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www crm umontreal ca, Thematic Semester Michigan State which took place from March 5 to March. continued from page 1 16 During the school Jean Berstel Universit de Marne la. Vall e and Christophe Reutenauer UQ M each gave a 10. The Aisenstadt chair was held by Richard Stanley MIT who hour lecture series In his series of lectures entitled Repetitions. gave a series of conferences in March on Increasing and in words Jean Berstel discussed the famous infinite word. decreasing subsequences Alternating permutations and of Prouhet Thue Morse discovered independently by those. Some enumeration problems involving GL n q three mathematicians in 1851 1912 et 1921 from very different. The semester started with points of view Prouhet to solve a problem in arithmetic that. the mini workshop Al became known as the problem of Tarry and Escott Thue for. gebraic Combinatorics aesthetical reasons and with the ultimate goal of solving dif. meets Inverse Systems ferent difficult problems as he himself puts it and Morse in. organized by F Bergeron relation with dynamical systems There are applications of the. UQ M K Dalili Dal Prouhet Thue Morse word to the towers of Hanoi problem. housie S Faridi Dal to magic squares to algebraic series over fields of characteris. housie and A Lauve tic p and to the Burnside problem Berstel also presented his. UQ M which was held Richard Stanley theorem on morphisms preserving square free words which. from January 19 to January allows the simplification of many constructions The work of. 21 The focus of the workshop which was a continuation of Crochemore which gives an algorithmic characterization of. a sequence of successful such workshops held in Kingston the square free morphisms and words was also discussed The. 2004 Ottawa 2005 and Toronto 2006 was to unite two lecture series concluded with a discussion of the notions of mo. research communities whose interests are overlapping with tive and of abelian square and the work of Dekking and Cur. increasing regularity Each year has brought new connections rie Visentin. and this year continued this trend On the first day both com In his lectures Christophe Reutenauer discussed the words of. munities presented their research while the second day saw Christoffel The main topics covered were their definition by. quivers applied to questions in each community As this se discretisation of finite intervals following Borel Laubie Bers. quence of workshops represents an attempt to get two com tel and Osborne Zieschang and their equivalent definition by. munities with different languages and different interests to the Cayley graph of a cyclic group following Christoffel the. talk to one another the tone and spirit has been by neces morphisms of Christoffel which preserve the conjucacy classes. sity friendly patient and cooperative and this year was no of the Christoffel words and form a monoid the theorem of. exception The impact of workshops such as this one which Mignosi S bold and Wen Wen which shows that the gener. create links between research communities and encourage the ators are the positive automorphisms of the free group on two. participants to acquire new languages cannot be overstated generators the construction by palindromisation of the words. This first workshop was followed by the school and workshop of Christoffel the Christoffel tree Berstel de Luca which can. dedicated to Statistical Mechanics and Combinatorics orga be used to generate all those words and its specialisation the. nized by M Bousquet M lou Bordeaux P Leroux UQ M Stern Brocot tree related to Farey sequences the conjugates. T Guttmann Melbourne and A Sokal New York which of the words of Christoffel which are the primitive elements of. took place from February 12 to February 23 The goal of the the free group on two generators the relations between contin. school was to introduce the basic methods of enumerative com ued fractions and words of Christoffel emphasizing their use. binatorics and the concepts of statistical mechanics which are in Markoff s theory of approximations of real numbers In all. at the heart of the interactions between these two areas The the proofs presented an effort was made towards discrete ge. school was intended for graduate students postdoctoral fel ometric approaches. lows and researchers wishing to be introduced to these ques Most of the 48 participants to the school were also present at the. tions It was held in the Laurentians region at the Far Hillsworkshop and there were about 20 new participants The main. Inn located in Val Morin Qu bec The workshop has given themes of the workshop were Sturmian words which were. rise to 24 expository and specialized talks on combinatorial discussed in the talks of J Berstel Marne la Vall e V Berth. problems raised by statistical mechanics as well as 6 poster Universit Montpellier II A Glen UQ M P Arnoux et. presentations The topics covered included enumerative prob S Ferenczi Institut de Math matique de Luminy A Luca. lems related to the classical models of statistical mechanics Universit degli Studi Napoli and L Vuillon Universit de. self avoiding walks tilings asymmetric exclusion processes Montr al group theory from the point of view of combina. alternating sign matrices plane partitions multiple partitions. torics on words which was discussed in the talks of A Mias. polymers and copolymers Mayer s theory and graph weights nikov McGill D Serbin McGill and A Juhasz Technion. Potts model on graphs and Feynman diagrams sets of words which were discussed in the talks of G Musiker. The next events of the theme semester were the school and the UCSD B Steinberg Carleton and D Perrin Marne la. workshop dedicated to Combinatorics on Words organized Vall e infinite classical words which were the subject. by S Brlek UQ M C Reutenauer UQ M and B Sagan continued on page 4. BULLETIN CRM 2,www crm umontreal ca,The 2007 Andr Aisenstadt Prize. by Gregory G Smith Queen s University, hat is a moduli space Roughly speaking a moduli space by its global sections In the first paper 5 we work with. W parametrizes a collection of objects there is a bijection. between the points in the space and the objects However not. modules over a polynomial ring graded by an abelian group. Our definition of regularity involves the vanishing of certain. any bijection will do The correspondence must be natural as graded components of local cohomology We establish the es. a point varies appropriately in a moduli space the correspond sential properties of regularity its connection with the mini. ing objects should form a suitably nice family For example the mal generators of a module and its behaviour in short exact. Grassmannian G k n parametrizes k dimensional subspaces sequences Using the dictionary between sheaves on a toric. of an n dimensional vector space over your favourite field A variety X and graded modules over the Cox ring an appro. point on G k n varies continuously if and only if the coeffi priate multigraded polynomial ring we prove that the regu. cients of the defining linear equations for the associated sub larity of an ideal sheaf on X bounds the multidegrees of the. space vary continuously in your favourite topology Working equations that cut out the associated subvariety The connec. over real numbers this makes the Grassmannian into a com tion to moduli spaces comes in the second paper 6 Returning. pact real manifold of dimension k n k The key observation to Mumford s original motivation for regularity we construct a. is that the geometry of a moduli spaces encodes families of space HilbX that parametrizes all subschemes of X with a given. objects Category theory especially representable functors is multigraded Hilbert polynomial P Q t1 tr This gener. needed to formalize these ideas Although the precise defini alizes Grothendieck s Hilbert scheme which parametrizes sub. tion is surprisingly short it is conceptually difficult to internal. schemes of projective space In a different direction Milena. ize so I will not include it here Hering Hal Schenck and I also use these techniques in 3 to. provide new insights into the syzygies of projective toric vari. Moduli spaces play a central r le in algebraic geometry By de. sign a moduli space provides geometric structure to the ob. jects it classifies For the Grassmannian this is relatively sim. ple because the naive notion of nearby subspaces corresponds. to neighbouring points on G k n Similar notions on the mod. uli space of curves M g which parametrizes all isomorphism. classes of smooth genus g projective curves are substantially. more involved they require deformation theory On the other. hand interpreting a given space as a moduli space elucidates. its properties For instance the fact that each point on the. Grassmannian corresponds to a vector space produces a tauto. logical vector bundle on G k n Moduli spaces also arise nat. urally in an ever widening number of fields including combi. natorics string theory complex analysis representation theory. and topology, Below I outline three projects that involve moduli spaces in dif. ferent ways The first constructs a parameter space the second. reinterprets certain varieties as moduli spaces and the third uti. lizes moduli spaces to define auxiliary gadgets The projects Gregory Smith receiving his prize from Fran ois Lalonde director of the. also illustrate my affinity for algebraic spaces arising from com CRM. binatorial structures such as toric varieties These spaces pro As the part of an ongoing project Alastair Craw and I real. vide an excellent computational environment I enjoy imple ize every projective toric variety X as a fine moduli space of. menting algorithms typically in Macaulay2 to collect heuristic quiver representations More precisely we introduce the mul. evidence and investigate conjectures Indeed all of the follow tilinear series associated to a list L L0 Lr of line. ing projects benefited from computer calculations bundles on X and a choice of section in H 0 X L j Li 1 for. In a pair of papers Diane Maclagan and I develop a multi 0 i j r Combinatorially the multilinear series is a. graded variant of Castelnuovo Mumford regularity Intu bound quiver Q R the vertices of Q correspond to the line. itively regularity measures the complexity of a module or bundles the arrows in Q correspond to the sections and the. sheaf The regularity of a module approximates the largest de ideal of relations R arises from binomial equations among the. gree of the minimal generators and the regularity of a sheaf tensor products of sections Generalizing classical linear series. estimates the smallest twist for which the sheaf is generated on X the quiver Q defines a smooth projective toric variety L. continued on page 6,BULLETIN CRM 3,www crm umontreal ca.

Thematic Semester The last activity of the theme semester was the workshop on. continued from page 2 Real Tropical and Complex Enumerative Geometry orga. nized by V Kharlamov Strasbourg and R Pandharipande. of the talks of A Blondin Mass and S Labb UQ M Princeton and described in detail on the article on page 11. L Balkova and P Ambroz Czech Technical University A Frid of the present bulletin. Novosibirsk and J Cassaigne Luminy Finally X Proven al. UQ M talked about tilings of the plane by polyominos and. A Fraenkel Weizmann Institute of Science applied combina SMS 2007. torics on words to a problem in game theory continued from page 1. The next scientific events to take place during the theme as well as the theory of Hamiltonian systems in infinite. semester were the school and workshop on Combinatorial dimensional phase space and its applications to problems in. Hopf Algebras and Macdonald Polynomials organized by continuum mechanics and partial differential equations Ap. M Aguiar Texas A M F Bergeron UQ M N Bergeron plications to several important areas of research were also pre. York M Haiman Berkeley and S van Willigenburg UBC sented including to celestial mechanics control theory partial. which took place from April 30 to May 11 During the school differential equations of fluid dynamics and adiabatic invari. the main speakers were M Aguiar Texas A M F Bergeron ants. UQ M and M Haiman Berkeley M Aguiar discussed ten, sor categories theory of species and graded Hopf algebras Physical laws are for the most part expressed in terms of dif. F Bergeron discussed symmetric functions representation the ferential equations and the most natural classes of these are. ory of finite groups and invariant theory with an emphasis in the form of conservation laws or of problems of the calcu. on the case of reflection groups coinvariant spaces harmonic lus of variations for an action functional These problems can. polynomials and diagonal coinvariant spaces M Haiman dis typically be posed as Hamiltonian systems whether dynamical. cussed the Weyl character formula algebraic representations systems on finite dimensional phase space as in classical me. of GL n Macdonald polynomials and Lascoux Leclerc chanics or partial differential equations PDE which are nat. Thibon polynomials and combinatorial models for Macdonald urally of infinitely many degrees of freedom For instance the. polynomials Written notes of all the lectures can be found at well known n body problem is still of great relevance to mod. karin math yorku ca nantel HopfAndMacdonald ern mathematics and more broadly to science indeed in appli. The workshop followed immediately the school Many inter cations the mission design of interplanetary exploration reg. Le Bulletin du CRM www crm umontreal ca Automne Fall 2007 Volume 13 No 2 Le Centre de recherches math matiques A Retrospective of the Winter 2007 Thematic Semester Recent Advances in Combinatorics by F Bergeron S Brlek P Leroux and C Reutenauer UQ M The 2007 theme semester Recent Advances in Combina torics took place at the CRM from January to June 2007 The

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