Effective Real Analytic Geometry
Organiser
Name | Institution |
|---|---|
| Vorobjov, Nicolai | University of Bath |
| Wilkie, Alex | University of Manchester |
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) | |
11.00 – 11.30 | Coffee/Tea | |
11.30 - 12.30 | Andre Gabrielov ( Purdue University) | |
13.00 – 14.30 | Lunch - buffet provided at 14 India Street | |
14.30 - 15.30 | Thierry Zell ( Vassar College) | |
15.30 – 16.00 | Coffee/Tea | |
16.00 - 17.00 | Jean-Philippe Rolin (Université de Bourgogne) | |
17.30 – 19.00 | Wine Reception at 14 India Street |
Tuesday 6 May
09.00 – 10.00 | Dmitry Novikov (Weizmann Institute of Science) |
10.00 – 11.00 | Marie-Francoise Roy (Université de Rennes 1) |
11.00 – 11.30 | Coffee/Tea |
11.30 – 12.30 | Vincent Grandjean (University of Bath & Universität Oldenburg) |
12.30 – 14.30 | Lunch |
14.30 – 15.30 | Frank Sottile (Texas A&M University) |
15.30 – 16.00 | Coffee/Tea |
16.00 – 17.00 | Frederic Bihan (Université de Savoie) |
Wednesday 7 May
09.00 – 10.00 | Krysztof Kurdyka (Université Savoie |
10.00 – 11.00 | Askold Khovanskii ( University of Toronto) |
11.00 – 11.30 | Coffee/Tea |
11.30 – 12.30 | Victor Palamodov ( Tel Aviv University & Potsdam University) |
12.30 – 13.00 | Bernd Martin (Brandenburgische Technische Universität) |
13.00 - | Free afternoon |
Thursday 8 May
09.00 – 10.00 | Angus Macintyre (University of London) |
10.00 – 11.00 | David Trotman ( University of Provence) |
11.00 – 11.30 | Coffee/Tea |
11.30 – 12.30 | Viacheslav Kharlamov (Université Louis Pasteur) |
12.30 – 14.30 | Lunch |
14.30 – 15.30 | Ilia Itenberg (Université Louis Pasteur) |
15.30 – 16.00 | Coffee/Tea |
16.00 – 17.00 | Saugata Basu (Georgia Institute of Technology) |
19.00 | Workshop dinner at First Coast, 99 – 101 Dalry Road, Edinburgh |
Friday 9 May
09.00 – 10.00 | Edward Bierstone ( University of Toronto) |
10.00 – 11.00 | Patrick Speissegger ( McMaster University) |
11.00 - 11.30 | Coffee/Tea |
11.30 – 12.30 | Gareth Jones ( McMaster University) |
12.30 – 14.30 | Lunch |
14.30 – 15.30 | Adam Parusinski ( University of Angers) |
15.30 – 16.00 | Coffee/Tea |
Presentations
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.
This is a joint work with F. Sottile
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).
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.
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.
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.
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.
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.
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.
Participants
Name | Institution |
|---|---|
| Saugata, Basu | Georgia Institute of Technology |
| David, Bew | University of Oxford |
| Edward, Bierstone | University of Toronto |
| Frederic, Bihan | Université de Savoie |
| Gareth, Boxall | University of Leeds |
| Michel, Coste | Université Rennes 1 |
| Alexandre, Eremenko | University of Purdue |
| Tom, Foster | University of Oxford |
| Andrei, Gabrielov | Purdue University |
| Vincent, Grandjean | University of Bath & Universität Oldenburg |
| Philipp, Hieronymi | University of Oxford |
| Ilia, Itenberg | Université Pierre et Marie Curie (Paris 6) |
| Gareth, Jones | McMaster University |
| Viacheslav, Kharlamov | Université Louis Pasteur, Strasbourg |
| Askold, Khovanskii | University of Toronto |
| Jonathan, Kirby | University of East Anglia |
| Krzysztof, Kurdyka | Université de Savoie |
| Leonard, Lipshitz | Purdue University |
| Angus, Macintyre | Queen Mary University of London |
| Bernd, Martin | Brandenburgische Technische Universität Cottbus |
| Dmitry, Novikov | Weizmann Institute of Science |
| Victor, Palamodov | Tel Aviv University & Potsdam University |
| Adam, Parusinski | Université Nice Sophia Antipolis |
| Jean-Philippe, Rolin | Université de Bourgogne |
| Marie-Francoise, Roy | Universite de Rennes 1 |
| Frank, Sottile | Texas A&M University |
| Patrick, Speissegger | McMaster University |
| Margaret, Thomas | University of Oxford |
| David, Trotman | Aix-Marseille University |
| Lou, van den Dries | University of Illinois |
| Nicolai, Vorobjov | University of Bath |
| Alex, Wilkie | University of Manchester |
| Thierry, Zell | Vassar College |