| 12.10.09, | 16:15 | S A Katre (University of Pune, India) |
| | | Waring's problem for Matrices
|
| | Abstract:
The problem is more related to Linear Algebra than to Algebraic
Geometry - it is about writing matrices as sums of powers.
The talk will be good for a general audience - and the results are
interesting, surprising, and new.
|
| | | |
| 19.10.09, | 16:15 | Rene Birkner (Berlin) |
| | | "Incoherent components of the Toric Hilbert scheme" |
| | Abstract
The classical Hilbert scheme is the scheme whose closed points are all subschemes of \IPn with the same Hilbert function. That is, for
S=k[x0,...,xn] and an ideal I\subset S the function H(t) = \dimk(S/I)_t whose value at d is the dimension over k of the degree
t part of S/I. Endowing the ring S with a multigrading, i.e. the degree of xi is ai \in \IZd, we construct the multigraded
Hilbert function. This is the analogon to the classical Hilbert function with degrees in \IZd for some d > 0. Then one can consider
all ideals I\subset S with the same multigraded Hilbert function, these are the closed points of the multigraded Hilbert scheme. We
consider the simplest case, taking the semigroup \IN \A=\{\sum ni ai \,|\, ni \in \IN\} and the multigraded Hilbert function
\[ \dim(S/I)a =
1 for a\in \IN\A\\
0 otherwise
This gives a so-called Toric Hilbert Scheme. This turns out to have many interesting properties, of which some will be presented in the
talk.
|
| | | |
| 26.10.09, | 16:15 | Wolfgang Ebeling (Hannover) |
| | | Singularitäten und Coxeter-Funktoren
|
| | Abstract:
Einer isolierten Singularität einer zweidimensionalen komplex-analytischen Varietät kann man zwei verschiedene Klassen von Graphen zuordnen. Die erste Klasse von Graphen ist nur definiert,
wenn die Varietät eine Hyperfläche ist, und ihre Definition benutzt eine Deformation der Singularität. Ein Graph dieser Art ist das Coxeter-Dynkindiagramm bezüglich einer ausgezeichneten Basis von
verschwindenden Zykeln des Milnorgitters. Auf der anderen Seite kann man den dualen Graphen einer minimalen Auflösung der Singularität oder ihrer Kompaktifizierung betrachten. Einem
Coxeter-Dynkindiagramm ist ein Coxeterelement zugeordnet, das der Monodromie der Singularität entspricht. Zwischen einigen dieser Singularitäten besteht eine Spiegelsymmetrie. Im Falle der Kleinschen
und Fuchsschen Singularitäten wurde entdeckt, dass das Coxeterelement einer dieser Singularitäten in einem gewissen Sinne dual zu einem Coxeterelement eines abstrakten Gitters ist, das zu dem
Auflösungsgraphen des Spiegelpartners in Beziehung steht. Wir geben eine geometrische Interpretation für diese Gitter und Coxeterelemente und heben sie zu triangulierten Kategorien von kohärenten
Garben hoch. Wir geben auch Beziehungen dieser Coxeterfunktoren mit den Poincarereihen der entsprechenden Singularitäten an. Es handelt sich um eine gemeinsame Arbeit mit David Ploog (z.Zt. Toronto).
|
| | | |
| 02.11.09, | 16:15 | Nathan Ilten (Berlin) |
| | |
|
| | Abstract:
Bericht über seiner Teilnahme am "Extremal Laurent polynomials --new approaches to mirror symmetry and classification of Fanos"
19. - 21. Oktober 2009, Warwick, UK
|
| | | |
| 09.11.09, | 16:15 | Alvaro Liendo (Chile / Grenoble) |
| | |
Ga-actions on affine T-varieties
|
| | Abstract:
Let X = spec A be a normal affine variety endowed with an effective action of a torus T of dimension n. To introduce a Ga-action on X is equivalent to fix a locally nilpotent derivation D on the Zn-graded algebra A.
In this talk we show some classification result of pairs (A,D) in the case where D is homogeneous and with special emphasis in the case where the general orbits of the Ga-action are contained in the orbit closures of the T-action. In particular, we give the birational characterization of normal affine varieties with trivial Makar-Limanov invariant.
|
| | | |
| | | Dieser Vortrag findet im SR 016, Königin-Luise-Str. 24-26 statt!
|
| 12.11.09, | 12:00 | Lutz Hille (Münster) |
| | | Tilting bundles, Hochschild (co)homology,
and resolutions of the diagonal
|
| |
|
| | | |
| 23.11.09, | 14:30 | Victor V. Przyjalkowski (Moskau/Wien) |
| | | Weak Landau--Ginzburg models and toric degenerations
|
| | Abstract
Given a smooth Fano variety, Mirror Symmetry predicts the existence
of a so called Landau--Ginzburg model --- one-dimensional family of
varieties whose symplectic
geometry reflects the algebraic geometry of the Fano variety, and
viceversa. We discuss this relation for mirror symmetry conjecture
of Hodge structure variations that translates this relation to a
quantitative level. Given
Landau--Ginzburg model one may predict some numerical invariants of Fano variety
and its birational type. We discuss relation (going back to Batyrev)
between Landau--Ginzburg
models for given Fano variety and its toric degenerations.
|
| | | |
| 30.11.09, | 16:15 | Jan Christophersen |
| | | Milnor fibers of cyclic quotient singularities
|
| |
I will review the recent results of Nemethi & Popescu-Pampu and
Lisca on the relationship between Milnor fibres of cyclic quotient
singularities and symplectic fillings of the standard contact structure
on lens spaces.
|
| | | |
| 07.12.09, | 16:15 | Stefan Günther (FU Berlin) |
| | | Valuation theory, Riemann varieties and birational geometry
|
| | Abstract:
In order to define the relevant classes of singularities needed in Mori theory such as terminal and canonical singularities, all one needs is to define valuations of rational top differential forms for arbitrary discrete algebraic valuation rings. We show how to extend this to arbitrary valuations of the function field, by first giving an explicit formula for Abhyankar places and then extend it to arbitrary places by means of density and continouity arguements. We are thus able to define log discrepancies of a log pair $(X,D)$\, for arbitrary valuations. Moreover, we find even for divisorial discrete rank one valuations an explicit formula for the valuation of a top differential form, that makes it in principle possible to calculate the discrepancy.
As applications we investigate the $\rm{NKLT}$-locus of the log pair $(X,D)$\,, i.e. the set of all valuations $\nu$\, for which the log discrepancy is $\leq 0$\,.
We can generalize the notion of log canonical centers and can prove a generalized adjunction formula for Abhyankar lc centers.
If there is enough time, we will consider sheaves on the Riemann variety of the function field associated to $b$-divisors in the sence of Shokurov and investigate their coherence and local freeness properties. We will give a simple criterion for a monotonous sequence of $\mathbb Q-b$-Cartier divisors to become stationary which is a basic question of birational geometry arising in connection with the finite generation of a divisorial algebra such as the flipping algebra of a flipping contraction or the log canonical algebra of a log pair $(X,D)$\,, and even in connection with the question whether a sequence of log flips termitates.
|
| | | |
| 14.12.09, | 16:15 | Ann Lemahieu (Paris) |
| | | The monodromy conjecture for nondegenerate surface singularities.
|
| | Abstract:
The monodromy conjecture predicts a relation between the geometry and the topology of a singularity. In particular, it says that a pole s_0 of the local topological zeta function in 0 induces an
eigenvalue of monodromy
e^{2i pi s_0} at a point in the neighbourhood of 0.
When the singularity is given by a polynomial that is nondegenerate with respect to its Newton polyhedron, then one can express the local topological zeta function and the zeta function of monodromy in terms
of the Newton polyhedron. We analyze these formulas for surface singularities: we provide a set of monodromy eigenvalues and a set of false candidate poles. In this way we obtain a proof for the monodromy
conjecture for nondegenerate surface singularities.
|
| | | |
| 04.01.10, | 16:15 | Frederik Witt (Regensburg/München) |
| | | Symplektische Orbifolds
|
| | Abstract:
Hier wird über eine Arbeit von Tolman/Lerman berichtet, in der T-Varietäten aus symplektischer Sicht betrachtet werden.
|
| | | |
| 11.01.10, | 16:15 | Swantje Gährs (Hannover) |
| | | GKZ-Systeme und Monodromie von Hyperflächen-Singularitäten
|
| | Abstract:
In diesem Vortrag soll eine Einführung in GKZ-Systeme gegeben werden. Insbesondere soll eine Beziehung zwischen GKZ-Systemen und Picard-Fuchs-Gleichungen von Hyperflächen-Singularitäten hergestellt
werden. Daraus lassen sich dann Beziehungen zur Monodromie herstellen und einige Aspekte der Spiegelsymmetrie ausnutzen.
| | | | |
| 18.01.10, | 16:15 | Ragnar-Olaf Buchweitz (Toronto) |
| | | The Koszul Complex blows up a point
|
| | Abstract:
We will report on the following results from joint work with Thuy Pham:
(1) The endomorphism ring of the syzygy modules
in the tautological Koszul complex is of finite global dimension
and its derived category is equivalent to that of the affine space
blown up in a point.
(2) For any Veronese embedding of a projective space
the cone over it admits a noncommutative desingularization
in that its canonical small desingularization, the total space of the
embedding ample line bundle, has its derived category equivalent to
that of an algebra.
Time permitting, we will also discuss in general the question of
existence of tilting objects on affine bundles and how it gives rise to
potentially new invariants and may help address some classical
open problems.
|
| | | |
| 25.01.10, | 16:15 | Vortrag fällt aus! |
| | |
|
| |
|
| | | |
| 01.02.10, | 16:15 | David Ploog (Hannover) |
| | | Comparing exceptional and spherical collections
|
| | Abstract:
Collections of exceptional and of spherical objects are of fundamental importance in the study of derived categories. It has been known for a while that they are not as antagonistic as it appears. In this
talk, I will present a way of comparing Coxeter functors from each collection
(joint with Chris Brav). This is inspired by recent results about lifts
of Milnor lattices of certain singularities in different ways.
|
| | | |
| 08.02.10, | 16:15 | Claus Hertling (Mannheim) |
| | | Eine Verallgemeinerung von Hodge-Strukturen und oszillierende Integrale.
|
| | Abstract:
Die Physiker Cecotti und Vafa haben 1991 eine Verallgemeinerung von
Hodge-Strukturen
betrachtet, die verwandt ist zu Simpsons harmonischen Bündeln. Von
Mathematikern
wurde sie seit 2002 unter verschiedenen Namen aufgegriffen (TERP-Struktur,
integrable Twistor-Struktur, wilde Hodge-Struktur, nichtkommutative
Hodge-Struktur).
Sie besteht aus einem holomorphen Vektorbündel auf P1C mit einem
flachen Zusammenhang
auf C* mit Polen der Ordnung 2 in 0 und unendlich, mit einer flachen
reellen Struktur
und einer gewissen flachen Paarung auf dem Bündel auf C*.
Sie tritt via oszillierende Integrale bei holomorphen Funktionen mit
isolierten Singularitäten auf,
das ist verwandt zum Auftreten in der Physik. Eine Reihe von Resultaten
zu Hodge-Strukturen
haben Analoga bei dieser Verallgemeinerung: der Begriffe nilpotente
Orbits und
gemischte Hodge-Strukturen und ihre Korrespondenz, die Krümmung von
klassifizierenden
Räumen. Bei zahmen Funktionen hat man reine polarisierte Strukturen.
Bei Funktionskeimen und ihren Entfaltungen hat man nilpotente Orbits und
gemischte Strukturen.
Nur erste Schritte sind getan bei potentiellen Anwendungen und bei der
Beschreibung
der Strukturen von den Stokes-Daten der irregulären Pole aus.
|
| | | |
| 15.02.10, | 16:15 | Mario Garcia (Bonn) |
| | | Some partial differential equations with moment map interpretation
|
| | Abstract:
We will recall the moment interpretation of two well-known partial differential equations: the Hermite-Yang-Mills equation for a connection on a bundle and the constant scalar curvature
equation for a Kähler metric. We will see that the theories concerning this equations merge into a unique one when we consider a suitable extension of the groups of symmetries. The system of equations
that arise as the zeros of the new moment map couples a Kähler metric with a Hermite-Yang-Mills connection. We will see then how, with the help of Algebraic Geometry, the moment map interpretation of
the equations can be used to find examples of solutions. In order to do (all) this, we will first recall the basic theory about moment maps, symplectic reduction and Kähler quotients.
|
| | | |
| 15.02.10, | 18:15 | Theo de Jong (Mainz) |
| | | Das Lebesgue-Integral
|