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Symposium on singularities and their topology

Date: Hannover, 14-17 July 2014

How to get here

Talks will take place in the main building of the University (Welfengarten 1). Room F309

Speakers include:

  • Enrique Artal
  • Wim Veys
  • Anatoly Libgober
  • Mathias Schulze
  • Christian Sevenheck
  • Eleonore Faber
  • Mihai Tibar
  • Vincent Florens
  • Ursula Ludwig
  • Alejandro Melle
  • Duco van Straten

Talks:

 

Enrique Artal:   On the topology of line arrangements II

The invariant introduced in Florens' talk is related with some twisted intersection forms which are helpful in the study of characteristic varieties of line arrangements; this study can be extended almost word-by-word to the case of rational arrangements. This is part of joint work with Vincent Florens and Benoît Guerville-Ballé

Wim Veys: Bounds for p-adic exponential sums and log-canonical thresholds

In joint work with Raf Cluckers, we propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo p^m lying p-adically close to y, and the proposed bounds are uniform in p, y, and m. We give evidence for the conjecture, by showing uniform bounds in p, y, and in some values for m. On the way, we prove new bounds for log-canonical thresholds which are closely related to the bounds predicted by the conjecture.

Anatoly Libgober: Elliptic genus of singularities.

I will discuss elliptic genus of orbifolds, elliptic genus of singularities and their iterrelation (Landau-Ginzburg-Calabi Yau correspondence). Mathematical generalizations (corresponding to phases of N=2 theories) will also be discussed.

Mathias Schulze: Derivations of negative degree and a generalized Euler sequence

Jonathan Wahl gave the following cohomological characterization of projective space: Let X be a smooth complex projective variety and L an ample line bundle on X. If the tangent sheaf
twisted by the inverse of L admits a global section then X must be projective space. One ingredient of his proof is a generalized Euler sequence that translates the global section to a derivation of degree -1 on the section ring of L. A key ingredient is a result of Mori and
Sumihiro that yields a degree 1 element of this section ring; then Zariski's lemma applies.
These investigations led Wahl to conjecture that normal graded isolated singularities do not admit negative degree derivations, at least for some choice of grading. It is this last reserve that makes the conjecture difficult to approach in general. We describe a proof of Wahl's conjecture for isolated complete intersection singularities of order at least 3 and a series of
counter-examples of order 2 (joint with Michel Granger). In order to explain these counter-examples cohomologically, in the sense of Wahl, we further generalize Wahl's Euler sequence using on results of Demazure-Watanabe (joint with Xia Liao).

Christian Sevenheck:  Hyperplane sections of free divisors and discriminants and potential applications in mirror symmetry.

About 10 years ago Buchweitz and Mond introduced so-called linear free divisors.They appear in particular as discriminants of quiver representations and yield many new examples of free divisors.The special and simplest case of the normal crossing divisor has an interpretation as Landau-Ginzburg model in mirror symmetry. We describe how to extend this construction to other discriminants, describe the Hodge theoretic properties of these functions and give some speculations on their appearence in mirror symmetry.

Eleonore Faber: Non-commutative resolutions and the global spectrum of commutative rings

Motivated by algebraic geometry, one studies non-commutative analogs of resolutions of singularities. In short, a non-commutative resolution of a commutative ring R is an endomorphism ring of a certain R-module of finite global dimension. However, it is not clear which finite values of global dimensions are possible, even for rings R of low Krull-dimension. This leads us to consider the so-called global spectrum of a commutative ring, that is, the set of all possible global dimensions of endomorphism rings of Cohen-Macaulay-modules.

In this talk we discuss several approaches to non-commutative resolutions of singularities, namely, Van den Bergh's non-commutative crepant resolution and the approach of Dao-Iyama-Takahashi-Vial. Then we study their relevance for non-normal rings, especially the question, under which conditions a non-commutative resolution exists. In particular, we consider the problem of the existence of non-commutative resolutions of free divisors. Finally, we will address some questions connected with the global spectrum. This is joint work with H. Dao and C. Ingalls.

Mihai Tibar  On the topology of singular hypersurfaces

We explain some new results based on the use of non-generic Lefschetz pencils or on the study of nonisolated singularities, where the central role is played by the monodromy and its variation.

Vincent Florens: On the topology of line arrangements

The boundary of a tubular neighborhood of a line arrangement is a graph 3-manifold, whose structure is determined by the combinatorics of the arrangement. We construct a new topological invariant of arrangements, derived from the inclusion map of this boundary manifold into the exterior. This invariant can be easily computed from a wiring diagram that encodes the braid monodromy of the arrangement. As an application, we present some new examples of Zariski pairs. Joint work with E.Artal, B.Guerville and a part with M.Marco.

Ursula Ludwig:  Witten deformation using stratified Morse functions and Gromov’s trick.

The Witten deformation, proposed by Witten in the 80’s, is an analytic proof of the Morse inequalities for compact manifolds and smooth Morse functions. The proof uses the deformation of the de Rham complex via the given Morse function f : M → R.

The main idea in generalising the so-called Witten deformation to singular spaces is to deform the complex of L2-forms instead of the de Rham complex. In this talk I will focus on deformations using so-called admissible Morse functions. Those comprise in particular the example of stratified Morse functions in the sense of the theory developed by Goresky and MacPherson on a singular complex curve.

In the beginning of the talk, I will recall the Witten deformation for smooth compact manifolds and smooth Morse functions.

Alejandro Melle: The generalized higher order Euler characteristics


For a complex quasi-projective manifold with a finite group action, we define its generalized higher order Euler characteristics with values in the Grothendieck ring of complex quasi-projective varieties extended by the rational powers of the class of the affine line. By using the geometric description of a power structure defined over such a ring the generating series of generalized Euler characteristics of a fixed order of the Cartesian products of the manifold with the wreath product actions on them is computed (joint work with S.M. Gusein-Zade, I. Luengo).

Duco van Straten: A resurgent Weyl-algebra

Abstract: The theory of plane curve singularities in the symplectic (p,q)-plane has a natural extension to the study of a version of "non-commutative singularity theory" where one uses a formal Heisenberg-algebra with relation pq-qp=h. In the talk I will describe aspects of an attempt to understand the divergences that appear naturally.
As usual, it turns out that the A_1-singularity is the modest hero. (joint work with M. Garay and A. de Goursac)

Remke Kloosterman Using Alexander polynomials in algebraic geometry

We revisit Dimca's method to compute the Alexander polynomial of a hypersurface in $\Ps^n$ with only isolated quasihomogeneous singularities. We present a variant of Dimca's method, which we use to prove several other  results. First, we use this method to relate the Alexander polynomial of a plane curve with ADE singularities with the Mordell-Weil rank of an isotrivial fibration of hyperelliptic Jacobians. Then we show that a hypersurface with nonconstant Alexander polynomial and only isolated quasihomogeneous singularities has a non $T$-smooth equianalytic deformation space. Moreover, for a finite set of points $\Sigma$ in $\Ps^n$ and an integer $m\geq n$, we give a lower bound for the degree of a form $f$ such that $f=0$ has isolated singularities and the multiplicity of $f$ at each point of $\Sigma$ is at least $m$

Xia Liao: normal crossing property for quasihomogeneous free divisors

The notion of normal crossing divisor is fundamental in algebraic geometry. However, its definition involves a choice of local analytic coordinate system. It is a natural question to ask whether normal crossing divisors can be characterized by a purely algebraic condition. Eleonore
Faber conjectured that a divisor is normal crossing if and only if it is a free divisor with radical Jacobian ideal. In this talk, I will briefly describe a joint work with Mathias Schulze, which gives a proof of the conjecture for quasihomogeneous divisors. Our approach is based on a study of the representation of a certain finite dimensional Lie algebra, which is constructed from a basis of the logarithmic vector fields along the divisor.

Christian Gorzel: The maximizing simple sextics with an elliptic component

Persson proved that the irreducible maximizing sextics are rational curves.  By Yang's work, we know that there are 519 combinations of simple singularities for maximizing sextics.
 We show that there are exactly twelve combinations for which it  exists a sextic with an elliptic component. In each of these cases there exists at least one other sextic with only rational components. We provide explicit equations for all these curves, complying with the predicted degree of the defining number field as found in Shimada's list.  We obtain characterizations of the sextics with an elliptic component in terms of the discriminant and j-invariant.

Jesse Kass: Coxeter--Dynkin diagrams for simple curve singularities

The simple curve singularities were classified by Giusti  and Frühbis-Krüger, and in important cases, Coxeter--Dynkin diagrams were described by Alpert, Ebeling, Ebeling--Guseĭn-Zade, and Mond--van Straten.  I will extend this work by describing Coxeter--Dynkin diagrams for all simple curve singularities.  The technique demonstrates an unexpected connection between c.i. singularities in 4-space and space curve singularities in 3-space.

Alexey Basaelev:  Mirror symmetry for the singularities with the group action
From the point of view of singularity theory mirror symmetry is the correspondence between Kyoji Saito's flat structure of a singularity and Gromov-Witten theory of the certain variety. This setting was generalized  by physicists who have proposed an generalized mirror conjecture that involves not only a singularity but also a symmetry group of it.
In the talk we present particular example of the mirror symmetry of this kind.
Mathias Zach: The topology of Cohen-Macaulay codimension 2 singularities

We present a way to compute the vanishing cycles of isolated Cohen-Macaulay codimension 2 singularities. This is done by exploiting its matrix structure to construct a second family from every deformation in which we reduce to complete intersections.

 

Schedule:


Monday :

Registration from 14
Faber   15 - 16
van Straten 16:15 - 17:15
Basalaev  17:45 - 18:15

Tuesday:

Sevenheck   9 - 10
Libgober 10:30 - 11:30
Kloostermann   11:45 - 12:15

Veys   15 - 16
Ludwig  16:30 - 17:30

Conference dinner 19:00 or later

Wednesday:

Florens   9 - 10
Tibar     10:30 - 11:30
Gorzel  11:45 - 12:15

Artal    15-16
Zach   16:30 - 17
Kass   17 - 17:30

Thursday:

Schulze  9 - 10
Liao  10:15 - 10:45
Melle    11:15 - 12:15

 

 

 

For registration or any other related question, contact Miguel Marco miguelmath.uni-hannover.de

Organizers:

Participants:

  • Enrique Artal
  • Christian Barz
  • Alexey Basalaev
  • Wolfgang Ebeling
  • Eleonore Faber
  • Vincent Florens
  • Anne Fruhbis- Krueger
  • Patrick Graf
  • Christian Gorzel
  • Sabir Gusein-Zade
  • Helmut Hamm
  • Jesse Kass
  • Remke Kloosterman
  • Xia Liao
  • Anatoly Libgober
  • Wenfei Liu
  • Michael Lönne
  • Ignacio Luengo
  • Ursula Ludwig
  • Miguel Marco
  • Alejandro Melle
  • Julio Moyano
  • Matteo Penegini
  • Mathias Schulze
  • Christian Sevenheck
  • Duco van Straten
  • Mihai Tibar
  • Orsola Tommasi
  • Davide Veniani
  • Wim Veys
  • Matthias Zach

Lodging:

This is a list of recommended hotels.

www.hotel-savoy.de
Hotel Savoy Hannover
Schlosswenderstr. 10
D-30159 Hannover
Manager: Marcel Lagershausen
Phone:0049 (0) 511-167487-0
Fax: 0049 (0) 511-167487-10
e-mail: info@hotel-savoy.de

www.hotel-schlafgut.de
Hotel im Werkhof
Kniestraße 33, 30167 Hannover
Phone: 0049 511 353560
email: booking@hotel-schlafgut.de

www.mit-h.de
Hotel Gästehaus am Herrenhäuser Garten
Herrenhäuser Kirchweg 17
30167 Hannover
url www.mit-h.de
email hotel@mit-h.de
Tel +49 (0)511 70072 0

www.loccumerhof.de
Hotel LOCCUMER HOF
GmbH & Co KG*
Kurt Schumacher Straße 14/16
30159 Hannover
Tel: +49 . 511 . 12 64 - 0
Fax: +49 . 511 . 13 11 92
E-Mail: info@loccumerhof.de