Foam categories from categorified quantum groups
Friday, 6.11.15, 10:15-11:15, Hörsaal II, Albertstr. 23b
About 15 years ago, Khovanov introduced an homological invariant of knots categoryfying the Jones polynomial. Though this polynomial can be viewed both from a representation-theoretic and a diagrammatic point of view, for long only the latter version had been categorified.\nI will explain how, inspired by the concept of skew-Howe duality developed by Cautis, Kamnitzer, Morrison and Licata, one can describe the cobordism categories used in Khovanov homology from categorified quantum groups. In turn, this method allowed us to precisely redefine the sln generalizations of these categories, yielding a combinatorial and integral description of Khovanov-Rozansky homologies.
Foam categories from categorified quantum groups
Friday, 6.11.15, 10:15-11:15, Hörsaal II, Albertstr. 23b
About 15 years ago, Khovanov introduced an homological invariant of knots categoryfying the Jones polynomial. Though this polynomial can be viewed both from a representation-theoretic and a diagrammatic point of view, for long only the latter version had been categorified.\nI will explain how, inspired by the concept of skew-Howe duality developed by Cautis, Kamnitzer, Morrison and Licata, one can describe the cobordism categories used in Khovanov homology from categorified quantum groups. In turn, this method allowed us to precisely redefine the sln generalizations of these categories, yielding a combinatorial and integral description of Khovanov-Rozansky homologies.
New counterexamples to Quillen's conjecture
Friday, 13.11.15, 10:15-11:15, Raum 404, Eckerstr. 1
In the talk I will explain the computation of cohomology of \(GL_3\) over function rings of affine elliptic curves. The computation is based on the study of the action of the group on its associated Bruhat-Tits building. It turns out that the equivariant cell structure can be described in terms of a graph of moduli spaces of low-rank vector bundles on the corresponding complete curve. The resulting spectral sequence computation of group cohomology provides very explicit counterexamples to Quillen's conjecture. I will also discuss a possible reformulation of the conjecture using a suitable rank filtration.
Six operations on dg enhancements of derived categories of sheaves and applications
Friday, 20.11.15, 10:15-11:15, Raum 404, Eckerstr. 1
We lift Grothendieck’s six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. If time permits we give applications concerning homological smoothness of derived categories of schemes.
The arc space of Grassmannians
Friday, 27.11.15, 10:00-11:00, Raum 404, Eckerstr. 1
Arc spaces can be used as an effective tool to compute invariants of\nsingularities of algebraic varieties. In this talk, I will explain how this can\nbe achieved for a classical example: the singularities of Schubert varieties\ninside the Grassmannian. This involves a delicate study of the combinatorics\ninside of the arc space of the Grassmannian. The main tool I will discuss is a\nstratification of the arc space which plays the role of a Schubert cell\ndecomposition for lattices. Analyzing the geometric structure of the resulting\nstrata leads to the computation of invariants, mainly the log canonical\nthreshold of pairs invoving Shubert varieties.