Ramsey classes Topological dynamics Model theory Ramsey theory - PowerPoint PPT Presentation

Introduction Questions Results Ramsey classes Miodrag Soki´ c Introduction Motivation Ramsey classes Topological dynamics Model theory Ramsey theory The KPT theory Questions Miodrag Soki´ c Posets Ultrametric spaces Caltech Results Posets Ultrametric April 1 st 2012, Madison Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Motivation Ramsey classes Miodrag Soki´ c Introduction topological dynamics Motivation Topological dynamics Model theory Ramsey theory The KPT theory Questions Posets Ultrametric spaces Results Posets Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Motivation Ramsey classes Miodrag Soki´ c Introduction topological dynamics Motivation Topological dynamics model theory Model theory Ramsey theory The KPT theory Questions Posets Ultrametric spaces Results Posets Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Motivation Ramsey classes Miodrag Soki´ c Introduction topological dynamics Motivation Topological dynamics model theory Model theory Ramsey theory The KPT theory combinatorics Questions Posets Ultrametric spaces Results Posets Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Motivation Ramsey classes Miodrag Soki´ c Introduction topological dynamics Motivation Topological dynamics model theory Model theory Ramsey theory The KPT theory combinatorics Questions KPT Posets Ultrametric spaces Results Posets Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Motivation Ramsey classes Miodrag Soki´ c Introduction topological dynamics Motivation Topological dynamics model theory Model theory Ramsey theory The KPT theory combinatorics Questions KPT Posets Ultrametric spaces Results Posets group actions on compact spaces Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Motivation Ramsey classes Miodrag Soki´ c Introduction topological dynamics Motivation Topological dynamics model theory Model theory Ramsey theory The KPT theory combinatorics Questions KPT Posets Ultrametric spaces Results Posets group actions on compact spaces Ultrametric Fraïssé theory Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Motivation Ramsey classes Miodrag Soki´ c Introduction topological dynamics Motivation Topological dynamics model theory Model theory Ramsey theory The KPT theory combinatorics Questions KPT Posets Ultrametric spaces Results Posets group actions on compact spaces Ultrametric Fraïssé theory Ramsey theory Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Topological dynamics Ramsey classes Miodrag Soki´ c Introduction continuous action on a compact Motivation G - ‡ow Topological dynamics Hausdor¤ space Model theory Ramsey theory The KPT theory G � X ! X Questions Posets Ultrametric spaces Results Posets Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Topological dynamics Ramsey classes Miodrag Soki´ c Introduction continuous action on a compact Motivation G - ‡ow Topological dynamics Hausdor¤ space Model theory Ramsey theory The KPT theory G � X ! X Questions Posets Ultrametric spaces Results Posets minimal ‡ow for all x 2 X Ultrametric G � x = X Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Topological dynamics Ramsey classes Miodrag Soki´ c Introduction continuous action on a compact Motivation G - ‡ow Topological dynamics Hausdor¤ space Model theory Ramsey theory The KPT theory G � X ! X Questions Posets Ultrametric spaces Results Posets minimal ‡ow for all x 2 X Ultrametric G � x = X Fact Every G -‡ow contains a minimal sub‡ow. Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Topological dynamics Ramsey classes Miodrag Soki´ c Introduction Theorem Motivation Topological dynamics Given a topological group G , there is a minimal G -‡ow M ( G ) Model theory Ramsey theory such that for any other minimal G -‡ow Y there exists surjective The KPT theory homomorphism Questions Posets Ultrametric spaces ϕ : M ( G ) ! Y Results Posets ϕ ( gx ) = g ϕ ( x ) Ultrametric Moreover such G -‡ow M ( G ) is uniquely determined up to isomorphism. Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Topological dynamics Ramsey classes Miodrag Soki´ c Introduction Motivation Topological dynamics Model theory Ramsey theory The minimal G -‡ow M ( G ) established in the previous theorem The KPT theory Questions is called the universal minimal G -‡ow . Posets Ultrametric spaces Results Posets Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Topological dynamics Ramsey classes Miodrag Soki´ c Introduction Motivation Topological dynamics Model theory Ramsey theory The minimal G -‡ow M ( G ) established in the previous theorem The KPT theory Questions is called the universal minimal G -‡ow . Posets A group G is extremely amenable (has …xed point on compacta Ultrametric spaces property ) if M ( G ) = f�g . Results Posets Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Fraïssé classes Ramsey classes Miodrag Soki´ c K -class of …nite structures in a countable signature L , closed Introduction Motivation under isomorphic images Topological dynamics Fraïssé class Model theory Ramsey theory The KPT theory contains structures of arbitrary large …nite cardinality Questions Posets Ultrametric spaces Results Posets Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Fraïssé classes Ramsey classes Miodrag Soki´ c K -class of …nite structures in a countable signature L , closed Introduction Motivation under isomorphic images Topological dynamics Fraïssé class Model theory Ramsey theory The KPT theory contains structures of arbitrary large …nite cardinality Questions Posets hereditary property (HP) A , ! B 2 K = ) A 2 K Ultrametric spaces Results Posets Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Fraïssé classes Ramsey classes Miodrag Soki´ c K -class of …nite structures in a countable signature L , closed Introduction Motivation under isomorphic images Topological dynamics Fraïssé class Model theory Ramsey theory The KPT theory contains structures of arbitrary large …nite cardinality Questions Posets hereditary property (HP) A , ! B 2 K = ) A 2 K Ultrametric spaces joint embedding property (JEP) ( 8 A , B 2 K )( 9 C 2 K ) Results Posets [ A , ! C & B , ! C ] Ultrametric Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Fraïssé classes Ramsey classes Miodrag Soki´ c K -class of …nite structures in a countable signature L , closed Introduction Motivation under isomorphic images Topological dynamics Fraïssé class Model theory Ramsey theory The KPT theory contains structures of arbitrary large …nite cardinality Questions Posets hereditary property (HP) A , ! B 2 K = ) A 2 K Ultrametric spaces joint embedding property (JEP) ( 8 A , B 2 K )( 9 C 2 K ) Results Posets [ A , ! C & B , ! C ] Ultrametric amalgamation property (AP) ( 8 A , B , C 2 K )( 8 f : A ! B , g : A ! C )( 9 D 2 K )( 9 f 0 : B ! D , g 0 : C ! D )[ f 0 � f = g 0 � g ] Miodrag Soki´ c Ramsey classes

Motivation Introduction Topological dynamics Questions Model theory Results Ramsey theory The KPT theory Fraïssé structures Ramsey classes Miodrag Soki´ c Introduction Motivation Topological Fraïssé structure (countable signature L ) dynamics Model theory Ramsey theory in…nite countable The KPT theory Questions Posets Ultrametric spaces Results Posets Ultrametric Age ( A ) -( age )class of …nite structures embedable into A Miodrag Soki´ c Ramsey classes