tba
Friday, 21.10.16, 10:15-11:15, Raum 404, Eckerstr. 1
tba
Elliptic surfaces
Friday, 4.11.16, 10:15-11:15, Raum 404, Eckerstr. 1
Elliptic surfaces form a central part of the classification of algebraic surfaces. In my talk, I will give a brief review of the theory of elliptic surfaces, especially those with section such that the theory of Mordell-Weil lattices applies. Then I will discuss old and new applications in several directions such as sphere packings, K3 surfaces of large Picard number, the maximum number of lines on quartic surfaces in P^3, Enriques surfaces containing a given configuration of smooth rational curves.
The b-semiampleness conjecture on surfaces
Friday, 11.11.16, 10:15-11:15, Raum 404, Eckerstr. 1
An lc-trivial fibration f:(X,B)->Y is, roughly speaking, a\nfibration such that the log-canonical divisor of the pair (X,B) is trivial along the fibres of f.\nAs in Kodaira’s canonical bundle formula for elliptic fibrations, the log-canonical divisor can be expressed as the sum of the pull-back of three divisors: the log-canonical divisor of Y; a divisor, called discriminant, containing informations on the singular fibres; and a\ndivisor called moduli part related to the birational variation of the fibres.\nBy analogy with the case of elliptic fibrations, the moduli part is conjectured to be semiample.\nAmber proved the conjecture when the base Y is a curve.\nIn this talk we will explain how to prove the conjecture when Y is a surface.\nThis is a joint work with Vladimir Lazić.
Application of homology in quantum fault-tolerance
Friday, 18.11.16, 10:15-11:15, Raum 404, Eckerstr. 1
It has been realized by Richard Feynman, Peter Shor and others that by exploiting the laws of quantum mechanics some computational problems may be solved exponentially faster than on 'classical' computers. Building a so-called quantum computer is a difficult undertaking due to the fragility of quantum mechanical systems.\n\nWe will discuss how homology can help in designing fault-tolerant quantum computing architectures. In particular, we introduce a simple procedure which turns a cell complex into a quantum mechanical system in which information can be protected against noise, a so-called homological quantum code. A nice feature of this construction is that it relates geometric properties of the cell complex to properties of the quantum code. We will focus on cell complexes which are tilings of closed 2D and 4D (hyperbolic) manifolds.\nLastly, we will discuss certain no-go theorems which prove that quantum codes with certain desirable properties can never be obtained by this procedure.
Hyperbolicity of moduli spaces of abelian varieties with a level structure
Friday, 25.11.16, 10:15-11:15, Raum 404, Eckerstr. 1
For any positive integers g and n, let Ag(n) be the moduli space of principally polarized abelian varieties with a level-n structure (it is a smooth quasi-projective variety for n>2). Building on works of Nadel and Noguchi, Hwang and To have shown that the minimal genus of a curve contained in Ag(n) grows with n. We will explain a generalization of this result dealing with subvarieties of any dimension. In particular, we show that all subvarieties of A_g(n) are of general type when n > 6g. Similar results are true more generally for quotients of bounded symmetric domains by lattices.
Non-Levi branching rules and Littelmann paths
Friday, 2.12.16, 10:15-11:15, Raum 404, Eckerstr. 1
Abstract: In recent work with Schumann we have proven a conjecture of\nNaito-Sagaki giving a branching rule for the decomposition of the\nrestriction of an irreducible\nrepresentation of the special linear Lie algebra to the symplectic Lie\nalgebra,\ntherein embedded as the fixed-point set of the involution obtained by\nthe folding of\nthe corresponding Dyinkin diagram. This conjecture had been open for\nover ten years,\nand provides a new approach to branching rules for non-Levi subalgebras\nin terms\nof Littelmann paths. In this talk I will introduce the path model,\nexplain the setting of the problem, our proof, and provide some\nexamples of other non-Levi branching situations.\n
The Cremona group of the real and the complex plane
Friday, 9.12.16, 10:15-11:15, Raum 404, Eckerstr. 1
Being the birational symmetry group of the simplest kind of variety, the Cremona groups are quite large, and, depending on the ground field, rather complicated. The classification of minimal surfaces over the complex numbers and over the real numbers is not the same, and from this some differences between the Cremona group of the plane over the complex numbers and over the real numbers arise. I would like to present some of them and motivate how they are related to the classification of minimal surfaces.
Relaxed highest weight representations from D-modules on the Kashiwara flag scheme
Friday, 16.12.16, 10:15-11:15, Raum 404, Eckerstr. 1
The relaxed highest weight representations introduced by Feigin,\nSemikhatov and Tipunin are a special class of representations of the Lie\nalgebra affine sl2, which do not have a highest (or lowest) weight.\nWe formulate a generalization of this notion for an arbitrary affine\nKac-Moody algebra g. We then\nrealize induced g-modules of this type and their duals as global\nsections of twisted D-modules\non the Kashiwara flag scheme associated to g. The D-modules that appear\nin our construction\nare direct images from subschemes given by the intersection of finite\ndimensional Schubert cells with their translate by a simple reflection.\nBesides the twist, they depend on a complex number describing the monodromy\nof the local systems we construct on these intersections. These results\ndescribe for the first time explicit\nnon-highest weight g-modules as global sections on the Kashiwara flag\nscheme and extend several\nresults of Kashiwara-Tanisaki to the case of relaxed highest weight\nrepresentations. This is based on the preprint arxiv:1607.06342 [math.RT].\n\n
Reciprocity functors and class field theory
Friday, 13.1.17, 10:15-11:15, Raum 404, Eckerstr. 1
Cartier crystals and perverse constructible étale p-torsion sheaves
Friday, 20.1.17, 10:15-11:15, Raum 404, Eckerstr. 1
In 2004, Emerton and Kisin established an analogue of the Riemann-Hilbert correspondence for varieties over fields with positive characteristic p. It is an anti-equivalence between the derived categories of so-called unit F-modules and etale constructible \(p\)-torsion sheaves, inducing an anti-equivalence between the abelian categories of unit F-modules and Gabber's perverse sheaves.\n\nIn the talk we explain how this Riemann-Hilbert correspondence can be generalized to singular varieties of positive characteristic which admit an embedding into smooth, F-finite varieties, and introduce the notion of Cartier crystals as a suitable alternative for unit F-modules in this context. Furthermore, we discuss possible further generalizations and the current situation with respect to compatibilities of the correspondence with pull-back and push-forward for certain morphisms.
Curvature of higher direct images
Friday, 3.2.17, 10:15-11:15, Raum 404, Eckerstr. 1
The differential geometric properties of the classical Hodge bundles were\nfirst studied by Griffiths in the context of the period map and variation of\nHodge structures. This can be used to show the hyperbolicity of the moduli\nspace of polarized Calabi-Yau manifolds. In the talk we consider generalized\nHodge bundles which are twisted by a relative ample line bundle. An intrinsic\ncurvature formula can be given. This generalizes a result of Berndtsson on\nthe\nNakano positivity of the direct image of the ample twisted relative canonical\nbundle of a fibration as well as the curvature formula for higher direct\nimages\nof Schumacher in the canonically polarized case.
tba.
Friday, 10.2.17, 10:15-11:15, Raum 404, Eckerstr. 1
Arithmetic hyperbolicity
Friday, 10.2.17, 10:15-11:15, Raum 404, Eckerstr. 1
I will explain what it means for a variety to be arithmetically hyperbolic. I will then explain that Lang-Vojta's conjecture implies that any variety with an immersive period map is arithmetically hyperbolic. In this joint work with Daniel Loughran we extend the latter statement to algebraic stacks by rigidifying stacky period maps.
Segal approach for algebraic structures
Friday, 10.2.17, 14:00-15:00, Raum 125, Eckerstr. 1
Abstract: The operads are considered today as a conventional tool to describe homotopy algebraic structures. However, for the original problem of delooping, another formalism exists, bearing the name of Segal. This approach has proven advantageous in certain situations, such as, for example, modelling higher categories.\n\nIn this talk, we will discuss how one can illuminate and arguably simplify the proof of Deligne conjecture, the existence of E_2-structure on Hochschild cochains, using the language of Segal objects and operator categories of Barwick. We will then elaborate on our solution to the problem of extending the Segal approach to arbitrary monoidal structures, which employs the language of Grothendieck fibrations and an extension of Reedy theorem to families of model categories.\n\nWhile the second part of the talk is technical, the first one will require only basic knowledge of categories and topology.\n