amenability definable groups and automorphism groups abstract we prove several theorems relating amenability of groups in various categories discrete definable topological automor phism group to model theoretic invariants quotients by connected components lascar galois group g compactness among the main tools which is possibly of independent interest is the adaptation and generalization of theorem 12 of 27 to other settings including amenable topological groups whose topology is generated by a family of subgroups on the side of definable groups we prove that if g is definable in a model m and g is definably amenable then the connected components g m 00 and g m 000 coincide answering positively a question from 22 we also prove some natural counterparts for topological groups using our generalizations of 27 by finding a dictionary relating quotients by connected components and galois groups of categorical theories we conclude that if m is countable and categorical and aut m is amenable as a topological group then t th m is g compact i e the lascar galois group gal l t is compact hausdorff equivalently the natural epimorphism from gal l t to gal k p t is an isomorphism we also take the opportunity to further develop the model theoretic approach to topological dynamics obtaining for example some new invariants for topological groups as well as allowing a uniform approach to the theorems above and the various categories

link invariant of the izergin korepin model the link invariant associated with the izergin korepin 19 vertex model is deduced using the method of statistical mechanics it is shown that the izergin korepin model leads to an invariant which is precisely the 3 state akutsu wadati polynomial previously known only for 2 and 3 braid knots we give a table of the invariant for all knots and links up to seven crossings

dynamical systems associated to separated graphs graph algebras and paradoxical decompositions abstract we attach to each finite bipartite separated graph e c a partial dynamical system e c f where e c is a zero dimensional metrizable compact space f is a finitely generated free group and is a continuous partial action of f on e c the full crossed product c algebra o e c c e c f is shown to be a canonical quotient of the graph c algebra c e c of the separated graph e c similarly we prove that for any field k the algebraic crossed product l k ab e c c k e c alg f is a canonical quotient of the leavitt path algebra l k e c of e c the monoid v l k ab e c of isomorphism classes of finitely generated projective modules over l k ab e c is explicitly computed in terms of monoids associated to a canonical sequence of separated graphs using this we are able to construct an action of a finitely generated free group f on a zero dimensional metrizable compact space z such that the type semigroup s z f k is not almost unperforated where k denotes the algebra of clopen subsets of z finally we obtain a characterization of the separated graphs e c such that the canonical partial action of f on e c is topologically free

one dimensional asymptotic classes of finite structures a collection c of finite l structures is a 1 dimensional asymptotic class if for every m n and every formula x y where y y 1 y m i there is a positive constant c and a finite set e c r 0 such that for every m c and a m m either m a c or for some e m a m c m 1 2 ii for every e there is an l formula y such that m m is precisely the set of a m m with m a m c m 1 2 one dimensional asymptotic classes are introduced and studied here these classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures connections with the model theory of infinite structures are also drawn

exchange leavitt path algebras and stable rank abstract we characterize those leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1 2 or concrete criteria in terms of properties of the underlying graph are given for each case

vapnik chervonenkis density in some theories without the independence property ii we study the vapnik chervonenkis vc density of denable families in certain stable rst order theories in particular we obtain uniform bounds on vc density of denable families in nite u rank theories without the nite cover property and we characterize those abelian groups for which there exist uniform bounds on the vc density of denable families

low lying excitations in the square triangle random tiling model we consider the sub dominant eigenstates of the transfer matrix for the square triangle random tiling model on an infinite strip of width l a numerical algorithm for generation of the corresponding solution of the bethe ansatz is developed numerical finite size scaling analysis of the associated eigenvalues reveals the presence of both integer and non integer critical exponents the analytical value of one of the non integer exponents is found it is also shown numerically that along with the leading correction to the free energy density for some excitations there is a term proportional to

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

finitely generated antisymmetric graph monoids abstract a graph monoid is a commutative monoid for which there is a particularly simple presentation given in terms of a quiver such monoids are known to satisfy various nonstable k theoretical representability properties for either von neumann regular rings or c algebras we give a characterization of graph monoids within finitely generated antisymmetric refinement monoids this characterization is formulated in terms of the prime elements of the monoid and it says that each free prime has at most one free lower cover we also characterize antisymmetric graph monoids of finite quivers in particular the monoid z 0 1 2 is a graph monoid but it is not the graph monoid of any finite quiver

the leavitt path algebra of a graph abstract for any row finite graph e and any field k we construct the leavitt path algebra l e having coefficients in k when k is the field of complex numbers then l e is the algebraic analog of the cuntz krieger algebra c e described in i raeburn graph algebras in cbms reg conf ser math vol 103 amer math soc 2005 the matrix rings m n k and the leavitt algebras l 1 n appear as algebras of the form l e for various graphs e in our main result we give necessary and sufficient conditions on e which imply that l e is simple

two dimensional o n model in a staggered field nienhuis truncated o n model gives rise to a model of self avoiding loops on the hexagonal lattice each loop having a fugacity of n we study such loops subjected to a particular kind of staggered field w which for n has the geometrical effect of breaking the three phase coexistence linked to the three colourability of the lattice faces we show that at t 0 for w 1 the model flows to the ferromagnetic potts model with q n2 states with an associated fragmentation of the target space of the coulomb gas for t 0 there is a competition between t and w which gives rise to multicritical versions of the dense and dilute loop universality classes via an exact mapping and numerical results we establish that the latter two critical branches coincide with those found earlier in the o n model on the triangular lattice using transfer matrix studies we have found the renormalization group flows in the full phase diagram in the t w plane with fixed n superposing three copies of such hexagonal lattice loop models with staggered fields produces a variety of one or three species fully packed loop models on the triangular lattice with certain geometrical constraints possessing integer central charges 0 c 6 in particular we show that benjamini and schramm s rgb loops have fractal dimension df 3 2

primely generated regular refinement monoids in the present note we prove a representation theorem for primely generated regular refinement monoids in terms of partially ordered systems of commutative groups this gives in particular a classification of all finite refinement monoids throughout all monoids and groups are commutative let m m 0 be a monoid m is called a refinement monoid 2 5 81 if the following conditions are satisfied

operator spectrum and exact exponents of the fully packed loop model we develop a coulomb gas description of the critical fluctuations in the fully packed loop model on the honeycomb lattice we identify the complete operator spectrum of this model in terms of electric and magnetic vector charges and we calculate the scaling dimensions of these operators exactly we also study the geometrical properties of loops in this model and we derive exact results for the fractal dimension and the loop size distribution function a review of the many different representations of this model that have recently appeared in the literature is given

a new exactly solvable case of an o n model on a hexagonal lattice a new solvable case of o n model on a hexagonal lattice is presented it is shown that the model has a phase transition at n 2 this phase transition can be interpreted as phase transition between percolating and non percolating phases

critical exponents of manhattan hamiltonian walks in two dimensions from potts and o n models we consider a set of hamiltonian circuits filling a manhattan lattice i e a square lattice with alternating traffic regulation we show that the generating function with fugacityz of this set is identical to the critical partition function of aq state potts model on an unoriented square lattice withq1 2 z the set of critical exponents governing correlations of hamiltonian circuits is derived using a coulomb gas technique these exponents are also found to be those of an o n vector model in the low temperature phase withn q1 2 z the critical exponents in the limitz 0 are then those of spanning trees q 0 and of dense polymers n 0 t tc corresponding to a conformal theory with central chargec 2 this shows that the manhattan orientation and the hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of hamiltonian walks

on minimal flows definably amenable groups and o minimality we study definably amenable groups in nip theories focusing on the problem raised in 10 of whether weak generic types coincide with almost periodic types equivalently whether the union of minimal subflows of a suitable type space is closed we give fairly definitive results in the o minimal context including a counterexample

regularity lemmas for stable graphs we develop a framework in which szemeredi s celebrated regularity lemma for graphs interacts with core model theoretic ideas and techniques our work relies on a coincidence of ideas from model theory and graph theory arbitrarily large half graphs coincide with model theoretic instability so in their absence structure theorems and technology from stability theory apply in one direction we address a problem from the classical szemeredi theory it was known that the irregular pairs in the statement of szemeredi s regularity lemma cannot be eliminated due to the counterexample of half graphs i e the order property corresponding to model theoretic instability we show that half graphs are the only essential difficulty by giving a much stronger version of szemeredi s regularity lemma for models of stable theories of graphs i e graphs with the non k order property in which there are no irregular pairs the bounds are significantly improved and each component satisfies an indivisibility condition in the other direction we take a more model theoretic approach and give several new szemeredi type partition theorems for models of stable theories of graphs the first theorem gives a partition of any such graph into indiscernible components meaning here that each component is either a complete or an empty graph whose interaction is strongly uniform this relies on a finitary version of the classic model theoretic fact that stable theories admit large sets of indiscernibles by showing that in models of stable theories of graphs one can extract much larger indiscernible sets than expected by ramsey s theorem the second and third theorems allow for a much smaller number of components at the cost of weakening the indivisibility condition on the components we also discuss some extensions to graphs without the independence property all graphs are finite and all partitions are equitable i e the sizes of the components differ by at most 1 in the last three theorems the number of components depends on the size of the graph in the first theorem quoted this number is a function ofonly as in the usual szemeredi regularity lemma

rigorous lower bound on the dynamic critical exponents of the swendsen wang algorithm we prove the rigorous lower bound z sw v for the dynamic critical exponent of the swendsen wang algorithm for two dimensional q state potts models with q 2 3 4 this implies z sw 0 2 5 1 we present numerical data indicating that z sw 0 55 0 03 0 89 0 05 for q 3 4 95 confidence limits statistical errors only the discrepancy for q 4 appears to be caused by multiplicative logarithmic corrections

critical properties of the izergin korepin and solvable o n models and their related quantum spin chains the central charge of the izergin korepin model the corresponding quantum spin chain and the o n model is calculated analytically via the bethe ansatz the calculation extends a technique recently developed for the zamolodchikov fateev model in addition critical exponents and the central charge for these models are obtained from numerical solutions of bethe ansatz equations for finite systems as a physical application the authors find the exponents v 12 23 and gamma 53 46 for the theta transition of polymers in two dimensions

the packing of two species of polygons on the square lattice we decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons we end up with a 24 vertex model which is known in the literature as the fully packed double loop model fpl2 in the particular case in which the fugacities of the polygons are the same the model admits an exact solution the solution is obtained using coordinate bethe ansatz and provides a closed expression for the free energy in particular we find the free energy of the four colouring model and the double hamiltonian walk and recover the known entropy of the ice model when both fugacities are set equal to 2 the model undergoes an infinite order phase transition

exact solution and surface critical behaviour of an o n model on the honeycomb lattice the authors obtain the exact bethe ansatz solution of an o n model on the honeycomb lattice with open boundaries across a finite strip several quantities relevant to the surface critical behaviour of the o n model are derived from this solution the exact surface critical exponents are in agreement with those obtained by cardy 1987 and by duplantier and saleur 1986 based on conformal invariance arguments

lebesgue measure and integration theory on non archimedean real closed fields with archimedean value group given a non archimedean real closed field with archimedean value group which contains the reals we establish for the category of semialgebraic sets and functions a full lebesgue measure and integration theory such that the main results from the classical setting hold the construction involves methods from model theory o minimal geometry and valuation theory we set up the construction in such a way that it is determined by a section of the valuation if the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism the range of the measure and integration is obtained in a controlled and tame way from the real closed field we start with the main example is given by the case of the field of puiseux series where the range is the polynomial ring in one variable over this field

the three state square lattice potts antiferromagnet at zero temperature we study the 3 state square lattice potts antiferromagnet at zero temperature by a monte carlo simulation using the wang swendsen koteck cluster algorithm on lattices up to 1024 1024 we confirm the critical exponents predicted by burton and henley based on the height representation of this model

the regular algebra of a quiver let k be a fixed field we attach to each column finite quiver e a von neumann regular k algebra q e in a functorial way the algebra q e is a universal localization of the usual path algebra p e associated with e the monoid of isomorphism classes of finitely generated projective right q e modules is explicitly computed

generically stable and smooth measures in nip theories we formulate the measure analogue of generically stable types in first order theories with nip without the independence property giving several characterizations answering some questions from 9 and giving another treatment of uniqueness results from 9 we introduce a notion of generic compact domination relating it to stationarity of keisler measures and also giving group versions we also prove the approximate definability of arbitrary borel probability measures on definable sets in the real and p adic fields

representing finitely generated refinement monoids as graph monoids abstract graph monoids arise naturally in the study of non stable k theory of graph c algebras and leavitt path algebras they play also an important role in the current approaches to the realization problem for von neumann regular rings in this paper we characterize when a finitely generated conical refinement monoid can be represented as a graph monoid the characterization is expressed in terms of the behavior of the structural maps of the associated i system at the free primes of the monoid

the six grothendieck operations on o minimal sheaves in this paper we develop the formalism of the grothendieck six operations on o minimal sheaves the grothendieck formalism allows us to obtain o minimal versions of i derived projection formula ii universal coefficient formula iii derived base change formula iv kunneth formula v local and global verdier duality

finite average lengths in critical loop models a relation between the average length of loops and their free energy is obtained for a variety of o n type models on two dimensional lattices by extending to finite temperatures a calculation due to kast we show that the number averaged loop length stays finite for all non zero fugacities n and in particular it does not diverge upon entering the critical regime n2 fully packed loop fpl models with n 2 seem to obey the simple relation 3lmin where lmin is the smallest loop length allowed by the underlying lattice we demonstrate this analytically for the fpl model on the honeycomb lattice and for the 4 state potts model on the square lattice and based on numerical estimates obtained from a transfer matrix method we conjecture that this is also true for the two flavour fpl model on the square lattice we present in addition numerical results for the average loop length on the three critical branches compact dense and dilute of the o n model on the honeycomb lattice and discuss the limit n0 contact is made with the predictions for the distribution of loop lengths obtained by conformal invariance methods

connected components of definable groups and o minimality i abstract we give examples of definable groups g in a saturated model sometimes o minimal such that g 00 g 000 yielding also new examples of non g compact theories we also prove that for g definable in a saturated o minimal structure g has a bounded orbit i e there is a type of g whose stabilizer has bounded index if and only if g is definably amenable giving a positive answer to a conjecture of newelski and petrykowski in this special case of groups definable in o minimal structures

relaxation time for a dimer covering with height representation this paper considers the monte carlo dynamics of random dimer coverings of the square lattice which can be mapped to a rough interface model two kinds of slow modes are identified associated respectively with long wavelength fluctuations of the interface height and with slow drift in time of the system wide mean height within a continuum theory the longest relaxation time for either kind of mode scales as the system sizen for the real discrete model an exactlower bound ofo n is placed on the relaxation time using variational eigenfunctions corresponding to the two kinds of continuum modes

algebraic bethe ansatz for the fpl2 model an exact solution of the model of fully packed loops of two colours on a square lattice has recently been proposed by dei cont and nienhuis using the coordinate bethe ansatz approach we point out here a simpler alternative in which the transfer matrix is directly identified as a product of r matrices this allows us to apply the nested algebraic bethe ansatz which leads to the same bethe equations we comment on some of the applications of this result

semilattices of groups and inductive limits of cuntz algebras we characterize in terms of elementary properties the abelian monoids which are direct limits of finite direct sums of monoids of the form z nz cup 0 where 0 is a new zero element for positive integers n the key properties are the riesz refinement property and the requirement that each element x has finite order that is n 1 x x for some positive integer n such monoids are necessarily semilattices of abelian groups and part of our approach yields a characterization of the riesz refinement property among semilattices of abelian groups further we describe the monoids in question as certain submonoids of direct products lambda times g for semilattices lambda and torsion abelian groups g when applied to the monoids v a appearing in the non stable k theory of c algebras our results yield characterizations of the monoids v a for c inductive limits a of sequences of finite direct products of matrix algebras over cuntz algebras o n in particular this completely solves the problem of determining the range of the invariant in the unital case of r o rdam s classification of inductive limits of the above type

groups measures and the nip we discuss measures invariant measures on definable groups and genericity often in an nip failure of the independence property environment we complete the proof of the third author s conjectures relating definably compact groups g in saturated o minimal structures to compact lie groups we also prove some other structural results about such g for example the existence of a left invariant finitely additive probability measure on definable subsets of g we finally introduce a new notion compact domination domination of a definable set by a compact space and raise some new conjectures in the o minimal case

the regular algebra of a poset let k be a fixed field we attach to each finite poset p a von neumann regular k algebra q k p in a functorial way we show that the monoid of isomorphism classes of finitely generated projective q k p modules is the abelian monoid generated by p with the only relations given by p p q q whenever q p in p this extends the class of monoids for which there is a positive solution to the realization problem for von neumann regular rings

diagonalisation of gl n invariant transfer matrices and quantum n wave system lee model the algebraic bethe ansatz is constructed for the gl n invariant transfer matrices with arbitrary gl n spin for this purpose the notion of vacuum subspace is introduced it is shown that the gl n magnet can be interpreted as an integrable discrete approximation of vector or matrix nonlinear schrodinger models or of the quantum n wave system

tiles and colors tiling models are classical statistical models in which different geometric shapes the tiles are packed together such that they cover space completely in this paper we discuss a class of two dimensional tiling models in which the tiles are rectangles and isosceles triangles some of these models have been solved recently by means of bethe ansatz we discuss the question why only these models in a larger family are solvable and we search for the yang baxter structure behind their integrability in this quest we find the bethe ansatz solution of the problem of coloring the edges of the square lattice in four colors such that edges of the same color never meet in the same vertex

linear groups definable in o minimal structures we study subgroups g of gl n r definable in o minimal expansions m r of a real closed field r we prove several results such as a g can be defined using just the field structure on r together with if necessary power functions or an exponential function definable in m b if g has no infinite normal definable abelian subgroup then g is semialgebraic we also characterize the definably simple groups definable in o minimal structures as those groups elementarily equivalent to simple lie groups and we give a proof of the kneser tits conjecture for real closed fields

gauge symmetry and non abelian topological sectors in a geometrically constrained model on the honeycomb lattice we study a constrained statistical mechanical model in two dimensions that has three useful descriptions they are i the ising model on the honeycomb lattice constrained to have three up spins and three down spins on every hexagon ii the three color and fully packed loop model on the links of the honeycomb lattice with loops around a single hexagon forbidden and iii three ising models on interleaved triangular lattices with domain walls of the different ising models not allowed to cross unlike the three color model the configuration space on the sphere or plane is connected under local moves on higher genus surfaces there are infinitely many dynamical sectors labeled by a noncontractible set of nonintersecting loops we demonstrate that at infinite temperature the transfer matrix admits an unusual structure related to a gauge symmetry for the same model on an anisotropic lattice this enables us to diagonalize the original transfer matrix for up to 36 sites finding an entropy per plaquette s k b ensuremath approx 0 3661 ensuremath cdots and substantial evidence that the model is not critical we also find the striking property that the eigenvalues of the transfer matrix on an anisotropic lattice are given in terms of fibonacci numbers we comment on the possibility of a topological phase with infinite topological degeneracy in an associated two dimensional quantum model

one basedness and reductions of elliptic curves over real closed fields building on the positive solution of pillay s conjecture we present a notion of intrinsic reduction for elliptic curves over a real closed eld k we compare such a notion with the traditional algebro geometric reduction and produce a classi cation of the group of k points of an elliptic curve e with three real roots according to the way e reduces algebro geometrically and the geometric complexity of the intrinsically reduced curve

fixed boundary octagonal random tilings a combinatorial approach some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied a geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems thus generalizing previous results on the more restricted class of codimension one tilings in particular this method gives access to counting formulas which are directly related to questions of entropy in these statistical systems methods and tools from the field of enumerative combinatorics are used

on pillay s conjecture in the general case abstract let m be an arbitrary o minimal structure let g be a definably compact definably connected abelian definable group of dimension n here we compute i the new intrinsic o minimal fundamental group of g ii for each k 0 the k torsion subgroups of g iii the o minimal cohomology algebra over q of g as a corollary we obtain a new uniform proof of pillay s conjecture an o minimal analogue of hilbert s fifth problem relating definably compact groups to compact real lie groups extending the proof already known in o minimal expansions of ordered fields

locally definable groups in o minimal structures abstract in this paper we develop the theory of locally definable groups in o minimal structures generalizing in this way the theory of definable groups

littlewood richardson coefficients for grothendieck polynomials from integrability abstract we study the littlewood richardson coefficients of double grothendieck polynomials indexed by grassmannian permutations geometrically these are the structure constants of the equivariant k theory ring of grassmannians representing the double grothendieck polynomials as partition functions of an integrable vertex model we use its yang baxter equation to derive a series of product rules for the former polynomials and their duals the littlewood richardson coefficients that arise can all be expressed in terms of puzzles without gashes which generalize previous puzzles obtained by knutson tao and vakil