NoGAGS Berlin 2012

Freie Universität Berlin December 06-07, 2012 - A joint seminar with Bremen, Humboldt University Berlin, FU Berlin, Groningen, Hamburg, and Hannover. (Other NoGAGS meetings can be found [here])

Organizers

Alexander Schmitt, Anna Wißdorf

Schedule

Unfortunately, the talk of S. Rollenske had to be canceled. It has been replaced by a talk of E. Franco Gomez (FU Berlin).

Time	Thursday, 06.12.	Friday, 07.12.
9:30		T. Finis (FU Berlin) "An approximation theorem for congruence subgroups"
10:30		Coffee
11:00		J. Kass (Hannover) "What is H_{1}(Abel map)? "
12:00		Lunch
14:00	E. Franco Gomez (FU Berlin) "Higgs-Bundles on Elliptic Curves"	P. Sosna (Hamburg) "On the Jordan-Hölder property for geometric derived categories "
15:00	Coffee	Coffee
15:30	F. Gounelas (HU Berlin) "Free curves on varieties"	F. Reede (Göttingen) "Line bundles on noncommutative surfaces"
16:30	Coffee
17:00	A. Anema (Groningen) "Covering spaces of an elliptic curve that ramify only above one point"
18:30	Dinner at Eierschale

All talks (60 min each) are at Großer Hörsaal in Zuse-Institut Berlin.

Abstracts

• A. Anema: "Covering spaces of an elliptic curve that ramify only above one point"

ABSTRACT: This talk deals with finite maps to elliptic curves E defined over the complex numbers. From algebraic topology and the theory of Riemann surfaces, one knows there exist curves D admitting a finite map g : D --> E such that g ramifies only above one point of E. We consider the problem of explicitly constructing such pairs (D, g). This is done by looking at torsion of the elliptic surface corresponding to y^2=x^3+ax+b over the curve E given by 4a^3+27b^2=1.

• T. Finis: "An approximation theorem for congruence subgroups"

ABSTRACT: By a classic theorem of Jordan (1878), every finite subgroup of GL (n, K), where K is a field of characteristic zero, contains an abelian normal subgroup of index at most J(n), where J(n) depends only on n. In characteristic p the situation is of course different. A theorem of Nori (1987) says that for all n > 0 and all primes p with p > N(n), where N is a suitable function, the subgroups of GL (n, F_p) which are generated by their elements of order p are described by connected algebraic subgroups of GL (n) defined over F_p. This result can be combined with Jordan's theorem to describe arbitrary subgroups (cf. also Larsen-Pink 2011).

Let G be a reductive algebraic group defined over Q. In the talk I will present an approximation theorem for subgroups of G (Z/p^N Z) (or, equivalently, for open subgroups of G (Z_p)), which provides a partial description of these subgroups in terms of connected algebraic subgroups of G defined over Q_p. The theorem has applications to the theory of congruence subgroups of arithmetic groups, in particular to the limit multiplicity problem. The results are joint work with Erez Lapid (Jerusalem/Rehovot).

• E. Franco Gomez: "Higgs-Bundles on Elliptic Curves"

ABSTRACT: In this talk we will describe G-Higgs bundles over an elliptic curve X. A key result on our study is that a Higgs bundle for G = GL(n,C) is (semi)stable if and only if the underlying vector bundle is (semi)stable. This fact can be extended to arbitrary complex reductive Lie groups and allows us to use the Atiyah's description of semistable vector bundles over X. We will give an explicit description of the moduli spaces of G-Higgs bundles and the Hitchin map. This is joint work with O. Garcia-Prada and P. Newstead.

• F. Gounelas: "Free curves on varieties"

ABSTRACT: We study various ways in which a variety can be "connected by curves of a fixed genus", mimicking the notion of rational connectedness. At least in characteristic zero, in the specific case of the existence of a single curve with a large unobstructed deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.

• J. Kass: "What is H_{1}(Abel map)? "

ABSTRACT: A smooth curve over the complex numbers admits an Abel map, that is, an embedding into the complex torus known as the Jacobian, and the homomorphism on homology induced by the Abel map can be identified with the Poincaré Duality isomorphism. I will describe how this result extends to singular curves. In doing so, I will describe the compactified Jacobian of a curve with axis-like singularities, a result that is of independent interest.

• F. Reede: "Line bundles on noncommutative surfaces"

ABSTRACT: In this talk we will shortly describe the concept behind noncommutative surfaces. We are interested in line bundles on these surfaces. There is a moduli space classifying such line bundles, which can be seen as a generalization of the usual Picard scheme. In examples we will study these moduli spaces and see that there is a lot of hidden classic geometry surrounding these surfaces and moduli spaces.

• P. Sosna: "On the Jordan-Hölder property for geometric derived categories"

ABSTRACT: We prove that the semiorthogonal decompositions of the derived category of the classical Godeaux surface X do not satisfy the Jordan-Hölder property. More precisely, we will show that there are two maximal exceptional sequences in this category, one of length 11, the other of length 9. This is joint work with C. Böhning und H.-C. Graf von Bothmer.

Registration

To register, please send an email mentioning your name, affiliation and whether you want to attend the conference dinner to Mrs Metzler.

Fee

There will be no conference fee.

Hotels

There is a limited capacity at Seminaris CampusHotel and Best Western Steglitz. Please make your own reservation. Details will be send to you with registration.

Other possibilities include Hotel Am Wilden Eber and Metropolitan Berlin.

Travel Information

How to get to the institute.

How to get to Zuse-Institut Berlin.

An additional map of the university.

Participants

• Klaus Altmann (FU Berlin)
• Ane Anema (Groningen)
• A. Apostolov (Hannover)
• Nurömür Hülya Argüz (Hamburg)
• Laura Bigalke (Bielefeld)
• Christian Böhning (Hamburg)
• Hans-Christian v. Bothmer (Hamburg)
• Nathan Broomhead (Hannover)
• Chiara Camere (Hannover)
• Joana Cirici (FU Berlin)
• Lennart Claus (FU Berlin)
• Ananyo Dan (HU Berlin)
• Wolfgang Ebeling (Hannover)
• Emilio Franco Gomez (FU Berlin)
• Tobias Finis (FU Berlin)
• Giovanni de Gaetano (HU Berlin)
• Frank Gounelas (HU Berlin)
• Andreas Hochenegger (Köln)
• Klaus Hulek (Hannover)
• Malek Joumaah (Hannover)
• Barbara Jung (HU Berlin)
• Sotiris Karanikolopoulos (FU Berlin)
• Jesse Kass (Hannover)
• Lars Kastner (FU Berlin)
• Stefan Keil (HU Berlin)
• Remke Kloosterman (HU Berlin)
• Sebastian Krug (Hamburg)
• Niels Lindner (HU Berlin)
• Michael Lönne (Hannover)
• Elena Martinengo (FU Berlin)
• Fabian Müller (HU Berlin)
• Nicola Pagani (Hannover)
• Stefano Pascolutti (Hannover)
• David Ploog (Hannover)
• Juan Pons Llopis (FU Berlin)
• Fabian Reede (Göttingen)
• Alexander Schmitt (FU Berlin)
• Frithjof Schulze (Hannover)
• Bernd Siebert (Hamburg)
• Pawel Sosna (Hamburg)
• Nicola Tarasca (Hannover)
• Matteo Tommasini (Hannover)
• Jaap Top (Groningen)
• Hung Ming Tsoi (Hamburg)
• Benjamin Wieneck (Hannover)
• Anna Wißdorf (FU Berlin)