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.