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# SoSe2017

Unless otherwise specified, all talks place in room 119 (Arnimallee 3) and begin at 16:15

Schedule:

24th April 2017
Arvid Seqveland (USN, Norway)
Noncommutative Geometric Invariant Theory

Abstract:

A scheme over $k$ can be defined by its local rings, and is a fine moduli for all its closed points $M$. It is well known that the local rings are pro-representing hulls for the deformation functor $Def_M$ from the category $l$ of local Artinian $k$-algebras to the category of sets. Earlier work gives an algorithm for computing pro-representing hulls for the deformation functor, and so gives a (formal) moduli for its objects. Here we will always work in a category of modules, but the deformation theory works for any additive abelian category.

We generalize the ordinary deformation theory: Let $A$ be an associative $k$-algebra, not necessarily commutative, and let $\mathcal M=\{M_1,...,M_r\}$ be a set of $r$ right $A$-modules. We will define the category $a_r$ of $r$-pointed Artinian $k$-algebras and the noncommutative deformation functor $Def_{\mathcal M}:a_ràSets$. We prove that the noncommutative deformation functor has a prorepresenting hull, acting as a semilocal ring for a noncommutative scheme which can be defined. Along the way, we obtain a structure theorem for finitely generated associative algebras from which we can conclude that the basic objects in noncommutative algebra are the matrix polynomial algebras.

Based on this noncommutative algebraic geometry, we study geometric invariant theory: The classification of orbits under group-actions. It turns out that we can construct noncommutative schemes that are fine moduli of the orbits, also for families of non-stable points. We give some examples where sufficient invariants and their relations are given.

Please note different time, date and place!!

27th April, 14:00 - 16:00: room 053 Takustr. 9 (Computer Science Building)

Barbara Bolognese (Sheffield)

Strange Duality on abelian surfaces
Abstract:

With the purpose of examining some relevant geometric properties of the moduli space of sheaves over an algebraic surface, Le Potier conjectured some unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on the moduli space. Such conjecture is widely known as Strange Duality conjecture. After having motivated the problem by looking at certain instances of quantization in physics, we will work in the setting of surfaces. We will then sketch the proof in the case of abelian surfaces, giving an idea of the techniques that are used. In particular, we will show how the theory of discrete Heisenberg groups and fiber wise Fourier-Mukai transforms, which might be applied to other cases of interest, enter the picture. This is joint work with Alina Marian, Dragos Opera and Kota Yoshioka.

This talk will take place in room 053, Takustr. 9 (Computer Science Building)

Please note different time, date and place!!

3rd Mai 2017, 14:00 - 16:00: room 046 Takustr. 9 (Computer Science Building)

Maria Evelina Rossi (Genoa) and Laura Tozzo (Kaiserslautern and Genoa)

Gorenstein k-algebras and the structure of Macaulay’s Inverse System

Abstract:
We  present an overview on some recent results concerning the structure of Gorenstein k-algebras.  The common tool of these talks is the so-called Inverse System, introduced by Macaulay at the beginning of the 20th century. The Inverse System is a specialization of Matlis duality, which allows us to compute explicitly the dual module associated to an Artinian (local or graded) Gorenstein k-algebra.

In the first part we give some applications to the Artinian Gorenstein k-algebras,  in particular we show how it is possible to use Macaulay's Inverse System to study isomorphism classes of local algebras. This tool has a strong interest in the study of the components of the Hilbert scheme of $d$ points in the affine space. We  will describe isomorphism classes of local Artinian Gorenstein k-algebras of given length (or given Hilbert function).

In the second part we present a recent generalization of Macaulay’s correspondence to higher dimensions. To date a general  structure for  Gorenstein   $k$-algebras of any dimension (and codimension)  is not understood. We extend Macaulay's correspondence  characterizing  the submodules of the divided powers ring in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras.  We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating the results are given.

15th May 2017

Immanuel van Santen (Hamburg)

Embeddings of Affine Spaces into Algebraic Groups

Abstract:

This is joint work with Peter Feller (Max Planck Institute for Mathematics in Bonn) and Jérémy Blanc (University of Basel). We consider embeddings X -> Y of affine varieties and study them up to automorphisms of Y. After recalling some classical results and examples in case Y is the affine space C^n, we focus on the following three results concerning embeddings where X is some affine space and Y is the underlying variety of a (linear) algebraic group:

(1) If G is an algebraic group with trivial character group and if G is not three-dimensional, then all embeddings C -> G are the same up to automorphisms of G.

(2) There exist infinitely many embeddings C^2 -> SL_2 with the same image as the standard embedding C^2 -> SL_2 up to automorphisms of SL_2. More precisely, these embeddings are parametrized by C^*.

(3) There exists an infinite-dimensional'' family of hypersurfaces in SL_2 that are isomorphic to C^2 such that distinct members of the family cannot be mapped to each other via automorphisms of SL_2.

We give some techniques used in the proofs of these results and describe the main strategies.

22nd May 2017

Bernd Kreussler (Limerick, Ireland)

Zariski decomposition on rational surfaces and the algebraic dimension of certain 3-folds

Abstract:

The anti-Kodaira dimension of rational surfaces plays an essential role in the calculation of the algebraic dimension of certain simply connected compact 3-folds. A powerful tool to study the anti-Kodaira dimension of rational surfaces is the Zariski decomposition of an anti-canonical divisor on such a surface. In this talk, which reports on joint work with N. Honda, our results on the Zariski decomposition of an anti-canonical divisor on a rational surface will be explained. The main application of these results is a proof that a certain type of twistor space cannot exist.

29th May 2017

Moduli spaces of generalized holomorphic bundles

Abstract:

Generalized holomorphic bundles are the analogues of holomorphic vector bundles in the generalized geometry setting. For some generalized complex structures, these bundles correspond to co-Higgs bundles, flat bundles or Poisson modules. I will give an overview of what is known about generalized holomorphic bundles, and describe their moduli spaces in some speci fic examples. Part of this is joint work with Shengda Hu and Mohamed El Alami.

12th June 2017

Duco van Straten (Mainz) 14:15 - 16:00

Elliptic curves and Calabi-Yau operators

Abstract:

An especially nice class of ordinary differential operators of order four was introduced by Almkvist and Zudilin and called Calabi-Yau operators. The name refers to Calabi-Yau varieties,  which lead to such operators and which played a major role in mirror symmetry. In the talk (which is joint work with S.Cynk from Krakow) I will explain how families of elliptic curves can be used to construct Calabi-Yau operators of great complexity which exhibit new features.

Andriy Regeta (Grenoble) 16:15 - 18:00

Automorphism groups of affine toric varieties

Abstract:

We are going to discuss the following problem:  to which extent the group of automorphisms of an affine algebraic variety determines the variety? In general the answer is negative. On the other hand, H. Kraft proved that the group of automorphisms of the affine n-space seen as an ind-group determines the affine n-space in the category of connected affine varieties.  In this talk we are going to discuss a similar result for affine toric varieties. In case of dimension two, we characterise a big class of affine surfaces by their automorphism groups viewed as abstract groups.

19th June 2017

Greg Stevenson

The homological algebra of complete intersection singularities

Abstract:
I'll give an overview of (some of) what is known about derived and singularity categories of (global) complete intersections. In particular, I'll discuss how to reduce the general case to understanding hypersurface singularities, the classification of thick subcategories, and consequences for duality and Ext-vanishing.

3rd July 2017

Priska Jahnke (Heilbronn)

Numerische Methoden in der algebraischen Geometrie

Abstract: Die Steuerung von Robotern oder andere kinematische Probleme lassen sich auf die Lösung oft komplizierter Systeme polynomialer Gleichungen über $\mathbb{R}$ zurückführen. Im allgemeinen wird eine Position im Raum durch endlich viele Stellungen der Roboterarme/-beine anfahrbar sein, die Lösungsmengen sind also endlich viele Punkte. Über $\mathbb{C}$ bzw. nach Übergang zu $\mathbb{P}_n(\mathbb{C})$ lässt sich bei allgemeiner Wahl aller Parameter zumindest die Anzahl der Punkte angeben - die Frage nach den physikalisch relevanten Lösungen ist jedoch nur schwer zu beantworten. Im Vortrag soll eine von A. Sommese und C. Wampler II entwickelte numerische Lösungsmethode des Problems vorgestellt werden, umgesetzt als Softwarepaket "Bertini" (D. Bates, J. Hauenstein, A. Sommese, C. Wampler II). Dabei werden die L\"osungen mittels "Pathtracking" gefunden, d.h. es werden reelle Wege von den Lösungen eines ähnlichen, aber bekannt lösbaren Startsystems zu den Lösungen des gesuchten Zielsystems konstruiert und verfolgt. Das System arbeitet mit einstellbarer Genauigkeit, kann projektiv und über $\mathbb{C}$ rechnen.