Workshop
Number Theory and Computability
Jun 25, 2007 - Jun 29, 2007
ICMS, 14 India Street, Edinburgh
Organisers
Name | Institution |
---|---|
Everest, Graham | University of East Anglia |
Short Report
Many people are fascinated to observe when a polynomial equation has a solution in integers or rational numbers (fractions), especially if the solution is not obvious. Witness the global interest in Simon Singh's book [4] about Fermat's Last Theorem. In 1900, David Hilbert, one of the giants of Mathematics, asked if it is possible to construct a general method which will determine if such an equation has an integral solution.
After several decades of hard thought, a surprising answer emerged from a surprising quarter. In 1970, logicians proved, using ideas that led directly to the invention of the modern computer, that no such general method can exist. This interplay between Logic and Number Theory led to a fantastic rapport developing between workers in these fields, one that has continued up to the present. The aim of the meeting was to build upon earlier workshops in bringing established as well as new workers into this fascinating dialogue.
Hilbert's question having been resolved for the integers, mathematicians naturally ask the same question about other arithmetic structures, such as polynomials or the rational numbers. Some new results have been obtained and the meeting provided a shop window for these.
However the central problem about solvability in rational numbers remains. In 2003, Poonen took a giant leap in that direction by giving a negative answer to Hilbert's question, now in the context of some large subrings of the rational numbers, whose denominators are constrained in some natural way. What is more, he used methods from the arithmetic theory of elliptic curves, itself an extremely exotic and powerful subject. The workshop also gave time over to studying Poonen's methods with the hope of eventually resolving this most difficult problem.
A large number of photographs were taken at this meeting. To view them please follow these links:
Participants list and links to available presentations are further down this page.
Download the pdf file of the full report
Original Details
Hilbert's Tenth Problem concerns the solution of Diophantine Equations; asking whether an algorithm exists which will determine if a system of polynomial equations in a finite number of variables, has an integral solution. This decision problem was eventually settled by Matijasevic who showed that no such algorithm exists. Much of the foundations for the proof were laid by J. Robinson, Davis and Putnam, using tools from Logic. Matijasevic, in the course of his proof brought about an interesting collision between Number Theory and Logic. Specifically, he reduced the problem to one in the arithmetic of integer linear recurrence sequences.
The rational version of Hilbert's Tenth Problem asks the same question as above except that one now seeks a rational solution. This is an unsolved problem.
Within the last six years, several workers have begun to ask whether the rational version of Hilbert's Tenth Problem might be resolved using the arithmetic of higher genus integer recurrence sequences. Similar questions have been formulated for rings of algebraic integers, rings of polynomials, fields of rational functions, rings of analytic and meromorphic functions. Many of these questions remain open.
Despite its apparent theoretical nature, the study of the subject has produced constructions of mathematical objects such as a polynomial whose set of positive values coincides with the set of prime numbers as well as examples of exotic varieties (over various domains of common interest). It has also produced very deep new questions such as Mazur's Question (which relates to the topology of rational points on a variety). Relations have been established with the Theory of Elliptic Curves and Abelian Varieties, Diophantine Geometry, Divisibility Sequences, Hyperbolic Varieties and much besides.
The field has brought together researchers from Number Theory, Logic, Algebraic Geometry and Theory of Computation and a substantial number of young researchers have chosen to work on the mentioned and related problems.
It is hoped that the workshop will contribute substantially to the cross-fertilization of all the mentioned areas and will benefit especially workers in Number Theory and Logic in the U.K. The Workshop will be a sequel to the meetings held at Ghent in 1999, Oberwolfach in 2003 Palo Alto and (American Institute of Mathematics) in 2005.
The expected outcomes, and the techniques developed to approach them, would be of interest to researchers in Mathematics, in particular in the areas of Number Theory and Logic and their interactions. Number Theory of this kind has links with Cryptography and the results may well be of interest to workers in that field and applicable in the short term.
An archive of background reading is available from http://www.mth.uea.ac.uk/~h090/icms.html
Clay Mathematics Institute is holding a meeting on Hilbert's Tenth Problem in March this year. You may also wish to look at some of the resources on their site at http://www.claymath.org/events/h10/ .
Arrangements
Participation
Participation is by invitation only. The workshop will begin on the morning of Monday 25 June and finish on the afternoon of Friday 29 June 2007. In preparation for the workshop you may wish to visit the archive of background reading mentioned above.
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.
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 £45.00 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 June. 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 June.
On Tuesday 26 June you are invited to attend an informal supper at Nargile Turkish Restaurant. The workshop dinner will take place on the evening of Thursday 28 June. The workshop grant will cover the cost of these 2 meals.
Registration
Registration will take place between 09.30 and 10.20 on Monday 25 June. The talks will start at 10.30.
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, the informal supper on Tuesday 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. 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 anticipate any difficulty covering the fee, please let me know.
Programme
Monday 25 June
09.30 - 10.20 | Registration |
10.20 - 10.30 | Welcome |
10.30 - 11.30 | Yuri Matiyasevich (Steklov Institute of Mathematics) |
11.30 - 12.30 | Kirsten Eisentraeger (University of Michigan) |
12.30 - 14.30 | Lunch (sandwich lunch provided) |
14.30 - 15.30 | Graham Everest (University of East Anglia) |
15.30 - 16.00 | Coffee/Tea |
16.00 - 17.00 | Jochen Koenigsmann (Max Planck Institut für Mathematik) |
17.00 - 18.30 | Wine reception at ICMS |
Tuesday 26 June
10.00 - 11.00 | Bjorn Poonen (University of California, Berkeley) |
11.00 - 11.30 | Coffee/Tea |
11.30 - 12.30 | Jeroen Demeyer (Ghent University) |
12.30 - 14.30 | Lunch |
14.30 - 15.30 | Alexandra Shlapentokh (East Carolina University) |
15.30 - 16.00 | Coffee/Tea |
16.00 - 17.00 | Aharon Razon (Elta Systems) |
19.00 | Informal group supper at Nargile Turkish Restaurant |
Wednesday 27 June
10.00 - 11.00 | Joseph H Silverman (Brown Univeristy) |
11.00 - 11.30 | Coffee/Tea |
11.30 - 12.30 | Anand Pillay (University of Leeds) |
12.30 - 14.00 | Lunch |
14.00 - 15.00 | Thomas Scanlon (University of California, Berkeley) |
15.00 - 15.30 | Coffee/Tea |
15.30 - 16.30 | Alexandra Shlapentokh (East Carolina University) |
17.00 - 17.45 | Screening of Julia Robinson and Hilbert's Tenth Problem |
Thursday 28 June
10.00 - 11.00 | Laurent Moret-Bailly (IRMAR, Université de Rennes 1) |
11.00 - 11.30 | Coffee/Tea |
11.30 - 12.30 | Moshe Jarden (Tel Aviv University) |
12.30 - 14.30 | Lunch |
14.30 - 15.30 | Katherine Stange (Brown University) |
15.30 - 16.00 | Coffee/Tea |
16.00 - 17.00 | Maxim Vsemirnov (Steklov Institute of Mathematics) |
19.00 | Workshop dinner at First Coast Restaurant |
Friday 29 June
10.00 - 11.00 | Marco Streng (Universiteit Leiden) |
11.00 - 11.30 | Coffee/Tea |
11.30 - 12.30 | Thanases C Pheidas (University of Crete) |
12.30 - 14.00 | Lunch |
14.00 - 15.00 | Gregory Cherlin (Rutgers University) |
15.00 - 15.30 | Coffee/Tea |
15.30 - 16.30 | Françoise Point (Mons-Hainaut University) |
16.30 | Close of workshop |
Presentations:
Presentation Details | |
---|---|
Cherlin, Gregory | |
Permutation groups of finite Morley rank | |
View Abstract | |
Demeyer, Jeroen | |
Diophantine sets of polynomials over a finite field | |
View Abstract | |
Eisentraeger, Kirsten | |
First-order undecidability and Hilbert's Tenth Problem for function fields of positive characteristic | |
View Abstract | |
Everest, Graham | |
The arithmetic of elliptic divisibility sequences | |
View Abstract | |
Jarden, Moshe | |
Undecidability of families of rings of totally real integers | |
View Abstract | |
Koenigsmann, Jochen | |
A Galois theoretic approach to decidability of fields | |
View Abstract | |
Matiyasevich, Yuri | |
Computation paradigms in the light of Hilbert's tenth problem | |
View Abstract | |
Moret-Bailly, Laurent | |
Positive-existential definability in Noetherian rings | |
View Abstract | |
Pheidas, Thanases C | |
Meromorphic maps of the punctured torus to the torus and undecidability | |
View Abstract | |
Pillay, Anand | |
A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields | |
View Abstract | |
Point, Françoise | |
Witt modules | |
View Abstract | |
Poonen, Bjorn | |
Characterizing Z in Q with a universal-existential formula | |
View Abstract | |
Razon, Aharon | |
On Rumely's local-global principle | |
View Abstract | |
Scanlon, Thomas | |
Defining valuations on curves | |
View Abstract | |
Shlapentokh, Alexandra | |
HTP over algebraic extensions of rational numbers: normforms vs. elliptic curves. Part 1: normforms, Part 2: elliptic curves | |
View Abstract | |
Silverman, Joseph H | |
P-adic properties of elliptic divisibility sequences | |
View Abstract | |
Stange, Katherine | |
From elliptic divisibility sequences to elliptic nets | |
View Abstract | |
Streng, Marco | |
Divisibility sequences for elliptic curves with complex multiplication | |
View Abstract | |
Vsemirnov, Maxim | |
Diophantine encoding and generalized Cantor's polynomials | |
View Abstract |
Participants
Name | Institution |
---|---|
Aliev, Iskander | University of Edinburgh |
Cherlin, Gregory | Rutgers University |
D'Aquino, Paola | Seconda Università di Napoli |
Demeyer, Jeroen | Ghent University |
Eisentraeger, Kirsten | University of Michigan |
Everest, Graham | University of East Anglia |
Flenner, Joseph | University of California, Berkeley |
Geyer, Wulf-Dieter | University of Erlangen-Nurnberg |
Gica, Alexandru | University of Bucharest |
Ingram, Patrick | University of Toronto |
Jarden, Moshe | Tel Aviv University |
Koenigsmann, Jochen | University of Oxford |
Macintyre, Angus | Queen Mary University of London |
Mahe, Valery | University of East Anglia |
Matiyasevich, Yuri | Steklov Institute of Mathematics |
Michaux, Christian | Université de Mons-Hainaut |
Moret-Bailly, Laurent | IRMAR, Université de Rennes 1 |
Pheidas, Thanases C | University of Crete |
Pillay, Anand | University of Leeds |
Point, Françoise | Mons University |
Poonen, Bjorn | University of California, Berkeley |
Razon, Aharon | Elta Systems |
Riviere, Cedric | University of Mons-Hainaut |
Scanlon, Thomas | University of California, Berkeley |
Shlapentokh, Alexandra | East Carolina University |
Silverman, Joseph H | Brown University |
Stange, Katherine | Brown University |
Stephens, Nelson | Bristol University |
Streng, Marco | Universiteit Leiden |
Swart, Christine | University of Cape Town |
Van Geel, Jan | Ghent University |
Vidaux, Xavier | Universidad de Concepción |
Videla, Carlos | Mount Royal College, Calgary |
Vsemirnov, Maxim | Steklov Institute of Mathematics |
Zahidi, Karim | University of Ghent |