In this paper we strengthen the results of [SV] by presenting their derived version. Namely, we define a derived Knizhnik - Zamolodchikov connection and identify it with a derived Gauss - Manin connection.
We prove general Dwork-type congruences for Hasse--Witt matrices attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and $p$-adic analytic properties of functions originating from polynomial solutions modulo $p^
s$ of Knizhnik--Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the $p$-adic KZ connection associated with the family of hyperelliptic curves $y^2=(t-z_1)dots (t-z_{2g+1})$ has an invariant subbundle of rank $g$. Notice that the corresponding complex KZ connection has no nontrivial subbundles due to the irreducibility of its monodromy representation.
We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy categories o
f DGLAs and SHLAs (L infinity algebras) considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible with the derived stacks of Toen--Vezzosi and Lurie. Another application is that the cohomology groups associated to any classical deformation problem (in any characteristic) admit the same operations as Andre--Quillen cohomology.
In order to develop the foundations of logarithmic derived geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log etale maps and use this to define derived log stacks.
We prove two existing conjectures which describe the geometrical McKay correspondence for a finite abelian G in SL3(C) such that C^3/G has a single isolated singularity. We do it by studying the relation between the derived category mechanics of comp
uting a certain Fourier-Mukai transform and a piece of toric combinatorics known as `Reids recipe, effectively providing a categorification of the latter.
We give three proofs that valuation rings are derived splinters: a geometric proof using the absolute integral closure, a homological proof which reduces the problem to checking that valuation rings are splinters (which is done in the second authors
PhD thesis and which we reprise here), and a proof by approximation which reduces the problem to Bhatts proof of the derived direct summand conjecture. The approximation property also shows that smooth algebras over valuation rings are splinters.