Résultats attendus:

Nous espérons obtenir des resultats significatifs dans tous les domaines indiques dans la description du projet. Concretement, les résultats attendus sont indiqués en detail dans cette description (B2).

Dans le domaine de recherche en question, l'énonce de résultats que "nous comptons obtenir" relève, nous en sommes convaincus, d'une certaine malhonnéteté scientifique. Par conséquent, nous ne promettons rien, mais avons indiqué, dans le descriptif detaillé du projet, quelques uns des résultats déjà obtenus par notre équipe, et les directions ou nous comptons continuer a travailler. Nous espérons que l'évaluation du projet se fait sur la base de ces descriptifs et sur la valeur des chercheurs du associés au projet, et non sur des promesses vagues et/ou optimistes de résultats.

Quelques publications recentes par les participants du projet:

I.A.Dynnikov, Three-page approach to knot theory. Encoding and local moves, (Russian) Funktsional'nyi Analiz i Prilozhenija, 33 (1999), no.4, 25-37; English translation in Functional Analysis and Its Appl. 34 (1999), no.4, 260-269

I.A.Dynnikov, Three-page approach to knot theory. Universal semigroup, Funktsional'nyi Analiz I Prilozhenija, 34 (2000), no.1, 29-40 (Russian); English translation in Functional Analysis and Its Appl. 34 (2000), no.1, 24-32.

I.A.Dynnikov, Finitely presented groups and semigroups in knot theory. (Russian) Tr. Mat. Inst. Steklova 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 231-248; English translation in Proc. Steklov Inst. Math. 2000, no. 4 (231), 220-237.

F.Hivert, Non-commutatifs et quasi-symétriques des fonctions de Hall-Littlewood et modules de Demazure d'une algèbre quantique dégénérée C. R. Acad. Sci. Paris, t. 362, Série I, p. 1-6, 1998.

F.Hivert Quasi-symmetric functions and Hecke algebras actions Proceeding of the 10-th congress "Formal Power series and algebraic combinatorics", Fields intitute, Toronto, june 1998 (N. Bergeron and F. Sottile eds.), p. 325-339.

F.Hivert, Hecke algebras, difference operators and quasi-symmetric functions Advances in Maths. t. 155, p. 181-238, 2000.

O.Karpenkov, Energy of a knot: variational principles, Russ. J. Math. Phys., vol.9 No.3, 2002

O.Karpenkov, Energy of a knot: some new aspects. Fundamental Mathematics Today, MCCME, 2003

V.Danilov, G.Koshevoy, Discrete convexity and unimodularity. I preprint : math.CO/0304291

V.Danilov, G.Koshevoy, Mathematics of Plott choice functions, preprint: math.CO/0304171

S.K.Lando, A.K.Zvonkin, Graphs on Surfaces and Their Applications, with an Appendix by D.Zagier, Springer, 2004, xv+455 pp.

S.K.Lando, On a Hopf algebra in graph theory, J. Comb. Theory, Ser. B, vol. 80, 104-121 (2000)

S.K.Lando, Ramified coverings of the two-dimensional sphere and the intersection theory in spaces of meromorphic functions on algebraic curves, Russ. Math. Surveys, vol. 57, no. 3, 463-533 (2002)

S.Matveev, Algorithmic Topology and Classification of 3-Manifolds (Springer: Berlin, 2003)

S.Matveev, T.Nowik, On 3-manifolds having the same Turaev-Viro invariants, Russian J. Math. Phys., 2, 317-324 (1994)

S.Matveev, Generalized graph manifolds and their efficient recognition, Sb. Math., 189, 1517-1531 (1998)

V.Prasolov, A.Sossinski, Knots, Links, Braids and 3-Manifolds (AMS: Providence R.I., 1997)

A.Sossinski, Feyman diagrams and Vassdiliev Invariants, IHES, Preprint vol. 13, No.M (1992)

A.Sossinski, Vassiliev Spaces and Classical Invariants, Russ.J.Math.Phys., 5, 47-62 (1996)

Erukhimovich I.Ya., Thamm M.V., Ermoshkin A.V. Theory of Sol-Gel Transition in Thermoreversible Gels with due Reguard for the Fundamental Role of Mesoscopic Cyclization Effects, Macromolecules, 34, 5653 (2001).

Tamm M.V., Lisachenko N., Erukhimovich I.Ya., Ivanov V.A., The finite size effects in the equilibrium ideal system of polymer rings: the formation of micro- and macroscopic cycles. Theory and computer simulation, Submitted to Pol. Sci.

Tamm M.V., Nechaev S.K., Erukhimovich I.Ya., Randomly branched copolymers near the impenetrable surface: statistical weights and the probability of branching, in preparation.

V.Vassiliev, Cohomology of knot spaces, Advances in Soviet Mathematics, vol.1, 23-69, 1990

V.Vassiliev, Complements of discriminants of smooth maps, Revised Edition, Amer.Math.Soc., Providence RI, 1994.

V.Vassiliev, Invariants and homology of spaces of knots in arbitrary manifolds. To appear.

A.Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples. Uspekhi Mat. Nauk 55, No.4(334), 59-128 (2000). English translation: Russian Math. Surveys 55, No.4, 667-733 (2000).

A.Vershik, Statistical properties of locally free groups with applications to braid groups and growth of random heaps (with S.Nechaev and R.Bikbov), Comm. Math. Phys. 212, No.2, 469-501 (2000)

A.Vershik, Numerical invariants of exponential groups, and a locally free approximation of braid groups (in Russian), Zapiski Nauchn. Semin. POMI. 266, 51-63 (2000). English translation to appear in J. Math. Sci.

Vershinin, Vladimir V. On Poisson-Malcev structures. Monodromy and differential equations (Moscow, 2001), Acta Appl. Math. 75 (2003), no. 1-3, 281--292.

Vershinin, Vladimir V. On homology of virtual braids and Burau representation. Knots in Hellas '98, Vol. 3 (Delphi). J. Knot Theory Ramifications 10 (2001), no. 5, 795-812.

Vershinin, V. V. Braid groups and loop spaces. (Russian) Uspekhi Mat. Nauk 54 (1999), no. 2(326), 3-84; translation in Russian Math. Surveys 54 (1999), no. 2, 273-350

P. Dehornoy, A fast method for comparing braids, Adv. in Maths., 125-2 (1997) 200-235

P. Dehornoy, Braids and self-distributivity, Progress in Math vol. 192, Birkhauser (2000)

P. Dehornoy, I. Dynnikov, D. Rolfsen, B. Wiest, Why are braids orderable? Panomramas & Syntheses vol 14, Soc. Math. de France (2002)

D.Krob, J.Mairesse, and I.Michos, Computing the average parallelism in trace monoids, Discrete Math., 273:131-162, 2003.

J.Mairesse and L.Vuillon, Asymptotic behavior in a heap model with two pieces, Theoret. Comput. Sci., 270:525-560, 2002.

J.Mairesse, Computations of Uniform Recurrence Equations Using Minimal Memory Size (with Bruno Gaujal and Alain Jean-Marie). SIAM J. on Computing, Vol. 30, n. 5, p. 1701-1738, 2000

A. Comtet, J. Desbois, S. N. Majumdar, The Local Time Distribution of a Particle Diffusing on a Graph, J. Phys. A: Math. Gen. 35, L687 (2002)

S.Majumdar, D.Dean, Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models, Phys. Rev. E 66, 041102 (2002)

S.N.Majumdar, S.Nechaev, Anisotropic ballistic deposition model with links to the Ulam problem and the Tracy-Widom distribution, Phys.Rev.E, 69, 011103 (2004)

O. Vasilyev, S. Nechaev, Thermodynamics and topology of disordered systems: Statistics of random knot diagrams on finite lattices, JETP, 93 (2001) 1119

O.Vasilyev, S.Nechaev, On topological correlations in trivial knots: New arguments in support of the crumpled globule concept, Theor.Math.Phys., 134 (2003) 142-159

S.Nechaev, O.Vasilyev, Topological percolation on square lattice, submitted to J.Phys.A.

S.Nechaev, R.Voituriez, On the plants leaves boundary, "jupe à godets" and conformal embeddings, J.Phys. (A): Math. Gen. 34 (2001) 11069-11082

S.Nechaev, R. Voituriez, Random walks on 3-strand braid and related hyperbolic groups, J. Phys. A: Math. Gen. 36 (2003), 43-66

G.Oshanin, R.Voituriez, Random walk generated by random permutations, preprint cond-mat/0312262

P.Zinn-Justin, The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links, Commun. Math. Phys., 238 (2003), 287-304

P.Zinn-Justin, J.-B.Zuber, Matrix Integrals and the Generation and Counting of Virtual Tangles and Links, accepté pour publication dans J. Knot Theor. Ramif.

G.Schaeffer, P.Zinn-Justin, On the Asymptotic Number of Plane Curves and Alternating Knots, accepté pour publication dans J. Exp. Math.