This is a joint work in progress with Sebastian Orsted.  Given an algebraic variety X acted by an affine algebraic group G, we make sense of  the derived category of DG-modules over the DG-algebra of differential forms on X  equivariant with respect to differential forms on G. The construction uses an explicit model for  a certain homotopy limit of a diagram of  DG-categories developed in our earlier work and generalizing a recent result of Block, Holstein and Wei. We compare the obtained category with a certain category of sheaves on the (shifted) cotangent bundle T^*X descending to the Hamiltonian reduction of the cotangent bundle. Two special cases are of interest. In the first, X is a point. Thus we compare comodules over Omega(G)  with G-equivariant coherent sheaves on Lie(G). In the second case, X is a simple algebraic group, with the action of the square of the upper triangular subgroup. We obtain a category closely related to the affine Hecke category.

We established an axiomatic version of Christ-Kiselev's multi-linear operator techniques.
As applications,  several spectral results of perturbed periodic Schrodinger operators are obtained, including WKB solutions, sharp transitions of preservation of absolutely continuous spectra, criteria of absence of singular spectra and sharp bounds of Hausdorff dimensions of singular spectra.

We study experimentally the Hermite factor of BKZ2.0 on uSVP lattices, with the motivation of understanding the concrete security of LWE in the setting of homomorphic encryption. We run experiments by generating instances of LWE in small dimensions, where we consider secrets sampled from binary, ternary or discrete Gaussian distributions. We convert each LWE instance into a uSVP instance and run the BKZ2.0 algorithm to find an approximation to the shortest vector. When the attack is successful, we can deduce a bound on the Hermite factor achieved for the given blocksize. This allows us to give concrete values for the Hermite factor of the lattice generated for the uSVP instance. We compare the values of the Hermite factors we find for these lattices with estimates from the literature and find that the Hermite factor may be smaller than expected for blocksizes 30, 35, 40, 45. Our work also demonstrates that the experimental and estimated values of the Hermite factor trend differently as we increase the dimension of the lattice, highlighting the importance of a better theoretical understanding of the performance of BKZ2.0 on uSVP lattices.

In 2012, Filoche and Mayboroda introduced the concept of the landscape function u, for an elliptic operator L, which solves the inhomogeneous equation Lu=1. This landscape function has remarkable power to predict the shape and location of localized low energy eigenfunction. These ideas led to beautiful results in mathematics, as well as theoretical and experimental physics. In this talk, we first briefly review these results of landscape theory for differential operators on R^d. We will then discuss some recent progress of extending landscape theory to tight-binding Hamiltonians on discrete lattice Z^d. In particular, we show that the effective potential 1/u creates barrier for appropriate exponential decay eigenfunctions of Agmon type for some discrete Schrodinger operators. We also show that the minimum of 1/u leads to a new counting function, which gives non-asymptotic estimates on the integrated density of states of the Schrodinger operators. This talk contains joint work in progress with S. Mayboroda and some numerical experiments with W. Wang.