We consider deformations of objects over non-commutative rings, which are more natural than commutative ones in some cases. As examples, we treat deformations of perverse coherent sheaves, and semi-orthogonal decompositions of derived categories corresponding to singularities.

The cycle map from Chow groups to Hodge cohomology has a long and illustrious history. This talk will concern a variant adapted to log-smooth pairs, mapping out of the “log Chow groups” and taking values in log Hodge cohomology (using differential forms with log poles).

One way to investigate whether two links are equivalent is to use link invariants. One of the most famous ones is Khovanov-Rozansky triply graded homology. It is defined algebraically and is notoriously hard to compute. Gorsky, Negut and Rasmussen conjectured that it can be calculated as cohomology of certain coherent sheaves on the flag Hilbert scheme. A link invariant of a similar nature has been constructed by Oblomkov and Rozansky in terms of matrix factorizations. The aim of this joint work in progress with Roman Bezrukavnikov, Alexei Oblomkov and Andrei Negut is to prove the conjecture by reformulating Oblomkov-Rozansky using work of Ben-Zvi and Nadler et al. together with work of Arkhipov and Kanstrup

The branching rule for representations of the symmetric group tells us that, over a field of characteristic zero, the dimension of the irreducible representation indexed by a partition \lambda is given by the number of directed lattice paths from \lambda (though of as an integer vector) to the origin that stay inside the dominant Weyl chamber. Over a field of positive characteristic (or for Hecke algebras at roots of unity) this is no longer true, and the dimension of an irreducible is in general unknown. We will see, however, that there is a nice class of irreducibles (called calibrated, completely splittable or tame by different authors) whose dimension is given by the number of lattice paths to the origin that stay within a dilation of the fundamental alcove. Then we will provide an explicit resolution of these modules by Specht modules, whose dimensions are as in the characteristic zero case. This gives a representation-theoretic interpretation of combinatorial results of Filasetta, Krattenthaler and others. Based on joint work with C. Bowman and E. Norton.

Abel initiated the  study of  the  polynomial  Pell equation $x^2-g(z)y^2=1$, where  $g(z)$ is a fixed polynomial. We are looking for solutions where $x, y, z$ are also polynomials in a variable  $t$. Then we use these results to exhibit some new  undecidable questions in complex algebraic geometry.

I will discuss applications of tropical geometry to the cohomology of moduli spaces of curves, with and without marked points.  Based on joint work with Melody Chan, Carel Faber, and Soren Galatius.

In the early 90s, Simpson established an equivalence between the category of local systems, and that of the semi-stable Higgs bundles whose Chern class is zero, over a smooth projective variety over the complex numbers. Such a correspondence between Higgs bundles and local systems is sometimes called the Simpson correspondence. In positive characteristic, an analogue of this result was first discovered by Ogus and Vologodsky in 2007, for bundles with nilpotent connections and Higgs fields. Later on, Groechenig, Chen and Zhu also established a full correspondence without the nilpotence conditions, for bundles over curves. In this talk, following the approach of Groechenig,  I will show that a full correspondence still holds for bundles over abelian varieties.

The group of invertible objects in a symmetric monoidal category is a basic invariant of the category. The Picard group of the category of spectra is known to consist only of the sphere spectrum and its suspensions. In chromatic homotopy theory we consider the categories of spectra localized with respect to the Morava K-theories K(n) (where we first fix a prime p), and these have much richer Picard groups. I will first describe some classical results about these groups. Then I will talk about current work on understanding the Picard group of the K(2)-local category at p=2. This is based on joint  
work with Beaudry, Goerss and Henn.