The combinatorial study of subsets of the set N of natural numbers and of functions from N to N leads to numerous cardinal numbers, uncountable but no larger than the continuum. For example, how many infinite subsets X of N must I take so that every subset Y of N or its complement includes one of my X’s? Or how many functions f from N to N must I take so that every function from N to N is majorized by one of my f’s? The main results about these cardinal characteristics of the continuum are of two sorts: inequalities involving two (or sometimes three) characteristics, and independence results saying that other such inequalities cannot be proved in ZFC. Other results concern, for example, the cofinalities of these cardinals or connections with other areas of mathematics. This survey concentrates on the combinatorial set-theoretic aspects of the theory.

Combinatorial Cardinal Characteristics of the Continuum

ALGEBRAIC VARIETIES By ALEXANDER GROTHENDIEGK It is less than four years since eohomologieal methods (i.e. methods of Homologieal Algebra) were introduced into Algebraic Geometry in Serre's fundamental paper, and it seems already certain that they are to overflow this part of mathematics in the coming years, from the foundations up to the most advanced parts. All we can do here is to sketch briefly some of the ideas and results. None of these have been published in their final form, but most of them originated in or were suggested by Serre's paper. Let us first give an outline of the main topics of eohomologieal investigation in Algebraic Geometry, as they appear at present. The need of a theory of cohomology for 'abstract' algebraic varieties was first emphasized by Weil, in order to be able to give a precise meaning to his celebrated conjectures in Diophantine Geometry. Therefore the initial aim was to find the 'Weil cohomology' of an algebraic variety, which should have as coefficients something 'at least as good as a field of characteristic 0, and have such formal properties (e.g. duality, Runneth formula) as to yield the analogue of Lefschetz's 'fixed-point formula '. Serre's general idea has been that the usual ' Zariski topology ' of a variety (in which the closed sets are the algebraic subsets) is a suitable one for applying methods of Algebraic Topology. His first approach was hoped to yield at least the right Betti numbers of a variety, it being evident from the start that it could not be considered as the Weil cohomology itself, as the coefficient field for cohomology was the ground field of the variety, and therefore not in general of characteristic 0. In fact, even the hope of getting the 'true ' Betti numbers has failed, and so have other attempts of Serre's to get Weil's cohomology by taking the cohomology of the variety with values, not in the sheaf of local rings themselves, but in the sheaves of Witt-vectors constructed on the latter. He gets in this way modules over the ring W{h) of infinite Witt vectors on the ground field h, and W{h) is a ring of characteristic 0 even if h is of characteristic p =t= 0. Unfortunately, modules thus obtained over W{h) may be infinitely generated, even when the variety V is an abelian variety. Although interesting relations must certainly exist between these cohomology groups and the 'true ones', it seems certain 104 ALEXANDER GROTHENDIECK now that the Weil cohomology has to be defined by a completely different approach. Such an approach was recently suggested to me by the connections between sheaf-theoretic cohomology and cohomology of Galois groups on the one hand, and the classification of unramified coverings of a variety on the other (as explained quite unsystematically in Serre's tentative Mexico paper), and by Serre's idea that a 'reasonable' algebraic principal fiber space with structure group G, defined on a variety V, if it is not locally trivial, should become locally trivial on some covering of V unramified over a given point of V. This has been the startingpoint of a definition of the Weil cohomology (involving both 'spatial' and Galois cohomology), which seems to be the right one, and which gives clear suggestions how Weil's conjectures may be attacked by the machinery of Homological Algebra. As I have not begun these investigations seriously as yet, and as moreover this theory has a quite distinct flavor from the one of the theory of algebraic coherent sheaves which we shall now be concerned with, we shall not dwell any longer on Weil's cohomology. Let us merely remark that the definition alluded to has already been the starting-point of a theory of eohomologieal dimension of fields, developed recently by Tate. The second main topic for eohomologieal methods is the cohomology theory of algebraic coherent sheaves, as initiated by Serre. Although inadequate for Weil's purposes, it is at present yielding a wealth of new methods and new notions, and gives the key even for results which were not commonly thought to be concerned with sheaves, still less with cohomology, such as Zariski's theorem on 'holomorphic functions' and his 'main theorem'—which can be stated now in a more satisfactory way, as we shall see, and proved by the same uniform elementary methods. The main parts of the theory, at present, can be fisted as follows: (a) General finiteness and asymptotic behaviour theorems. (6) Duality theorems, including (respectively identical with) a eohomologieal theory of residues. (c) Riemann-Roch theorem, including the theory of Chern classes for algebraic coherent sheaves. (cu) Some special results, concerning mainly abelian varieties. The third main topic consists in the application of the eohomologieal methods to local algebra. Initiated by Koszul and Cartan-Eilenberg in connection with Hubert's 'theorem of syzygies', the systematic use of COHOMOLOGY THEORY 105 these methods is mainly due again to Serre. The results are the characterization of regular local rings as those whose global eohomologieal dimension is finite, the clarification of Cohen-Macaulay's equidimensionality theorem by means of the notion of eohomologieal codimension\ and especially the possibility of giving (for the first time as it seems) a theory of intersections, really satisfactory by its algebraic simplicity and its generality. Serre's result just quoted, that regular local rings are the only ones of finite global eohomologieal dimension, accounts for the fact that only for such local rings does a satisfactory theory of intersections exist. I cannot give any details here on these subjects, nor on various results I have obtained by means of a local duality theory, which seems to be the tool which is to replace differential forms in the case of unequal characteristics, and gives, in the general context of commutative algebra, a clarification of the notion of residue, which as yet was not at all well understood. The motivation of this latter work has been the attempt to get a global theory of duality in cohomology for algebraic varieties admitting arbitrary singularities, in order to be able to develop intersection formulae for cycles with arbitrary singularities, in a nonsingular algebraic variety, formulas which contain also a 'Lefschetz formula mod.p'. In fact, once a proper local formalism is obtained, the global statements become almost trivial. As a general fact, it appears that, to a great extent, the 'local' results already contain a global one; more precisely, global results on varieties of dimension n can frequently be deduced from corresponding local ones for rings of Krull dimension n + l. We will therefore turn now to giving some main ideas in the second topic, that is, the cohomology theory of algebraic coherent sheaves. First, I would like, however, to emphasize one point common to all of the topics considered (except perhaps for {d)), and in fact to all of the standard techniques in Algebraic Geometry. Namely, that the natural range of the notions dealt with, and the methods used, are not really algebraic varieties. Thus, we know that an affine algebraic variety with ground field h is determined by its co-ordinate ring, which is an arbitrary finitely generated ^-algebra without nilpotent elements; therefore, any statement concerning affine algebraic varieties can be viewed also as a statement concerning rings A of the previous type. Now it appears that most of such statements make sense, and are true, if we assume only J. to be a commutative ring with unit, provided we sometimes submit it to some mild restriction, as being noetherian, for instance. In 106 ALEXANDER GROTHENDIECK the same way, most of the results proved for the local rings of algebraic geometry, make sense and are true for arbitrary noetherian local rings. Besides, frequently when it seemed at first sight that the statement only made sense when a ground field k was involved, as in questions in which differential forms are considered, further consideration of the matter showed that this impression was erroneous, and that a better understanding is obtained by replacing k by a ring B such that A is a finitely generated J5-algebra. Geometrically, this means that instead of a single affine algebraic variety V (as defined by A) we are considering a 'regular map' or 'morphism' of V into another affine variety W, and properties of the variety V then are generalized to properties of a morphism V -> W (the 'absolute' notion for V being obtained from the more general 'relative' notion by taking W reduced to a point). On the other hand, one should not prevent the rings having nilpotent elements, and by no means exclude them without serious reasons. Now just as arbitrary commutative rings can be thought of as a proper generalization of affine algebraic varieties, one can find a corresponding suitable generalization of arbitrary algebraic varieties (defined over an arbitrary field). This was done by Nagata in a particular case, yet following the definition of schemata given by Chevalley he had to stick to the irreducible case, and with no nilpotent elements involved. The principle of the right definition is again to be found in Serre's fundamental paper, and is as follows. If A is any commutative ring, then the set Spec {A) of all prime ideals of A can be turned into a topological space in a classical way, the closed subsets consisting of those prime ideals which contain a given subset of A. On the other hand, there is a sheaf of rings defined in a natural way on Spec {A), the fiber of this sheaf at the point p being the local ring Ap. More generally, every module M over A defines a sheaf of modules on Spec {A), the fiber of which at the point p is the localized moduleMp over Ap. Now we call pre-schema a topological space X with a sheaf of rings Gx on X, called its structure sheaf, such that every point of X has an open neighborhood isomorphic to some Spec {A). If X and Y are two pre-schemas, a morphism f fro

Proceedings of the International Congress of Mathematicians

The study of permutation groups has always been closely associated with that of highly symmetric structures. The objects considered here are countably infinite, but have only finitely many different substructures of any given finite size. They are precisely those structures which are determined by first-order logical axioms together with the assumption of countability. This book concerns such structures, their substructures and their automorphism groups. A wide range of techniques are used: group theory, combinatorics, Baire category and measure among them. The book arose from lectures given at a research symposium and retains their informal style, whilst including as well many recent results from a variety of sources. It concludes with exercises and unsolved research problems.

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Oligomorphic permutation groups

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups
of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological
Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class
(answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has
a comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we study Polish groups
that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample
generics). We show that Polish groups with ample generics have the small index property (generalizing results
of Hodges, Hodkinson, Lascar and Shelah) and arbitrary homomorphisms from such groups into separable
groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have
uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of
many ω-stable, ℵ0-categorical structures and of the random graph. In this connection, we also show that the
infinite symmetric group S∞ has a unique non-trivial separable group topology. For several interesting groups
we also establish Serre’s properties (FH) and (FA).

/pdf/turbulence-amalgamation-and-generic-automorphisms-of-5e0hrghg9u.pdf

Turbulence, amalgamation, and generic automorphisms of homogeneous structures

https://ambio1.leeds.ac.uk/Pure/staff/macpherson/homog7.pdf

A survey of homogeneous structures

Let G be a permutation group on the set Ω. Usually we take |Ω|=N o . We seek a notion of genericity for members of G. The idea is that a member of G should be generic if it is in some sense typical. We argue that the following is the correct definition. Suppose that G is endowed with a metric so that it becomes a complete metric space. It follows that the Baire category theorem holds, and we may use the notions of meagre and comeagre sets

Generic Automorphisms of Homogeneous Structures

From the Publisher:
Key Features
Thorough coverage of all area of discrete mathematics, including logic, natural numbers, coding theory, combinatorics, sets, algebraic functions, partially ordered structures, graphs, formal machines & complexity theory Special emphasis on the central role of propositional & predicate logic Full chapters on algorithm analysis & complexity theory Introductory coverage of formal machines & coding theory Over 700 exercises Flexible structure so that the material can be easily adapted for different teaching styles 
New to this Edition
Improved treatment of induction Coverage of more 'basic' algebra List of symbols including page references for definition/explantion Modern text design and new exercises to aid student comprehension

Discrete mathematics for computer scientists

Let C be a set with at least two, and at most ℵ 0 , members, and for any set X let [ X ] 2 denote the set of its 2-element subsets. If Γ is a countable set, and F c is a function from [Γ] 2 into C, then the structure Γ c = (Γ, F c ) is called the countable universal C-coloured graph if the following condition is satisfied: Whenever α is a map from a finite subset of Γ into C there is x eΓ–dom α such that (∀ y edom α) F c { x, y } = α(y).

The group of the countable universal graph

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Classes of Dedekind finite cardinals

1. The method of hypersequents in the proof theory of propositional non-classical logic 2. Church-Rosser lambda theories, infinite lambda-terms and consistency problems 3. Baire category for monotone sets 4. Equality in substructural logics 5. Substructural predicates 6. Critical points in an algebra of elementary embeddings II 7. O-minimality nd tame topology 8. Embeddings of Heyting algebras 9. Embedding normal forms 10. Analysing proofs in analysis 11. Subalgebras of Cohen Algebras need not be Cohen 12. Independence structures in set theory 13. Recovering the action of an automorphism group 14. On the logical strength of Nash-Williams' theorem on transfinite sequences 15. Open questions around Buchi and Presburger arithmetics 16. A growth dichotomy for O-minimal expansions of ordered fields 17. Lukasiewicz normal forms and toric desingularizations 18. Fine hierarchy and definability in the Lindenbaum algebra 19. From logic problems to inductive definitions 20. Hyperstable theories 21. Quasi-Riemann Surfaces

John K. Truss

Papers

Generic Automorphisms of Homogeneous Structures

Discrete mathematics for computer scientists

The group of the countable universal graph

Classes of Dedekind finite cardinals

Logic: from foundations to applications: European logic colloquium