the regular spaces with countably based models the regular spaces which may be realized as the set of maximal elements in an continuous dcpo are the polish spaces in addition we give a new and conceptually simple model for complete metric spaces these results enable us to prove that the probabilistic powerdomain of a countably based model of a metric space always contains a copy of the normalized borel measures in their weak topology and to establish the hierarchy for countably based models

real congruences of complex subspaces of 2 2 symmetric complex matrices abstract consider complex subspaces of symmetric 2 2 complex matrices under real congruences sending m to ama t with a real and invertible the orbits for dimension one were classified in different language two decades ago those for dimension two were recently classified by coffman it turns out that these two problems are in fact dual to each other and coffman s classification yields a simpler expression for the orbits in dimension one

polish group actions dichotomies and generalized elementary embeddings the results in this paper involve two different topics in the descriptive theory of polish group actions the book becker kechris 6 is an introduction to that theory our two topics and two collections of theorems are rather unrelated but the proofs for both topics are essentially the same locally compact polish groups i e second countable locally compact groups are the traditional objects of study in the field known as topological groups more recently there has been interest in nonlocally compact polish groups such as s the symmetric group on n topologized as a subspace of nn much of this paper is concerned with a proper subclass of the class of polish groups namely those polish groups which admit a complete left invariant metric we call these cli groups the class of cli groups includes all locally compact groups proposition 3 c 2 d but also much more for example all solvable groups are cli hjorth solecki see proposition 3 c 2 f and a closed subgroup g of s is cli iff g is closed in nn see example 3 c 3 a we study continuous actions by polish groups on polish spaces and the associated orbit equivalence relation for locally compact groups all of our results are known or trivial or both but some of our results can be viewed as generalizations of known theorems about locally compact groups to the larger class of cli groups one of the two topics considered in this paper is dichotomy theorems for equivalence relations these are theorems which assert that either the quotient space is small or else it contains a copy of a specific large set we prove some dichotomy theorems for the orbit space of actions by cli groups there are two types of dichotomies which we call the silver vaught dichotomy and the glimm effros dichotomy the silver vaught dichotomy asserts that either there are only countably many equivalence classes or else there is a perfect set of pairwise inequivalent elements assuming the negation of the continuum hypothesis the silver vaught dichotomy is equivalent to the proposition that there are either countably many equivalence classes or 2 0 equivalence classes the topological vaught conjecture which is open is for any continuous action by a polish group g on a polish space x the orbit equivalence relation satisfies the silver vaught dichotomy this conjecture implies and was motivated by the vaught conjecture in model theory indeed

totally real immersions of surfaces first published in transactions of the american mathematical society in volume 362 and issue 1 published by the american mathematical society

on the number of permutatively inequivalent basic sequences in a banach space let x be a banach space with a schauder basis en n n the relation e0 is borel reducible to permutative equivalence between normalized block sequences of en n n or x is c0 or lp saturated for some 1 p if en n n is shrinking unconditional then either it is equivalent to the canonical basis of c0 or lp 1 p or the relation e0 is borel reducible to permutative equivalence between sequences of normalized disjoint blocks of x or of x if en n n is unconditional then either x is isomorphic to l2 or x contains 2 subspaces or 2 quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

ideal models of spaces ideal domains have an elementary order theoretic structure every element is either compact or maximal despite this we establish that 1 they can model any space currently known to possess a countably based model and 2 the metric spaces with ideal models are exactly the completely metrizable spaces

on an n manifold in mathbf c n near an elliptic complex tangent in this paper we will be concerned with the local biholomorphic properties of a real n manifold m in c at a generic point such a manifold basically has the nature of the standard r in c near a complex tangent however the consideration can be much more complicated and the manifold may acquire a nontrivial local hull of holomorphy and many other biholomorphic invariants the study of such a problem was first carried out in a celebrated paper of e bishop bis where for each sufficiently non degenerate complex tangent he attached a biholomorphic invariant called the bishop invariant when the complex tangent is elliptic i e when 0 12 for a more precise definition see 2 he showed the existence of families of complex analytic disks with boundary on m that shrink down to the locus of points in m with complex tangents in particular using the well known continuity principle one sees that the image m of such families is contained in the holomorphic hull of the manifold at the time he asked whether m gives precisely the local holomorphic hull of m as well as certain uniqueness properties of the attached disks he also proposed the problem of determining the fine structure of m near such complex tangents later there appeared a sequence of papers concerning the smooth character of m in case m c here we would like to mention in particular the famous theorem proved by kenig webster in their deep work kw1 which states that the local hull of holomorphy m near an elliptic complex tangent is a smooth levi flat hypersurface with m c as part of its smooth boundary in another important paper of moser webster mw a systematic normal form theory was employed for the understanding of the local biholomorphic invariants of m in case m is real analytic when the bishop invariant 6 0 their method works in any dimension and even for some hyperbolic complex tangents but it breaks down for complex tangents with 0 among other things they showed that m can be biholomorphically mapped into the affine space

the smooth ideal we give a classical proof of the generalization of the characterization of smoothness to quotients of polish spaces by borel equivalence relations as an application we describe the extent to which any given borel equivalence relation on a polish space is encoded by the corresponding ideal generated by the family of borel sets on which it is smooth

on totally real spheres in complex space let m n be two totally real and real analytic submanifolds in cn we say that m and n are biholomorphically equivalent if there is a biholomorphic mapping f defined in a neighborhood of m such that f m n as a standard fact of complexification one knows that all totally real and real analytic embeddings of m in c n are biholomorphically equivalent if m is of maximal dimension n however the topology of the manifold plays a major role in the existence of totally real immersions or embeddings for instance r o wells 23 proved that if an n dimensional compact and orientable manifold m admits a totally real embedding in cn then its euler number must vanish it was also observed by wells that if m is a manifold of dimension n and it admits a totally real immersion in cn then its complexified tangent bundle t cm tm c is trivial conversely the triviality of t cm also implies the existence of totally real immersions of m in cn this was obtained by m l gromov in 11 through the method of convex integration a stronger result due to j a lees 17 says that m also admits lagrangian immersions in cn the sphere s k x 2 1 x 2 k 1 1 in r k 1 gives us a trivial totally real embedding of s k in ck 1 on the other hand the works of gromov 11 ahernrudin 1 and stout zame 21 tell us that s k admits a totally real and real analytic embedding in ck if and only if k 1 3 our main result is the following theorem 1 1 if k 4 all totally real and real analytic embeddings of s k in cn are biholomorphically equivalent if k 5 and nk k 2 k 1 4 there exist totally

flattening of cr singular points and analyticity of the local hull of holomorphy i this is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in mathbb c n with n ge 3 near a non degenerate cr singular point the problem is motivated from the study of the complex plateau problem that seeks for the levi flat hypersurface bounded by a given real submanifold and is motivated by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space the present article is focused on the case of cr singular points with at least one elliptic direction we solve the holomorphic flattening problem and thus provide a complete description of the local hull of holomorphy in this setting the results in this paper and those in flattening of cr singular points and analyticity of the local hull of holomorphy ii p 60 2014 are taken from our arxiv post flattening of cr singular points and analyticity of the local hull of holomorphy 2012 we split flattening of cr singular points and analyticity of the local hull of holomorphy 2012 into two independent articles to avoid it being too long

real congruence of complex matrix pencils and complex projections of real veronese varieties quadratically parametrized maps from a real projective space to a complex projective space are constructed as projections of the veronese embedding a classification theorem relates equivalence classes of projections to real congruence classes of complex symmetric matrix pencils the images of some low dimensional cases include certain quartic curves in the riemann sphere models of the real projective plane in complex projective 4 space and some normal form varieties for real submanifolds of complex space with cr singularities

essential countability of treeable equivalence relations we establish a dichotomy theorem characterizing the circumstances under which a treeable borel equivalence relation e is essentially countable under additional topological assumptions on the treeing we in fact show that e is essentially countable if and only if there is no continuous embedding of e1 into e our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations and allow us to show that up to continuous kakutani embeddability there is a minimum borel function which is not essentially countable to one

genericity and amalgamation of classes of banach spaces abstract we study universality problems in banach space theory we show that if a is an analytic class in the effros borel structure of subspaces of c 0 1 of non universal separable banach spaces then there exists a non universal separable banach space y with a schauder basis that contains isomorphs of each member of a with the bounded approximation property the proof is based on the amalgamation technique of a class c of separable banach spaces introduced in the paper we show among others that there exists a separable banach space r not containing l 1 0 1 such that the indices and r n d are unbounded on the set of baire 1 elements of the ball of the double dual r of r this answers two questions of h p rosenthal we also introduce the concept of a strongly bounded class of separable banach spaces a class c of separable banach spaces is strongly bounded if for every analytic subset a of c there exists y c that contains all members of a up to isomorphism we show that several natural classes of separable banach spaces are strongly bounded among them the class of non universal spaces with a schauder basis the class of reflexive spaces with a schauder basis the class of spaces with a shrinking schauder basis and the class of spaces with schauder basis not containing a minimal banach space x

the classification problem for torsion free abelian groups of finite rank in 1937 baer 5 introduced the notion of the type of an element in a torsion free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion free abelian groups of rank 1 since then despite the efforts of such mathematicians as kurosh 23 and malcev 25 no satisfactory system of complete invariants has been found for the torsion free abelian groups of finite rank n 2 so it is natural to ask whether the classification problem is genuinely more difficult for the groups of rank n 2 of course if we wish to show that the classification problem for the groups of rank n 2 is intractible it is not enough merely to prove that there are 2 such groups up to isomorphism for there are 2 pairwise nonisomorphic groups of rank 1 and we have already pointed out that baer has given a satisfactory classification for this class of groups in this paper following friedman stanley 11 and hjorth kechris 15 we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n 2 recall that up to isomorphism the torsion free abelian groups of rank n are exactly the additive subgroups of the n dimensional vector space qf which contain n linearly independent elements thus the collection of torsion free abelian groups of rank 1 r n can be naturally identified with the set s qn of all nontrivial additive subgroups of qn notice that s qn is a borel subset of the polish space p qn of all subsets of qn and hence s qn can be regarded as a standard borel space i e a polish space equipped with its associated o algebra of borel subsets here we are identifying p qn with the space 2 qn of all functions h qn 0 1 equipped with the product topology furthermore the natural action of gln q on the vector space qn induces a corresponding borel action on s qn and it is easily checked that if a b c s qn then a v b iff there exists an element f e gln q such that p a b it follows that the isomorphism relation on s qn is a countable borel equivalence relation if x is a standard borel space then a borel equivalence relation on x is an equivalence relation e c x2 which is a borel subset of x2 the borel equivalence relation e is said to be countable iff every e equivalence class is countable

complete analytic equivalence relations we prove that various concrete analytic equivalence relations arising in model theory or analysis are complete i e maximum in the borel reducibility ordering the proofs use some general results concerning the wider class of analytic quasi orders

turbulence representations and trace preserving actions we establish criteria for turbulence in certain spaces of c algebra repre sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space x and on the hypernite ii1 factor r we also prove that the conjugacy action on the space of free actions of a countably innite amenable group on r is turbulent and that the conjugacy action on the space of ergodic measure preserving ows on x is generically turbulent

a dichotomy characterizing analytic digraphs of uncountable borel chromatic number in any dimension we study the extension of the kechris solecki todorcevic dichotomy on analytic graphs to dimensions higher than 2 we prove that the extension is possible in any dimension finite or infinite the original proof works in the case of the finite dimension we first prove that the natural extension does not work in the case of the infinite dimension for the notion of continuous homomorphism used in the original theorem then we solve the problem in the case of the infinite dimension finally we prove that the natural extension works in the case of the infinite dimension but for the notion of baire measurable homomorphism

the classification of hypersmooth borel equivalence relations this paper is a contribution to the study of borel equivalence relations in standard borel spaces i e polish spaces equipped with their borel structure a class of such equivalence relations which has received particular attention is the class of hyperfinite borel equivalence relations these can be defined as the increasing unions of sequences of borel equivalence relations all of whose equivalence classes are finite or as it turns out equivalently those induced by the orbits of a single borel auto morphism hyperfinite equivalence relations have been classified in djk under two notions of equivalence borel bi reducibility and borel isomorphism

the classification problem for von neumann factors abstract we prove that it is not possible to classify separable von neumann factors of types ii 1 ii or iii 0 1 up to isomorphism by a borel measurable assignment of countable structures as invariants in particular the isomorphism relation of type ii 1 factors is not smooth we also prove that the isomorphism relation for von neumann ii 1 factors is analytic but is not borel

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

non classification of cartan subalgebras for a class of von neumann algebras abstract we study the complexity of the classification problem for cartan subalgebras in von neumann algebras we construct a large family of ii1 factors whose cartan subalgebras up to unitary conjugacy are not classifiable by countable structures providing the first such examples additionally we construct examples of ii1 factors whose cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures finally we show directly that the cartan subalgebras of the hyperfinite ii1 factor up to unitary conjugacy are not classifiable by countable structures and deduce that the same holds for any mcduff ii1 factor with at least one cartan subalgebra

vaught s conjecture and the glimm effros property for polish transformation groups we extend the original glimm effros theorem for locally compact groups to a class of polish groups including the nilpotent ones and those with an invariant metric for this class we thereby obtain the topological vaught conjecture

descriptive complexity of some isomorphism classes of banach spaces abstract we present a topological frame in which it is possible to estimate the complexity of some borel families of separable banach spaces we use this frame for evaluating the complexity of the isomorphism class of the hilbert space l 2 of certain asymptotically hilbertian spaces and of the l p spaces under the condition 1 p the complexity appears to increase with the distance to the hilbert space commented open problems conclude the article

the space of composants of an indecomposable continuum abstract the family of all composants of an indecomposable continuum is studied we investigate the equivalence relation induced on an indecomposable continuum by its partition into composants we show that up to borel bireducibility such an equivalence relation can be of only two types e 0 and e 1 the e 0 type being simple and the e 1 type being complicated as a consequence of this we show that each indecomposable continuum carries a borel probability measure which assigns 0 to each composant and 0 or 1 to each borel set which is the union of a family of composants in particular it follows that there is no borel transversal for the family of all composants this solves an old problem in the theory of continua we prove that all hereditarily indecomposable continua are of the complicated type that is they fall into the e 1 type we analyze the properties of being of type e 1 or of type e 0 we show using effective descriptive set theory that the first of these properties is analytic and so the second one is coanalytic we construct examples of continua of both types in fact we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic respectively hence non borel so they do not admit simple topological characterizations we also use continua from this family to show that an indecomposable continuum may be of type e 1 only because of the behavior of composants on a small subset of the continuum this in particular shows that certain natural approaches to solving kuratowski s problem on generic ergodicity of the composant equivalence relation will not work we finish with some open problems

on the classification of positions and complex structures in banach spaces abstract a topological setting is defined to study the complexities of the relation of equivalence of embeddings or position of a banach space into another and of the relation of isomorphism of complex structures on a real banach space the following results are obtained a if x is not uniformly finitely extensible then there exists a space y for which the relation of position of y inside x reduces the relation e 0 and therefore is not smooth b the relation of position of l p inside l p or inside l p p 2 reduces the relation e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group c the relation of position of a space inside another can attain the maximum complexity e max d there exists a subspace of l p 1 p 2 on which isomorphism between complex structures reduces e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group

stein structures and holomorphic mappings we prove that every continuous map from a stein manifold x to a complex manifold y can be made holomorphic by a homotopic deformation of both the map and the stein structure on x in the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank the analogous result holds for any compact hausdorff family of maps but it fails in general for a noncompact family our main results are actually proved for smooth almost complex source manifolds x j with the correct handlebody structure the paper contains another proof of eliashberg s int j math 1 29 46 1990 homotopy characterization of stein manifolds and a slightly different explanation of the construction of exotic stein surfaces due to gompf ann math 148 2 619 693 1998 j symplectic geom 3 565 587 2005

on isometry and isometric embeddability between ultrametric polish spaces abstract we study the complexity with respect to borel reducibility of the relations of isometry and isometric embeddability between ultrametric polish spaces for which a set d of possible distances is fixed in advance these are respectively an analytic equivalence relation and an analytic quasi order and we show that their complexity depends only on the order type of d when d contains a decreasing sequence isometry is borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi orders if d is well ordered the situation is more complex for isometry we have an increasing sequence of borel equivalence relations of length 1 which are cofinal among borel equivalence relations classifiable by countable structures while for isometric embeddability we have an increasing sequence of analytic quasi orders of length at least 3 we then apply our results to solve various open problems in the literature for instance we answer a long standing question of gao and kechris by showing that the relation of isometry on locally compact ultrametric polish spaces is borel bireducible with countable graph isomorphism

uniformisations partielles et criteres a la hurewicz dans le plan we give characterizations of the borel sets potentially in some wadge class among the borel sets with countable vertical sections of a product of two polish spaces to do this we use some partial uniformization results

smooth sets for a borel equivalence relation we study some properties of smooth borel sets with respect to a borel equivalence relation showing some analogies with the collection of countable sets from a descriptive set theoretic point of view we found what can be seen as an analog of the hyperarithmetic points in the context of smooth sets we generalize a theorem of weiss from z actions to actions by arbitrary countable groups we show that the ci ideal of closed smooth sets is ill non borel

decomposing borel sets and functions and the structure of baire class 1 functions all spaces considered are metric separable and are denoted usually by the letters x y or z w stands for the set of all natural numbers if a metric separable space is additionally complete we call it polish if it is a continuous image of ww or equivalently of a polish space it is called souslin the main subject of the present paper is the structure of baire class 1 functions recent developments in banach space theory in particular discoveries of rosenthal and bourgain fremlin and talagrand see r for a survey of these developments stimulated investigations into the structure of baire class 1 functions the interested reader may consult papers by haydon odel rosenthal hor kechris louveau kl rosenthal r2 and references quoted there these investigations have already provided new applications in banach space theory see for example ri in our study of baire class 1 functions we will be interested in two decomposition properties one of them defined by lusin the other one by jayne and rogers first however we want to consider a more general problem of determining how difficult it is to represent a borel set as a union of simpler borel sets or the graph of a borel function as a union of the graphs of simpler borel functions using effective descriptive set theory in particular louveau s theorem we show that if a c x is borel x polish then a e e or there is a continuous injection q ww a such that q 1 b is meager for any b c a which is e2 this gives a new proof of j stern s result that if a borel set a is the union of cov m sets in e then a is itself e cov m is the smallest cardinality of a family of meager sets covering ill we prove similar results for functions put for f x y and a family of functions g

on normal forms of complex points of codimension 2 submanifolds abstract in this paper we present some linear algebra behind quadratic parts of quadratically flat complex points of codimension two real submanifold in a complex manifold assuming some extra nondegenericity and using the result of hong complete normal form descriptions can be given and in low dimensions we obtain a complete classification without any extra assumptions

linear algebraic groups and countable borel equivalence relations this paper is a contribution to the study of borel equivalence relations on standard borel spaces i e polish spaces equipped with their borel structure in mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants frequently these objects can be viewed as elements of a standard borel space x and the equivalence turns out to be a borel equivalence relation e on x a complete classification of x up to e consists of finding a set of invariants i and a map c x i such that xey c x c y for this to be of any interest both i and c must be explicit or definable and as simple and concrete as possible the theory of borel equivalence relations studies the set theoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems

borel actions of polish groups we show that a borel action of a polish group on a standard borel space is borel isomorphic to a continuous action of the group on a polish space and we apply this result to three aspects of the theory of borel actions of polish groups universal actions invariant probability measures and the topological vaught conjecture we establish the existence of universal actions for any given polish group extending a result of mackey and varadarajan for the locally compact case we prove an analogue of tarski s theorem on paradoxical decompositions by showing that the existence of an invariant borel probability measure is equivalent to the nonexistence of paradoxical decompositions with countably many borel pieces we show that various natural versions of the topological vaught conjecture are equivalent to each other and in the case of the group of permutations of n with the model theoretic vaught conjecture for infinitary logic this depends on our identification of the universal action for that group

on the complexity of the uniform homeomorphism relation between separable banach spaces we investigate the uniform homeomorphism relation between separable banach spaces and the related relation of local equivalence we completely characterize the descriptive complexity of local equivalence in the borel reducibility hierarchy this also provides a lower bound for the complexity of the uniform homeomorphism

completeness of the isomorphism problem for separable c algebras this paper studies the descriptive set theoretical complexity of the isomorphism problem for separable c algebras we prove that the isomorphism problem for separable simple ai c algebras is complete in the class of orbit equivalence relations this means that any isomorphism problem arising from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple separable ai c algebras

real classification of complex quadrics abstract canonical forms are given for complex quadric surfaces and conics under real changes of variable the basic idea is to use known results on pairs of real quadratic forms

levi foliations in pseudoconvex boundaries and vector fields that commute approximately with boas and straube proved a general sufficient condition for global regularity of the neumann problem in terms of families of vector fields that commute approximately with in this paper we study the existence of these vector fields on a compact subset of the boundary whose interior is foliated by complex manifolds this question turns out to be closely related to properties of interest from the point of view of foliation theory

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

dichotomy theorems for families of non cofinal essential complexity we prove that for every borel equivalence relation e either e is borel reducible to mathbb e 0 or the family of borel equivalence relations incompatible with e has cofinal essential complexity it follows that if f is a borel equivalence relation and cal f is a family of borel equivalence relations of non cofinal essential complexity which together satisfy the dichotomy that for every borel equivalence relation e either e in cal f or f is borel reducible to e then cal f consists solely of smooth equivalence relations thus the dichotomy is equivalent to a known theorem

on the isomorphism problem for measures on boolean algebras abstract the paper investigates possible generalizations of maharam s theorem to a classification of boolean algebras that support a finitely additive measure we prove that boolean algebras that support a finitely additive non atomic uniformly regular measure are metrically isomorphic to a subalgebras of the jordan algebra with the lebesgue measure we give some partial analogues to be used for a classification of algebras that support a finitely additive non atomic measure with a higher uniform regularity number we show that some naturally induced equivalence relations connected to metric isomorphism are quite complex even on the cantor algebra and therefore probably we cannot hope for a nice general classification theorem for finitely additive measures we present an example of a boolean algebra which supports only separable measures but no uniformly regular one

rationally convex domains and singular lagrangian surfaces in mathbb c 2 we give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in mathbb c 2 we recall some classical obstructions and prove some deeper ones related to symplectic and contact topology we explain the close connection to lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces our proof also gives a complete characterization of lagrangian surfaces with open whitney umbrellas answering a question first posed by givental in 1986

borel chromatic numbers we study in this paper graph coloring problems in the context of descriptive set theory we consider graphs g x r where the vertex set x is a standard borel space i e a complete separable metrizable space equipped with its algebra of borel sets and the edge relation r x 2 is definable i e borel analytic co analytic etc a borel n coloring of such a graph where 1 n n 0 is a borel map c x y with card y n such that xry c x c y if such a borel coloring exists we define the borel chromatic number of g in symbols x b g to be the smallest such n otherwise we say that g has uncountable borel chromatic number in symbols x b g n 0