Homology of Lie algebras of generalized Jacobi matrices and its orthogonal, symplectic variants.
Montag, 7.5.18, 14:15-15:15, Hörsaal FRIAS, Albertstr. 19
In 1983, B. Feigin and B. Tsygan related the Homology of the\nLie algebra of generalized Jacobi matrices over a field k of\ncharacteristic 0. In this talk, I will explain my work with A. Fialowski on extension of their result to i) over any associative unital k-algebra R and ii) their orthogonal and symplectic subalgebras over R.
Some global analytic properties of VHS (…and of Hodge Modules)
Freitag, 11.5.18, 10:30-11:30, FRIAS
Hyperbolicity of moduli spaces of CY
Donnerstag, 17.5.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Mixed Hodge theory and representations of fundamental groups of algebraic varieties
Freitag, 18.5.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Number Theory Day
Freitag, 1.6.18, 00:00-01:00, Basel
TBA
Freitag, 8.6.18, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
Quantised dihedral angles and quantum dilogarithms
Montag, 11.6.18, 13:15-14:15, Hörsaal FRIAS, Albertstr. 19
I will describe a relation between quantum dilogarithms and\n3-dimensional hyperbolic geometry obtained by quantising the dihedral\nangles of an ideal hyperbolic tetrahedron with respect to the\nNeumann—Zagier symplectic structure. In this way, one constructs a\n(metaplectic) quantum operator \(Q\) realising the 3-3 Pachner move for\n4-dimensional triangulations. This realisation admits a natural\ngeneralisation to any self-dual locally compact abelian group, together\nwith a fixed gaussian exponential. The 5-term operator identity,\nsatisfied by a quantum dilogarithm over such a group, is equivalent to\nan integral identity involving the operator kernel of \(Q\).
TBA
Freitag, 15.6.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Moduli spaces of sheaves on K3 surfaces and irreducible symplectic varieties
Freitag, 22.6.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Irreducible symplectic manifolds are one of the three building blocks of compact K\b"ahler manifolds with numerically trivial canonicl bundle (together with abelian varieties and Calabi-Yau manifolds), thanks to the Beauville-Bogomolov decomposition theorem. A recent result of A. H\b"oring and T. Peternell has completed the extension of this decomposition theorem to singular projective varieties: irreducible symplectic varieties are the singular analogue of irreducible symplectic manifolds, and they are one of the building blocks of normal, projective varieties having canonical singularities and numerically trivial canonical bundle. In a recent joint work with A. Rapagnetta we prove that all moduli spaces of semistable sheaves over projective K3 surfaces (with respect to a generic polarization) are irreducible symplectic varieties, with the only excption of those isomorphic to symmetric products of K3 surfaces, and compute their Beauville form and Fujiki constant. Similar results are shown to hold for the Albanese fiber of moduli spaces of sheaves over Abelian surfaces.
A variety that cannot be dominated by one that lifts
Mittwoch, 27.6.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
In the sixties, Serre constructed a smooth projective variety in\ncharacteristic p that cannot be lifted to characteristic 0. If a variety\ndoesn't lift, a natural question is whether some variety related to it does\nlift. We construct an example of a smooth projective variety that cannot be\nrationally dominated by a smooth projective variety that lifts.\n
The rigidity theorem for motives of non-archimedean analytic spaces
Freitag, 6.7.18, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
I will give a quick introduction to the notions of motive and of motivic sheaves. Then, after recalling the main ideas of non-Archimedean analytic geometry I will define the category of motivic sheaves of non-Archimedean analytic spaces. Finally, I will state the Rigidity Theorem in this context and if time permits I will briefly sketch the main ideas about its proof and mention some applications.
Global Serre dualities
Montag, 9.7.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Serre equivalences are important autoequivalences of k-linear categories appearing in different fields of mathematics. In this talk we will ask the following question. In which way are Serre equivalences compatible with k-linear functors? For this we first review the situation in the case of algebraic geometry, where some compatibilty results are known. This motivates us to introduce the notion of a "global Serre duality", which is an abstract framework encoding the naturality of Serre equivalences. Afterwards we show the existence of global Serre dualities in the case of (abstract) representation theory. In interesting special cases, we obtain explicit descriptions of Serre equivalences. This last step will require some techniques from abstract cubical homotopy theory. This is part of an on-going project with Moritz Groth.\n
Sheaves on the alcoves and modular representations
Freitag, 13.7.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
I will give an overview of some recent results obtained jointly with Martina Lanini. While trying to understand the intricacies of the combinatorial category of Andersen, Jantzen and Soergel we came up with a new category that consists of ordinary sheaves on the space of alcoves of an affine Weyl group. I will show how this category provides new methods and tools for the problem of determining rational characters of algebraic groups in positive characteristics.
Variations on the theme of moment graphs
Freitag, 20.7.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Naturally arising as the 1-skeletons of torus actions on\n(nice) complex projective algebraic varieties, moment graphs were\noriginally introduced by Goresky, Kottwitz and MacPherson to compute\nequivariant cohomology of such varieties. In this talk, I will review\nsome applications of moment graph theory, starting from the equivariant\ncohomology of the flag variety, and the representation theory of a\ncomplex finite dimensional simple Lie algebra. Time permitting, I will\nalso discuss some ongoing joint work with Tomoyuki Arakawa on a certain\nclass of modules ("admitting a Wakimoto flag") for an affine Kac-Moody\nalgebra at a negative level.