Workshop on Singularities

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Workshop on Singularities

 25 - 29 Aug 2008
 ICMS, 14 India Street, Edinburgh

Organiser

Name
Institution
Goryunov, VictorUniversity of Liverpool
Siersma, DirkUniversiteit Utrecht
Vassiliev, VictorSteklov Institute, Moscow


The purpose of the meeting is to bring together experts (both international leaders and promising young researchers) in geometrical and topological branches of Singularity Theory and in some adjacent areas where these branches find applications. The major aims of the workshop are to disseminate new results and ideas, to foster new developments within Singularity Theory and to look for new areas of application and interaction.

The following are likely to be major topics of the meeting:

  • Global singularity theory: topology and geometry
  • Connection between analytic and topological invariants of singularities
  • Non-isolated singularities
  • Singularities in differential geometry
  • Computer packages on singularities

Numbers will be limited to a maximum of 40, there will be a contribution to local costs (including accommodation and subsistence) from awards to ICMS. A Registration Fee of £30 will be payable by all delegates.

Participation is by invitation only: those interested in attending should contact Victor Goryunov (goryunov@liv.ac.uk).

 

Arrangements

Participation
Participation is by invitation only. The workshop will begin on Monday 25 August and finish on Friday 29 May 2008.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one. If you do require a visa, ICMS can provide a signed invitation letter.

Venue
The workshop will take place at the head-quarters of ICMS, 14 India Street, Edinburgh. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, near the city centre.

The ICMS travel pages contain advice on how to travel to Edinburgh. For local information the finding ICMS page shows the location of ICMS and contains useful maps of the city centre.

The seminar room at ICMS has whiteboards, 2 overhead projectors, a data projector and laptop. We ask that you use our laptop for your presentation which you may bring on a memory stick or CD.

Wireless access is available throughout the ICMS building. There are also 7 public PCs which may be used at any time for internet access and to check email.

Accommodation
ICMS will arrange single en-suite rooms in local guest houses for those who require it. Accommodation is typically about 15 to 30 minutes walk from ICMS. Participants are also free to make their own arrangements and may claim back the cost, with receipts, up to a maximum of £60.00 per night bed and breakfast. A list of Edinburgh accommodation of various sorts and prices is available here . Sections 1-3 are particularly relevant.

Meals and Refreshments
A sandwich lunch will be provided on the first day of the workshop, Monday 25 August. For the remainder of the days, participants are free to go out for lunch and explore the many cafes, restaurants, sandwich shops and bars in the surrounding area. On arrival we will provide you with a ‘welcome’ pack which will contain information about eating places nearby.

Morning and afternoon refreshments will be provided throughout the workshop.

There will be an informal wine reception after the close of lectures on Monday 25 August.

The workshop dinner will take place on the evening of Thursday 28 August.

Registration
Registration will take place from 09.00 to 09.45 on Monday 25 August.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee throughout the workshop, lunch on the first day, the wine reception and the Workshop Dinner on Thursday evening.

If we have agreed to pay some of your travel costs, you will be informed in the 'final information' email. Reimbursement will take place after the workshop. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with receipts. Please note that we cannot reimburse any item without a receipt. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world should supply their IBAN and SWIFT/BIC code.

Under the terms of our EPSRC funding we are required to charge a 30.00 GBP registration fee to cover costs not admissible under the grant. The fee will be payable on arrival at the workshop payment may be by cash, sterling cheque or credit/debit card. If you anticipate any difficulty covering the fee, please let me know.

 

Programme

Monday 25 August

09.00 - 09.45

Registration and coffee

09.45 - 10.00

Introduction and welcome

10.00 - 11.00

Duco van Straten (Johannes Gutenberg-Universität, Mainz)
Periods and Frobenius polynomials

11.00 - 11.30

Coffee/tea break

11.30 - 12.30

Mihai Tibar (Université de Lille 1)
On the simplicity of surfaces

12.30 - 14.30

Lunch provided in the Exhibition Room

14.30 - 15.30

Sabir Gusein-Zade (Moscow State University)
Filtrations corresponding to ideals and their Poincaré series

15.30 - 16.00

Coffee/tea break

16.00 - 17.00

Wolfgang Ebeling (Leibniz Universität Hannover)
McKay correspondence for the Poincaré series of Kleinian and Fuchsian singularities
17.00 - 19.00 Wine Reception in the Exhibition Room at ICMS

 

Tuesday 26 August

09.00 - 10.00

Maxim Kazarian (Steklov Institute of Mathematics)
Stabilization of Thom polynomials

10.10 - 11.10

Andras Szucs (Eötvös Loránd University)
Multiplicative structure on the cobordism groups of Morin maps

11.10 - 11.40

Coffee/tea break

11.40 - 12.40

David Mond (University of Warwick)
Functions on the Milnor fibrations of linear free divisors

12.40 - 14.30

Lunch

14.30 - 15.30

Bradford Hovinen(University of Toronto)
Matrix factorizations of the classical discriminant

15.30 - 16.00

Coffee/tea break

16.00 - 17.00

Anne Fruehbis-Krueger (Leibniz Universität Hannover)
Computational desingularization and the strict transform

19.00

Informal evening meal at Nargile Turkish Restaurant, 73 Hanover Street

 

Wednesday 27 August

09.00 - 10.00

Grigory Mikhalkin (University of Toronto)
Singularities of effective tropical cycles

10.10 - 11.10

Ilia Itenberg (Université Louis Pasteur, Strasbourg)
Recursive formulas for Welschinger invariants

11.10 - 11.40

Coffee/Tea

11.40 - 12.40

Ragnar-Olaf Buchweitz (University of Toronto Scarborough)
Classifying determinantal presentations of hypersurface singularities

12.40

Lunch break and free afternoon

 

Thursday 28 August

09.00 - 10.00

Terry Wall (University of Liverpool)
Geometry of space curves

10.10 - 11.10

Gert-Martin Greuel (University of Kaisersalautern)
Invariants and classification of singularities in positive characteristic

11.10 - 11.40

Coffee/Tea

11.40 - 12.40

Alexey Gorinov (University of Liverpool)
Mixed Hodge structures on configuration spaces of algebraic varieties and applications to moduli spaces of pointed genus 1 curves

12.40 - 14.30

Lunch break

14.30 - 15.30

Anna Pratoussevitch (University of Liverpool)
Higher spin structures on Riemann surfaces

15.30 - 16.00

Coffee/Tea

16.00 - 17.00

Alejandro Melle Hernández (Universidad Complutense de Madrid)
On a conjecture by A Durfee

19.00

Workshop dinner at First Coast Restaurant, 99-101 Dalry Road, Edinburgh

 

Friday 29 August

09.30 - 10.30

Ilya Tyomkin (Massachusetts Institute of Technology)
Geometry of Severi varieties on Hirzebruch surfaces

10.30 - 11.00

Coffee/Tea

11.00 - 12.00

Elizabeth Gasparim (University of Edinburgh)
Characteristic classes of sheaves on singular varieties

12.00 - 14.00

Lunch break

14.00 - 15.00

Eugenii Shustin (Tel Aviv University)
On deformations of real plane curve singularities

15.00 - 15.30

Coffee/Tea

15.30 - 16.30

Vladimir Zakalyukin (University of Liverpool)
Non-standard equivalence relations for singularities


Presentations

Buchweitz, Ragnar-Olaf
Classifying determinantal presentations of hypersurface singularities
View Abstract
We discuss how to effect the classification in the title by means of liftings of maximal Cohen-Macaulay modules of rank one from low-dimensional singularities. We present a simple algorithm and discuss in detail the case of the 3--discriminant, treated in Hovinen's thesis. (Joint work with Bradford Hovinen, University of Toronto)
Ebeling, Wolfgang
McKay correspondence for the Poincaré series of Kleinian and Fuchsian singularities
View Abstract
A Kleinian singularity is the quotient of the complex plane by a binary polyhedral group. A Fuchsian singularity is obtained by contracting the image of the zero section in the quotient of the cotangent bundle of the upper half plane by a Fuchsian group of the first kind. We consider the Poincaré series of the graded coordinate algebras of these singularities. For a Kleinian singularity not of type $A_{2n}$ we have derived from the McKay correspondence that this series is the quotient of the characteristic polynomials of two Coxeter elements. A similar resultat was shown for the Fuchsian singularities of genus 0. This result was obtained independently by H. Lenzing and J. de la P~e{n}a in the realm of representation theory of algebras. Here we give a uniform proof of both these results following G. Gonzalez-Sprinberg's and J.-L. Verdier's geometric construction of the McKay correspondence and using results of Lenzing. This is joint work with David Ploog.
Fruehbis-Krueger, Anne
Computational desingularization and the strict transform
View Abstract
Implementations of desingularization mainly focus on Villamayor's approach (consequently using the weak transform), because the Bierstone-Milman approach seems impractical due to the use of the Hilbert-Samuel function. In this talk, I shall introduce a hybrid approach which allows the use of the strict transform and the Hilbert-Samuel function, but avoids the excessive cost of naively computing the Hilbert-Samuel stratification at each step.
Gasparim, Elizabeth
Characteristic classes of sheaves on singular varieties
View Abstract
I will describe applications of characteristic classes of sheaves on singular varieties to: 1. the topology of moduli spaces, 2. mathematical physics.
Gorinov, Alexey
Mixed Hodge structures on configuration spaces of algebraic varieties and applications to moduli spaces of pointed genus 1 curves
View Abstract
We'll describe some results and conjectures about the rational mixed Hodge structures on the cohomology of the configuration spaces of smooth complex algebraic varieties. As an application, we'll give an explicit (but complicated) description of the rational cohomology of the moduli spaces of smooth genus 1 curves with any given number of marked points and level N structures.
Greuel, Gert-Martin
Invariants and classification of singularities in positive characteristic
View Abstract
This is a report about new results on isolated hypersurface singularities in positive characteristic. Most of the well known invariants of such singularities which are classical over the complex numbers, have to be reconsidered and modified over algebraically closed fields in positive charactereistic. This is in particular the case for the Milnor and Tjurina number, the determinacy, the Newton number and various notions of nondegeneracy for semi piecewise-homogeneous singularities. These invaraints are used to classify the K-unimodal singularities in positive characteristic. The results are joint work with Yousra Boubakri and Thomas Markwig.
Gusein-Zade, Sabir
Filtrations corresponding to ideals and their Poincare series
View Abstract
Earlier there were considered and, in some cases, computed Poincare series of two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular on the complex plane). A filtration from the first class was defined by a curve (with several branches) on the surface singularity. The other one (so called divisorial filtration) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. For an ideal or for a set of ideals in the ring of germs of functions on a surface singularity one defines a corresponding filtration. Its Poincare series can be computed in some cases. For the complex plane this notion unites the two classes of filtrations described above. The talk describes a joint work with A.Campillo and F.Delgado.
Hovinen, Bradford
Matrix factorizations of the classical discriminant
View Abstract
The classical discriminant detects whether a univariate polynomial over a field has repeated roots. Classical results of Cayley, Legendre, Sylvester, and Bézout show that there exist nontrivial determinantal formulae for the classical discriminant, allowing for more efficient evaluation. However, the classical formulae are all equivalent in the sense that the cokernels of the matrices are all isomorphic. This begs the question of whether there are any nontrivial inequivalent determinantal formulae.

This talk will describe a new determinantal formula for the classical discriminant which is inequivalent to the classical formulae: the presentation matrix of the "open swallowtail" studied by Arnol'd and Givental. The associated matrix is smaller than that of any of the classical formulae and carries information on the root structure of the polynomial which the classical formulae cannot detect.

Some work on using deformation theory to classify determinantal formulae for the classical discriminant will also be described. In particular, by constructing versal deformations of maximal Cohen-Macaulay modules on certain curve singularities, it is possible to construct a moduli space of graded rank-1 MCM modules on the discriminant. A moduli space of such modules on the discriminant of degree 4 polynomials will be shown.
Itenberg, Ilia
Recursive formulas for Welschinger invariants
View Abstract
Welschinger invariants are designed to bound from below the number of real rational curves which pass through a given real generic collection of points on a real rational surface. In some cases these invariants can be calculated using Mikhalkin's approach which deals with a corresponding count of tropical curves. Using this approach and tropical Caporaso-Harris type formulas, we establish a logarithmic equivalence of Welschinger and Gromov-Witten invariants in several situations. The proofs are based on new versions of correspondence theorems. These versions deal in particular with tropicalizations of curves having ordinary multiple points.
Kazarian, Maxim
Stabilization of Thom polynomials
View Abstract
Thom polynomials describe cohomology classes dual to singularity loci of a generic map between two smooth manifolds. Thom polynomial is a polynomial in the Chern classes of manifolds determined uniquely by the singularity type. It independent of the particular manifolds participating in the map but it does depend on the dimensions of these manifolds. Rimanyi and Feher observed that the coefficients of Thom polynomials stabilize with the growth of the dimensions. Peculiar properties of this stabilization will be discussed in the talk. Some properties of this stabilization are proved and some of them are still conjectural.
Melle Hernández, Alejandro
On a conjecture by A Durfee
View Abstract
We show how superisolated surface singularities can be used to find a counterexample to the following conjecture by A Durfee [1]: for a complex polynomial F(x,y,z) in three variables vanishing at 0 with an isolated singularity there, "the local complex algebraic monodromy is of finite order if and only if a resolution of the germ (\{F=0\},0) has no cycles".

[1] A.Durfee, The monodromy of a degenerating family of curves, Inv. Math. (1975)
Mikhalkin, Grigory
Singularities of effective tropical cycles
View Abstract
We consider effective cycles in tropical manifolds. Like in the classical case each point of such a cycle can be prescribed a positive multiplicity. Unlike the classical case the points of multiplicity 1 do not always locally correspond to smooth tropical submanifolds. We pay a special attention to such multiplicity 1 points.
Mond, David
Functions on the Milnor fibrations of linear free divisors
View Abstract
We study the singularity theory of functions on the Milnor fibres of linear free divisors. Reductivity of the Lie group of automorphisms of the divisor turns out to play a key role in the existence of non-degenerate functions. We explain some of the representation theory involved, and go on to describe the construction of a Frobenius manifold structure on the base space of the versal deformation when the group is reductive. (This is joint work with Ignacio de Gregorio and Christian Sevenheck).
Pratoussevitch, Anna
Higher spin structures on Riemann surfaces
View Abstract
A higher spin structure on a Riemann surface is a line bundle with m-th tensor power isomorphic to the cotangent bundle. In the talk I will show how the space of m-spin structures on a Riemann surface can be interpreted as a finite affine space of Z/mZ-valued functions on the fundamental group of the surface.

Using this results one can describe the topology of the moduli space: Any connected component of the space of m-spin structures on compact Riemann surfaces with finite number of punctures and holes is homeomorphic to a quotient of R^d by a discrete group action.

The more general classification of singular higher spin structures on Riemann surfaceas leads to results on quasi-homogeneous Gorenstein surface singularities.
Shustin, Eugenii
On deformations of real plane curve singularities
View Abstract
We discuss two examples denostrating the difference between real algebraic and real symplectic curves and between algebraic and symplectic deformations of real plane curve singularities. The first example, obtained jointly with S. Orevkov, is a real plane non-singular symplectic curve of degree 9 which is not isotopic to any real algebraic curve of the same degree. This symplectic curve appears in an algebraically forbidden deformation of an existing singular curve. Another example exhibits a symplectic deformation of a real multiple point of order 8, modeled by a real singular symplectic curve of degree 8, found by Vic. S. Kulikov. We prove that this collection of singularities cannot appear in a versal deformation of a multiple point of order 8. This seems to be the first example of a symplectic non-algebraic deformation of a plane curve singular point.
Szucs, Andras
Multiplicative structure on the cobordism groups of Morin maps
View Abstract
We define a multiplication on the direct sum of all cobordism groups of Morin maps from n-manifolds to the n+k dimensional Euclidean space, where the summation goes for all n and k. This multiplication behaves very nice with respect to the singularity strata. Namely the product of the A_r strata give that of the product map (up to cobordism). This allows us to compute this ring completely. (Joint work with Gabor Lippner).
Tibar, Mihai
On the simplicity of surfaces
View Abstract
One of the Mumford's well known results tells that the link of a normal surface is simply connected if and only if this surface is non-singular. The "simplicity'' of the surface is understood here as the simplicity of its link. We expand a little bit this idea and try to point out a few old and new open questions.
Tyomkin, Ilya
Geometry of Severi varieties on Hirzebruch surfaces
View Abstract
Severi varieties are the varieties parameterizing irreducible curves of a given geometric genus in a given linear system on a projective surface. They were introduced by Severi in 20s in order to prove irreducibility of moduli spaces of curves. He proved that if the surface is the projective plane then the Severi varieties are irreducible, and deduced that moduli spaces are also irreducible. However Severi's proof contained a gap. The irreducibility of moduli spaces was obtained using other tools, but the irreducibility problem of Severi varieties remained open till 80s when Harris gave a correct proof in the plane case. His proof is based on a result of Zariski characterizing Severi varieties in terms of their dimensions. In this talk I will discuss Severi varieties on Hirzebruch surfaces. I will prove that they are irreducible. If time permits, I’ll say few words about the case of positive characteristic. In this case the irreducibility is not known for any genus other than zero. The main missing ingredient is the analog of Zariski’s result. I will discuss an example of a rational (toric) surface for which the analog of Zariski’s result is no longer true.
van Straten, Duco
Periods and Frobenius polynomials
View Abstract
Fifty years ago Dwork discovered the beautiful relation between the periods and the zeta-function of a family of elliptic curves. In the talk we extend Dworks method to describe a similar relation for the case of a family of Calabi-Yau threefolds. (Joint work with K Samol).
Wall, Terry
Geometry of space curves
View Abstract
I will describe the local geometry of the stratification associated to a space curve, and methods to calculate the degrees of the strata in the algebraic case.
Zakalyukin, Vladimir
Non-standard equivalence relations for singularities
View Abstract
We describe several classifications of functions and mapping which play an
intermediate role between standard right-left action and boundary preserving action. Some new interconnections of well known classifications be described.

Participants

Name
Institution
Fawaz, AlharbiUniversity of Liverpool
Ayse, AltintasUniversity of Warwick
Jean-Paul, BrasseletCentre National de la Recherche Scientifique (CNRS)
Ragnar-Olaf, BuchweitzUniversity of Toronto Scarborough
Paul, CadmanUniversity of Warwick
Andrew, du PlessisAarhus Universitet
Wolfgang, EbelingLeibniz Universität Hannover
Anne, Fruehbis-KruegerLeibniz Universität Hannover
Elizabeth, GasparimUniversity of Edinburgh
Alexey, GorinovUniversity of Liverpool
Victor, GoryunovUniversity of Liverpool
Gert-Martin, GreuelMathematisches Forschungsinstitut Oberwolfach
Sabir, Gusein-ZadeMoscow State University
Joel, HaddleyUniversity of Liverpool
Helmut, HammWestfälische Wilhelms-Universität Münster
Bradford, HovinenUniversity of Toronto
Ilia, ItenbergUniversité Pierre et Marie Curie (Paris 6)
Maxim, KazarianSteklov Institute of Mathematics
Remke, KloostermanLeibniz Universität Hannover
Dũng Tráng, LêInternational Centre for Theoretical Physics (ICTP)
Ignacio, LuengoUniversidad Complutense de Madrid
Alejandro, Melle HernándezUniversidad Complutense de Madrid
Grigory, MikhalkinUniversité de Genève
David, MondUniversity of Warwick
Anna, PratoussevitchUniversity of Liverpool
Eugenii, ShustinTel Aviv University
Dirk, SiersmaUniversiteit Utrecht
Jan, StevensGöteborgs Universitet
Andras, SzucsEötvös Loránd University (ELTE)
Mihai, TibarUniversité de Lille 1
Orsola, TommasiLeibniz Universität Hannover
Ilya, TyomkinMassachusetts Institute of Technology
Duco, van StratenJohannes Gutenberg-Universität, Mainz
Terry, WallUniversity of Liverpool
Vladimir, ZakalyukinUniversity of Liverpool