsevich it is formal (over $\mathbb{R}$), i.e., there is a zigzag of (homotopy) Hopf cooperads relating the cooperad of differential forms $\Omega(D_n)$ with the cohomology cooperad $H(D_n)$.

The SYZ mirror of the moduli space of Higgs bundles for a group $G$ is interpreted as the moduli space for the Langlands dual group. The talk will discuss some of the outcomes and problems related to this, with particular reference to holomorphic Lagrangian submanifolds.  Some instructive examples arise from a recent paper of Gaiotto.