Entrance hall of the ICMS Workshop

Effective Real Analytic Geometry

May 05, 2008 - May 09, 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.
View Abstract Down
Bierstone, Edward
Problems on resolution of singularities
View Abstract Down
Bihan, Frederic
Bounds on the topology of fewnomial hypersurfaces
View Abstract Down
Gabrielov, Andrei
Dessins d'Enfants for the eigenfunctions of the quartic oscillator
View Abstract Down
Grandjean, Vincent
Geodesics at singular points of non-conical isolated surface singularities
View Abstract Down
Itenberg, Ilia
Recursive formulas for Welschinger invariants
View Abstract Down
Jones, Gareth
Model completeness results for polynomially bounded o-minimal structures.
View Abstract Down
Kharlamov, Viacheslav
Real cubic four-folds; deformations and chirality
View Abstract Down
Khovanskii, Askold
Degree or rational mappings and simple proofs of the Sturm and Tarski theorems
View Abstract Down
Kurdyka, Krzysztof
Trajectories of horizontal gradients of polynomials
View Abstract Down
Lipshitz, Leonard
Real Closed Fields with Analytic Structure
View Abstract Down
Macintyre, Angus
Elementary Theory of Elliptic Functions
View Abstract Down
Martin, Bernd
Modular strata of unimodal singularities are corresponding to Milnor algebras
View Abstract Down
Novikov, Dmitry
Pseudoabelian integrals: toward an analogue of Varchenko-Khovanskii theorem
View Abstract Down
Palamodov, Victor
Moduli of hypersurface germs
View Abstract Down
Parusinski, Adam
Equivalence relations of two variable real analytic function germs
View Abstract Down
Rolin, Jean-Philippe
An o-minimal structure which does not admit C-infinite cellular decomposition
View Abstract Down
Roy, Marie-Francoise
Certificates of positivity in the multivariate Bernstein basis
View Abstract Down
Sottile, Frank
Fewnomial bounds from Gale dual systems
View Abstract Down
Speissegger, Patrick
Transition maps of non-resonant hyperbolic singularities are o-minimal
View Abstract Down
Trotman, David
Valette's generic definable bi-Lipschitz equisingularity
View Abstract Down
Zell, Thierry
Around the descent spectral sequence
View Abstract Down

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 University of Purdue
Foster, Tom University of Oxford
Gabrielov, Andrei Purdue University
Grandjean, Vincent University of Bath & Universität Oldenburg
Hieronymi, Philipp University of Oxford
Itenberg, Ilia Université Pierre et Marie Curie (Paris 6)
Jones, Gareth McMaster University
Kharlamov, Viacheslav Université Louis Pasteur, Strasbourg
Khovanskii, Askold University of Toronto
Kirby, Jonathan University of East Anglia
Kurdyka, Krzysztof Université de Savoie
Lipshitz, Leonard Purdue University
Macintyre, Angus Queen Mary University of London
Martin, Bernd Brandenburgische Technische Universität Cottbus
Novikov, Dmitry Weizmann Institute of Science
Palamodov, Victor Tel Aviv University & Potsdam University
Parusinski, Adam Université Nice Sophia Antipolis
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 Aix-Marseille University
van den Dries, Lou University of Illinois
Vorobjov, Nicolai University of Bath
Wilkie, Alex University of Manchester
Zell, Thierry Vassar College