A new proof of the Global Torelli Theorem for holomorphic symplectic varieties
Freitag, 10.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In a joint work with Benjamin Bakker, we develop a theoretical framework to approach the global moduli theory of certain singular symplectic varieties. Our work is based on new results about the deformation theory of these varieties together with the notion of ergodic complex structures which has been introduced by Verbitsky and used to study for example hyperbolicity questions. I will explain how to use these techniques to prove a Global Torelli theorem for the varieties in question. Our result in particular gives a new proof of Verbitsky's Global Torelli Theorem for irreducible symplectic manifolds as soon as the second Betti number is at least 5.
A new proof of the Global Torelli Theorem for holomorphic symplectic varieties
Freitag, 10.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In a joint work with Benjamin Bakker, we develop a theoretical framework to approach the global moduli theory of certain singular symplectic varieties. Our work is based on new results about the deformation theory of these varieties together with the notion of ergodic complex structures which has been introduced by Verbitsky and used to study for example hyperbolicity questions. I will explain how to use these techniques to prove a Global Torelli theorem for the varieties in question. Our result in particular gives a new proof of Verbitsky's Global Torelli Theorem for irreducible symplectic manifolds as soon as the second Betti number is at least 5.
Deformations of path algebras of quivers with relations
Freitag, 17.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk I will present ongoing joint work with Zhengfang Wang on deformations of path algebras of quivers with relations. Such path algebras naturally appear in many different guises in algebraic geometry and representation theory and I would like to explain how one can obtain concrete descriptions of their deformations. For example, deformations of path algebras of quivers with relations can be used to describe deformations of the Abelian category of coherent sheaves on any quasi-projective variety X, deformation quantizations of Poisson structures on affine n-space, or PBW deformations of graded algebras.
Dual complexes of log Calabi-Yau pairs and Mori fibre spaces
Freitag, 24.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Dual complexes are CW-complexes, encoding the combinatorial data of how the irreducible components of a simple normal crossing pair intersect. They have been finding useful applications for instance in the study of degenerations of projective varieties, mirror symmetry and nonabelian Hodge theory. In particular, Kollár and Xu conjecture that the dual complex of a log Calabi-Yau pair should be a sphere or a finite quotient of a sphere. It is natural to ask whether the conjecture holds on the end products of minimal model programs. In this talk, we will validate the conjecture for Mori fibre spaces of Picard rank two.
On automorphism groups of fields with operators
Freitag, 31.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In 1993 Lacar showed with model-theoretical techniques that the group of field automorphisms of the complex numbers which fix pointwise the algebraic closure of the rationals is simple, assuming the continuum hypothesis. He later on provided a different proof without assuming CH. There are two main ingredients in Lascar's proof: First, isolating those automorphisms such that the image of a point is algebraic over the point, and secondly, amalgamating field extensions with prescribed automorphisms.\n\nIn this talk, we will present a sketch of Lascar's proof and explain how the techniques can be used in order to determine the simplicity of the automorphism group of algebraically closed fields (in all possible characteristics) with additional structure (such as a derivation or a transformal map, often arising in algebraic dynamical systems). No prior knowledge of model theory or mathematical logic is required for this talk.\n\n\n