ICMS

May 5, 2008 - May 9, 2008

ICMS, 14 India Street, Edinburgh

Scientific Organisers

Professor Nicolai Vorojov, University of Bath
Professor Alex Wilkie, University of Manchester
Professor Alexandre Eremenko, Purdue University

Short Report

Effective real analytic geometry has its origins in several sources. First, in the theory of semi- and subanalytic sets and their complements founded in the late 60s, and since developed further by many authors. It was suggested that the o-minimal structure of the classes of semialgebraic or subanalytic sets makes precise the Grothendieck's vision of a "tame topology". O-minimality links subanalytic geometry to modern applied model theory. Model-theoretic methods allowed us to obtain the first proofs of complement theorems (model completeness, in the language of logic) for some classes of real analytic functions, which were unassailable by geometric techniques of the previous stage. Another origin of effective analytic geometry lies in the theory of fewnomials, with its underlying ideology that the geometric objects described by "short" formulae should have a "simple" topology. Finally, very significant progress was achieved in the semialgebraic case, where the defining functions are polynomials, in particular in Thom-Milnor style upper bounds on the homological complexity of semialgebraic sets, and in computational real algebraic geometry. All these strands are now becoming very close to one another, which helps to produce significant new results. The workshop brought together leading international experts, as well as younger researchers, in real analytic geometry, o-minimality, algebraic computational complexity, and related areas of topology and model theory. Many excellent talks reported on the most recent developments. We expect that the lively discussions that followed will result in a noticeable progress in the current outstanding problems, and will stimulate the younger researches to study and apply the variety of the available mathematical techniques.

Participants list and links to available presentations are further down this page.

Download the pdf file of the full report

Original details

The workshop is dedicated to Professor Andrei Gabrielov on the occasion of his 60th birthday. Its aim is to bring together leading international experts in real analytic geometry, o-minimality, algebraic computational complexity and related areas of topology, model theory and theoretical computer science. Professor Gabrielov made fundamental contributions to all these areas. With the growing tendency of these strands to share concepts and techniques, there is a clear need for specialists to meet and discuss the state of affairs. The meeting will provide an overview of the current state of research in real analytic geometry with the emphasis on “tameness” and effectivity.

The main objective will be to discuss and advance the research in this modern and rapidly developing interdisciplinary field. Special attention will be paid to the study of new explicit quantitative bounds on the geometric characteristics of definable sets, and their applications.

Numbers will be limited to a maximum of 35, there will be a contribution to local costs (including accommodation and subsistence) from awards to ICMS and a Registration Fee of £30 will be payable by all delegates.

Participation is by invitation only: those interested in attending should contact Nicolai Vorobjov (nnv@cs.bath.ac.uk).

 

Arrangements

Participation
Participation is by invitation only. The workshop will begin on the morning of Monday 5 May and finish on Friday 9 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 4 whiteboards, 2 overhead projectors, a data projector and laptop.

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. 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 5 May. 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 5 May.

The workshop dinner will take place on the evening of Thursday 8 May. The workshop grant will cover the cost of this meal.

Registration
Registration will take place on Monday 5 May.

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 by 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. Your expenses will be paid directly into your bank account so you will need to provide your account details plus IBAN, SWIFT or Routing numbers as applicable. Please note that we cannot reimburse any item without a receipt.

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 wish to pay this fee in advance please e-mail and request a credit/debit card payment form. If you anticipate any difficulty covering the fee, please let me know.

 

Programme

Monday 5 May

09.00 – 10.00	Registration
10.00 – 11.00	Leonard Lipshitz ( Purdue University) Real closed fields with analytic structure
11.00 – 11.30	Coffee/Tea
11.30 - 12.30	Andre Gabrielov ( Purdue University) Dessins d'Enfants for the Eigenfunctions of the Quartic Oscillator
13.00 – 14.30	Lunch - buffet provided at 14 India Street
14.30 - 15.30	Thierry Zell ( Vassar College) Around the descent spectral sequence
15.30 – 16.00	Coffee/Tea
16.00 - 17.00	Jean-Philippe Rolin (Université de Bourgogne) An o-minimal structure which does not admit C-infinite cellular decomposition
17.30 – 19.00	Wine Reception at 14 India Street

 

Tuesday 6 May

09.00 – 10.00	Dmitry Novikov (Weizmann Institute of Science) Pseudoabelian integrals: toward an analogue of Varchenko-Khovanskii theorem
10.00 – 11.00	Marie-Francoise Roy (Université de Rennes 1) Certificates of positivity in the Multivariate Bernstein basis
11.00 – 11.30	Coffee/Tea
11.30 – 12.30	Vincent Grandjean (University of Bath & Universität Oldenburg) Geodesics at singular points of non-conical isolated surface singularities
12.30 – 14.30	Lunch
14.30 – 15.30	Frank Sottile (Texas A&M University) Fewnomial bounds from Gale dual systems
15.30 – 16.00	Coffee/Tea
16.00 – 17.00	Frederic Bihan (Université de Savoie) Bounds on the topology of fewnomial hypersurfaces

 

Wednesday 7 May

09.00 – 10.00	Krysztof Kurdyka (Université Savoie Trajectories of horizontal gradients of polynomials
10.00 – 11.00	Askold Khovanskii ( University of Toronto) Degree or rational mappings and simple proofs of the Sturm and Tarski theorems
11.00 – 11.30	Coffee/Tea
11.30 – 12.30	Victor Palamodov ( Tel Aviv University & Potsdam University) Moduli of hypersurface germs
12.30 – 13.00	Bernd Martin (Brandenburgische Technische Universität) Modular strata of unimodal singularities are corresponding to Milnor algebras
13.00 -	Free afternoon

 

Thursday 8 May

09.00 – 10.00	Angus Macintyre (University of London) Elementary theory of elliptic functions
10.00 – 11.00	David Trotman ( University of Provence) Valette's generic definable bilipschitz equisingularity
11.00 – 11.30	Coffee/Tea
11.30 – 12.30	Viacheslav Kharlamov (Université Louis Pasteur) Real cubic four-folds; deformations and chirality
12.30 – 14.30	Lunch
14.30 – 15.30	Ilia Itenberg (Université Louis Pasteur) Recursive formulas for Welschinger invariants
15.30 – 16.00	Coffee/Tea
16.00 – 17.00	Saugata Basu (Georgia Institute of Technology) Combinatorial complexity in o-minimal geometry
19.00	Workshop dinner at First Coast, 99 – 101 Dalry Road, Edinburgh

 

Friday 9 May

09.00 – 10.00	Edward Bierstone ( University of Toronto) Problems on resolution of singularities
10.00 – 11.00	Patrick Speissegger ( McMaster University) Transition maps of non-resonant hyperbolic singularities are o-minimal
11.00 - 11.30	Coffee/Tea
11.30 – 12.30	Gareth Jones ( McMaster University) Model completeness results for polynomially bounded o-minimal structures
12.30 – 14.30	Lunch
14.30 – 15.30	Adam Parusinski ( University of Angers) Equivalence relations of two variable real analytic function germs
15.30 – 16.00	Coffee/Tea

Presentations:

Presentation Details
Basu, Saugata
Combinatorial complexity in o-minimal geometry.
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In this talk I will state some tight bounds on the combinatorial and topological complexity, as well as on the number of homotopy types, of sets defined in terms of $n$ definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semi-algebraic and semi-Pfaffian sets, and as a result increases the applicability of results on combinatorial and topological complexity of arrangements studied in discrete and computational geometry. As a sample application, we extend a Ramsey-type theorem due to Alon et al., originally proved for semi-algebraic sets of fixed description complexity to this more general setting.
Bierstone, Edward
Problems on resolution of singularities
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I will discuss several problems on resolution of singularities in characteristic zero that seem to be unsolved, though some are stated as theorems in the classical literature. The problems include questions concerning algorithms, desingularization of a metric, and equisingularity.
Bihan, Frederic
Bounds on the topology of fewnomial hypersurfaces
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This talk will follow the talk by F. Sottile in which new fewnomial bounds for the number of real solutions to polynomial systems are presented. Using these bounds and stratified Morse theory, we obtain new bounds for the total Betti numbers of fewnomial hypersurfaces. This is a joint work with F. Sottile
Gabrielov, Andrei
Dessins d'Enfants for the eigenfunctions of the quartic oscillator
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Eigenfunctions of the Schrodinger operator on the real line with a polynomial potential are associated with properly embedded infinite planar trees. The braid group action on the trees helps to understand the dependence of the eigenfunctions and the corresponding eigenvalues on the coefficients of the potential.
Grandjean, Vincent
Geodesics at singular points of non-conical isolated surface singularities
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The local Riemannian geometry of embedded singular spaces in an Euclidean space, when equipped with the induced metric, in the neighbourhood of singularities, is hardly known. But the asymptotically conical case, understanding how the geodesics fall on a singular point is still to be explored, even for real algebraic isolated surface singularities of the Eucilidean 3-space. In this work we study a class of non (asymptotically) conic real algebraic isolated surface singularities, for which we build an exponential map based at the singular point O and are able to give a precise asymptotics, in a suitable sense, of the length of a geodesic falling on O. This is a joint work with D. Grieser (Universität Oldenburg).
Itenberg, Ilia
Recursive formulas for Welschinger invariants
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(joint work with V. Kharlamov and E. Shustin) 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.
Jones, Gareth
Model completeness results for polynomially bounded o-minimal structures.
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I shall give a model completeness result for polynomially bounded o-minimal expansions of the real field, with some consequences for certain exponential structures. The proof combines Gabrielov's methods with ideas from Wilkie's exponentiation paper.
Kharlamov, Viacheslav
Real cubic four-folds; deformations and chirality
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This is a report on a joint work with Sergey Finashin. According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Now we can show how to eliminate the projective equivalence and to obtain a pure deformation classification, that is how to respond to the chirality question: which cubics are not deformation equivalent to their image under a mirror reflection. We provide an arithmetical criterion of chirality, in terms of the eigen-sublattices of the complex conjugation involution in homology, and show how this criterion can be effectively applied taking as examples M-cubics (that is those for which the real locus has the richest topology) and (M-1)-cubics (the next case with respect to complexity of the real locus). It happens that there is one chiral class of M-cubics and three chiral classes of (M-1)-cubics, contrary to two achiral classes of M-cubics and three achiral classes of (M-1)-cubics.
Khovanskii, Askold
Degree or rational mappings and simple proofs of the Sturm and Tarski theorems
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I will present a simple explicit formula which computes the mapping degree of a rational mapping from the real projective space to itself. As a corollary one can obtain simple proofs of Sturm and Tarski theorems. Similarly a version of Tarski theorem can be proved over any algebraically closed field.
Kurdyka, Krzysztof
Trajectories of horizontal gradients of polynomials
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(joint work with S-T. Dinh and P. Orro) We consider a class of non-holonomic codimension 1 distributions on affine space. This class contains contact structure, Heisenberg and Martinet distributions. We equip these distributions with a sub Riemannian metric by fixing polynomial vector fields which give orthonormal basis for the distribution. Thus for a given polynomial on the affine space we obtain its horizontal gradient with respect to the sub Riemannian metric. We show that the behaviour of trajectories of horizontal gradient differs from the known results of Lojasiewicz on the Riemannian gradient of polynomials, in particular Lojasiewicz's gradient inequality does not hold. But we show that for generic polynomials the trajectories of the horizontal gradient approaching critical set have limits.
Lipshitz, Leonard
Real Closed Fields with Analytic Structure
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The talk will present a general framework for studying real closed fields which in addition to algebraic functions also carry a class of analytic functions.
Macintyre, Angus
Elementary Theory of Elliptic Functions
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I analyze the fine detail of definitions and decidability in the theory of Weierstrass elliptic functions, using methods of Gabrielov-Vorobjov and Macintyre-Wilkie, and assuming Andre's Conjecture.
Martin, Bernd
Modular strata of unimodal singularities are corresponding to Milnor algebras
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We find and describe unexpected isomorphisms between two different objects associated to hypersurface singularities: its modular stratum and its Milnor algebra. These phenomenon is checked by computer based computations for all unimodal singularities.
Novikov, Dmitry
Pseudoabelian integrals: toward an analogue of Varchenko-Khovanskii theorem
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Limit cycles appearing in perturbations of planar polynomial Hamiltonian vector fields are closely related to zeros of abelian integrals (i.e. of periods of a restriction of a polynomial one-form omega in CP2 to a level curve {H=t} of a bivariate polynomial H, considered as a function of t). Classical Varchenko-Khovanskii theorem gives a uniform upper bound for this number in terms of degrees of H, omega. We consider the same question for perturbations of Darboux integrable vector fields. Most of the good algebraic properties of the abelian integrals are lost in this case. However, we succeed to prove local uniform boundedness on a dense subset of such fields, using completely different, more analytic tools.
Palamodov, Victor
Moduli of hypersurface germs
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Recent machine computations of deformations of non-contractible hypersurface germs (B. Martin, H. Suss) show a striking similarity (up to a isomorphism) between the algebra of the modular stratum and the Milnor algebra of the germ. Only a weak form of the conjecture allows now a rigorous proof.
Parusinski, Adam
Equivalence relations of two variable real analytic function germs
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We give complete characterisations of blow-analytic equivalence for two variable real analytic function germs in terms of their minimal resolution and their real tree models. We compare also this equivalence to other natural equivalence relations, particularly to the bi-Lipschitz and C1 ones.
Rolin, Jean-Philippe
An o-minimal structure which does not admit C-infinite cellular decomposition
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Most of the known o-minimal expansions of the real fields have the analytic cell decomposition property. We show the existence an o-minimal expansion of the reals which does not admit C-infinite cell decomposition (joint work with Olivier Legal)
Roy, Marie-Francoise
Certificates of positivity in the multivariate Bernstein basis
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I discuss multivariate certificates of positivity. New results of R. Leroy on the distance between control polygon and graph expressed in terms of relevant "second differences of the Bernstein coefficients" introduced in 1992 by Goodman and Peters will be presented. Finally the current state of the art on the size of certificates of positivity in the multivariate case will be described and open problems presented.
Sottile, Frank
Fewnomial bounds from Gale dual systems
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In 1980, Askold Khovanskii established his fewnomial bound for the number of real solutions to a system of polynomials, showing that the complexity of the set of real solutions to a system of polynomials depends upon the number of monomials and not on the degree. This fundamental finiteness result in real algebraic geometry is believed to be unrealistically large. Recent work with Bihan has led to a new and substantially lower fewnomial bound which is asymptotically optimal. The bound is obtained by reducing the original system to a Gale dual system, and then bounding the number of solutions to a Gale system. In this talk, I will describe the history of this problem and outline our new results, as well as some applications of this new bound.
Speissegger, Patrick
Transition maps of non-resonant hyperbolic singularities are o-minimal
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It has been a long-standing hope, first spread by Van den Dries, that the concept of o-minimality might lead to new insights into Dulac's problem. The o-minimal structure mentioned in the title is a small step in this direction. I will give a quick review of Dulac's problem, and I will outline how we obtain o-minimality for the structure generated by all transition maps near non-resonant hyperbolic singularities. If time permits, I will outline our strategy to include transition maps of all hyperbolic singularities. (Joint work with Tobias Kaiser and Jean-Philippe Rolin.)
Trotman, David
Valette's generic definable bi-Lipschitz equisingularity
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Mostowski (1985) proved that every complex analytic variety admits a locally bi-Lipschitz trivial stratification refining the canonical Whitney-Kuo-Verdier stratification, and Parusinski (1994) proved the corresponding statement for subanalytic sets. The trivialisations are obtained by integrating Lipschitz stratified vector fields and so are not definable. Guillaume Valette (2005) has proved, via triangulation, the existence of locally definably bi-Lipschitz trivial stratifications of definable sets in polynomially bounded o-minimal structures. We describe properties and applications of Mostowski stratifications of subanalytic sets, and some striking applications of Valette's triangulations including his proof of a well-known conjecture of Siebenmann and Sullivan (1977).
Zell, Thierry
Around the descent spectral sequence
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A survey of the spectral sequence associated to a continuous surjection and its applications in establishing upper-bounds on Betti numbers of definable sets.

Participants

Name	Institution
Basu, Saugata	Georgia Institute of Technology
Bew, David	University of Oxford
Bierstone, Edward	University of Toronto
Bihan, Frederic	Université de Savoie
Boxall, Gareth	University of Leeds
Coste, Michel	Université Rennes 1
Eremenko, Alexandre	Purdue University
Foster, Tom	University of Oxford
Gabrielov, Andrei	Purdue University
Grandjean, Vincent	University of Bath & Universität Oldenburg
Hieronymi, Philipp	University of Oxford
Itenberg, Ilia	Université Louis Pasteur, Strasbourg
Jones, Gareth	McMaster University
Kharlamov, Viacheslav	Université Louis Pasteur, Strasbourg
Khovanskii, Askold	University of Toronto
Kirby, Jonathan	University of Oxford
Kurdyka, Krzysztof	Université de Savoie
Lipshitz, Leonard	Purdue University
Macintyre, Angus	University of London
Martin, Bernd	Brandenburgische Technische Universität Cottbus
Novikov, Dmitry	Weizmann Institute of Science
Palamodov, Victor	Tel Aviv University & Potsdam University
Parusinski, Adam	University of Angers
Rolin, Jean-Philippe	Université de Bourgogne
Roy, Marie-Francoise	Universite de Rennes 1
Sottile, Frank	Texas A&M University
Speissegger, Patrick	McMaster University
Thomas, Margaret	University of Oxford
Trotman, David	University of Provence
van den Dries, Lou	University of Illinois
Vorobjov, Nicolai	University of Bath
Wilkie, Alex	University of Manchester
Zell, Thierry	Vassar College