Outline O-minimal structures 1 Dimension 2 Definable groups and t-topology 3 Short course on definable groups: part I Euler characteristic 4 Existence of torsion in definably compact groups 5 Alessandro Berarducci Maximal tori 6 Dipartimento di Matematica Counting the torsion elements 7 Università di Pisa Higher homotopy 8 Leeds, 17-19 Jan 2015 Simple groups 9 10 Pillay’s conjectures 11 Abelian case 12 General case Alessandro Berarducci (Dipartimento di Matematica Università di Pisa) Short course on definable groups: part I Leeds, 17-19 Jan 2015 1 / 134 Alessandro Berarducci (Dipartimento di Matematica Università di Pisa) Short course on definable groups: part I Leeds, 17-19 Jan 2015 2 / 134 Structures Examples Three of the most important structures in mathematics are: A structure M is a non-empty set M = dom ( M ) equipped with some functions, constants, and relations. 1 ( Z , + , · ) , a “gödelian” structure; Groups, rings, modules, ordered sets, boolean algebras, are examples of 2 ( C , + , · ) , a “stable” structure; structures. 3 ( R , <, + , · ) , an “o-minimal” structure. Each structure has a language L , consisting of the “symbols” (or “names”) of its functions, constants, and relations. In the early foundational period logicians were mostly interested in gödelian The language of ordered rings consists of the symbols ≤ , + , · , 0 , 1. structures, focusing on indecidability and incompleteness results. Each symbols of L has a “type”, speficying whether it has to be interpreted as The study of stable structures brought to light connections with algebraic a function, a relation or a constant, and its “arity” (numer of arguments). geometry (e.g. [Hru96]). O-minimal structures, have an order < and a topology induced by the order. Given two structures M , N in the same language L , a morphisms from M to Real-algebraic and subanalytic geometry, as well as PL topology, fit into this N is a function f : M → N which preserves the intepretation of the symbols context. They recently have found applications to number theory of L . (e.g. [Wil04, PZ08, PW06]). Sometimes it is convenient to consider many-sorted structures, with more NIP structures encompass both the stable and the o-minimal structures. Keisler than one domain and functions and relations between the various domains measure play a key role in their study. (for instance a valued field). For the moment we consider one-sorted structures. Alessandro Berarducci (Dipartimento di Matematica Università di Pisa) Short course on definable groups: part I Leeds, 17-19 Jan 2015 3 / 134 Alessandro Berarducci (Dipartimento di Matematica Università di Pisa) Short course on definable groups: part I Leeds, 17-19 Jan 2015 4 / 134 Formulas Definable sets Given a language L , the L -formulas are expressions built up from: Given an L -structure M , and an L -formula ϕ (¯ x ) with free variables included in x = ( x 1 , . . . , x n ) , we write ¯ 1 the symbols of the language L (namely the names of the functions, constant and relations); a ∈ M n : M | { ¯ = ϕ (¯ a ) } , 2 the equality sign; 3 variables and parenthesis; for the set of n-tuples from M satisfying the formula (also denoted ϕ ( M ) ). a ∈ M n : M | 4 the boolean connectives, and the quantifier ∀ x and ∃ x A ∅ -definable set in M is a set of the form { ¯ = ϕ (¯ a ) } for some L -formula ϕ (¯ x ) ; According to the following grammar: a ∈ M n : M | a , ¯ A definable set in M is a set of the form { ¯ = ϕ (¯ b ) } for some y ) and parameters ¯ L -formula ϕ (¯ x , ¯ b from M . Term ::= variable | constant | function symbol applied to terms If the parameters ¯ b belong to a subset B of M we say that the set is Formula ::= (Term = Term) | relation symbol applied to terms | (Formula x , ¯ B -definable (so “definable” means “ M -definable”). We consider ϕ (¯ b ) as a ∧ Formula) | ¬ Formula | ∀ x Formula | etc. formula with parameters from ¯ b , or “over B ”. Terms are generalizations of polynomials. Given a structure, they represent Formulas with no free variables are called sentences. They are either true or functions on the structure. Formulas represent statements about the structure and false in M . its elements. The variables range in the domain of the structure. Alessandro Berarducci (Dipartimento di Matematica Università di Pisa) Short course on definable groups: part I Leeds, 17-19 Jan 2015 5 / 134 Alessandro Berarducci (Dipartimento di Matematica Università di Pisa) Short course on definable groups: part I Leeds, 17-19 Jan 2015 6 / 134 Example Adding constants to the language 1 In ( R , + , · ) the non-negative elements are ∅ -definable: x ≥ 0 iff ∃ y ( y 2 = x ) . 2 A circle of radious r is definable with parameter r ∈ R (by the formula x 2 + y 2 = r 2 ). In the structure ( N ; + , · , 0 , 1 ) the set P of primes is definable: 3 If r is real algebraic, the circle of radious r is ∅ -definable. ✜ n ∈ P ⇐ ⇒ N | = ∀ x , y ( x · y = n → x = 1 ∨ y = 1 ) . 4 The positive elements are not definable in ( R , + , 0 ) . ✜ The factorial function is also definable in the same structure. Indeed, by Gödel’s theorems, every computable function is definable in ( N ; + , · , 0 , 1 ) , as well as many Given a subset A ⊆ M , we can turn A -definable subsets of M n into ∅ -definable non-computable ones. sets by working in a bigger language L ( A ) ⊇ L obtained by adding constants for the elements of A , and considering M as an L ( A ) -structure. Formally we should use a different notation, so we denote by M A , or ( M , a ) a ∈ A , the expansion of M to the bigger language. Alessandro Berarducci (Dipartimento di Matematica Università di Pisa) Short course on definable groups: part I Leeds, 17-19 Jan 2015 7 / 134 Alessandro Berarducci (Dipartimento di Matematica Università di Pisa) Short course on definable groups: part I Leeds, 17-19 Jan 2015 8 / 134
Recommend
More recommend
