automorphism groups and ramsey properties of sparse graphs
play

Automorphism Groups and Ramsey Properties of Sparse Graphs Jan Hubi - PowerPoint PPT Presentation

Jun 09, 2023 •109 likes •1.26k views

Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Automorphism Groups and Ramsey Properties of Sparse Graphs Jan Hubi cka Computer Science Institute of Charles University Charles

  1. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme Ramsey classes = ⇒ amalgamation classes ⇑ ⇓ expansions of homogeneous ⇐ = homogeneous structures ⇓ ⇑ ⇓ extremely amenable groups = ⇒ universal minimal flows Kechris, Pestov, Todorˇ cevi` c: Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups (2005)

  2. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme Ramsey classes = ⇒ amalgamation classes ⇑ ⇓ expansions of homogeneous ⇐ = homogeneous structures ⇓ ⇑ ⇓ extremely amenable groups = ⇒ universal minimal flows Kechris, Pestov, Todorˇ cevi` c: Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups (2005) Definition Let L ′ be language containing language L . A expansion (or lift) of L -structure A is L ′ -structure A ′ on the same vertex set such that all relations/functions in L ⊆ L ′ are identical.

  3. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Nešetˇ ril’s Classification Programme, 2005 Classification Programme Ramsey classes = ⇒ amalgamation classes ⇑ ⇓ expansions of homogeneous ⇐ = homogeneous structures ⇓ ⇑ ⇓ extremely amenable groups = ⇒ universal minimal flows Kechris, Pestov, Todorˇ cevi` c: Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups (2005) Definition Let L ′ be language containing language L . A expansion (or lift) of L -structure A is L ′ -structure A ′ on the same vertex set such that all relations/functions in L ⊆ L ′ are identical. Theorem (Nešetˇ ril, 1989) All homogeneous graphs have Ramsey expansion.

  4. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Milliken tree theorem: C-relations Ramsey’s theorem: rationals Product arguments

  5. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Milliken tree theorem: C-relations Permutations Equivalences Ramsey’s theorem: rationals Product arguments

  6. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Boolean algebras Semilattices Milliken tree theorem: C-relations Permutations Interval graphs Cyclic orders Equivalences Ramsey’s theorem: rationals Product arguments Interpretations

  7. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Boolean algebras Semilattices Milliken tree theorem: C-relations Permutations Interval graphs Unary functions Cyclic orders Equivalences Ramsey’s theorem: rationals Product arguments Interpretations Adding unary functions

  8. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Graham Rotschild Theorem: Parametric words Boolean algebras Semilattices Milliken tree theorem: C-relations Permutations Interval graphs Unary functions Free amalgamation classes Cyclic orders Equivalences − → Ramsey’s theorem: rationals Rel( L ) Product arguments Interpretations Adding unary functions Partite construction

  9. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Graham Rotschild Theorem: Parametric words Boolean algebras Semilattices Partial Steiner systems Milliken tree theorem: C-relations Metric spaces Permutations Interval graphs Unary functions Free amalgamation classes Cyclic orders Equivalences − → Ramsey’s theorem: rationals Rel( L ) Product arguments Interpretations Adding unary functions Partite construction

  10. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems Milliken tree theorem: C-relations Metric spaces Permutations Interval graphs Structures with unary functions Unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Product arguments Interpretations Adding unary functions Partite construction

  11. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Does every amalgamation class have a Ramsey expansion? Question (Bodirsky, Nešetˇ ril, Nguyen van Thé, Pinsker, Tsankov 2010) Does every amalgamation class have a Ramsey expansion? Yes: extend language by infinitely many unary relations; assign every vertex to unique relation.

  12. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Does every amalgamation class have a Ramsey expansion? Question (Bodirsky, Nešetˇ ril, Nguyen van Thé, Pinsker, Tsankov 2010) Does every amalgamation class have a Ramsey expansion? Yes: extend language by infinitely many unary relations; assign every vertex to unique relation. Definition (Nguyen van Thé) Let K be class of L -structures and K ′ be class of expansions of K . • K ′ is precompact wrt K if for every A ∈ K there are only finitely many expansions of A in K ′ .

  13. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Does every amalgamation class have a Ramsey expansion? Question (Bodirsky, Nešetˇ ril, Nguyen van Thé, Pinsker, Tsankov 2010) Does every amalgamation class have a Ramsey expansion? Yes: extend language by infinitely many unary relations; assign every vertex to unique relation. Definition (Nguyen van Thé) Let K be class of L -structures and K ′ be class of expansions of K . • K ′ is precompact wrt K if for every A ∈ K there are only finitely many expansions of A in K ′ . • K ′ has expansion property if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ .

  14. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Does every amalgamation class have a Ramsey expansion? Question (Bodirsky, Nešetˇ ril, Nguyen van Thé, Pinsker, Tsankov 2010) Does every amalgamation class have a Ramsey expansion? Yes: extend language by infinitely many unary relations; assign every vertex to unique relation. Definition (Nguyen van Thé) Let K be class of L -structures and K ′ be class of expansions of K . • K ′ is precompact wrt K if for every A ∈ K there are only finitely many expansions of A in K ′ . • K ′ has expansion property if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Theorem (Kechris, Pestov, Todorˇ cevi` c 2005, Nguyen van Thé 2012) For every amalgamation class K there exists, up to bi-definability, at most one Ramsey class K ′ of precompact expansions of K with expansion property.

  15. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Existence of precompact expansions Better question (Nguyen Van Thé) Does every ω -categorical structure have a precompact Ramsey expansion?

  16. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Existence of precompact expansions Better question (Nguyen Van Thé) Does every ω -categorical structure have a precompact Ramsey expansion?

  17. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Existence of precompact expansions Better question (Nguyen Van Thé) Does every ω -categorical structure have a precompact Ramsey expansion?

  18. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems Milliken tree theorem: C-relations Metric spaces Permutations Interval graphs Structures with unary functions Unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Product arguments Interpretations Adding unary functions Partite construction

  19. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems Models (Structures with functions) Milliken tree theorem: C-relations Metric spaces Permutations Interval graphs Structures with unary functions Unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Locally finite subclass Product arguments Interpretations Adding unary functions Partite construction

  20. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems S -metric spaces Models (Structures with functions) Milliken tree theorem: C-relations Metric spaces Permutations Cherlin Shelah Shi classes Metrically homogeneous graphs Interval graphs Structures with unary functions Unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Locally finite subclass Product arguments Interpretations Adding unary functions Partite construction

  21. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Gower’s Ramsey Theorem Dual structural Ramsey theorem Lelek fans Graham Rotschild Theorem: Parametric words Boolean algebras Boolean algebras with ideals Semilattices Partial Steiner systems Models (Structures with functions) Milliken tree theorem: C-relations S -metric spaces Metric spaces Permutations Cherlin Shelah Shi classes Metrically homogeneous graphs Interval graphs Structures with unary functions Structures with unary functions Free amalgamation classes Cyclic orders Partial orders Equivalences Acyclic graphs Line graphs − → Ramsey’s theorem: rationals Rel( L ) Locally finite subclass Product arguments Interpretations Adding unary functions Partite construction

  22. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Existence of precompact expansions Theorem (Evans, 2015+) There is a countable, ω -categorical structure M F no precompact Ramsey expansion. Counter-example was given by Hrushovski construction. In this talk we explore properties of this example.

  23. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Three variants of David’s example • C 0 : The easy example • C 1 : The kindergarten example • C F : The actual counter-example

  24. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 .

  25. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0.

  26. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. • G ⊆ H is self-sufficient, G ≤ s H , iff ∀ G ⊆ G ′ ⊆ H δ ( G ) ≤ δ ( G ′ ) .

  27. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . Example δ ( K 1 ) = 2 δ ( K 2 ) = 4 − 1 = 3 δ ( K 3 ) = 6 − 3 = 3 δ ( K 4 ) = 8 − 6 = 2 δ ( K 5 ) = 10 − 10 = 0 δ ( K 6 ) = 12 − 30 = − 18 . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. • G ⊆ H is self-sufficient, G ≤ s H , iff ∀ G ⊆ G ′ ⊆ H δ ( G ) ≤ δ ( G ′ ) . Lemma C 0 is closed for free amalgamation over self-sufficient C B substructures. A B ′ Proof. δ ( C ) = δ ( B ) + δ ( B ′ ) − δ ( A ) .

  28. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski class C 0 • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. Lemma (By marriage theorem) • G ∈ C 0 iff it has 2 -orientation (out-degrees at most 2). • H ≤ s G iff G can be 2 -oriented with no edge from H to G \ H .

  29. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski class C 0 • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. Lemma (By marriage theorem) • G ∈ C 0 iff it has 2 -orientation (out-degrees at most 2). • H ≤ s G iff G can be 2 -oriented with no edge from H to G \ H .

  30. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski class C 0 • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. Lemma (By marriage theorem) • G ∈ C 0 iff it has 2 -orientation (out-degrees at most 2). • H ≤ s G iff G can be 2 -oriented with no edge from H to G \ H .

  31. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski class C 0 • Predimension of a graph G = ( V , E ) is δ ( G ) = 2 | V | − | E | . • Finite graph G is in C 0 iff ∀ H ⊆ G δ ( H ) ≥ 0. Lemma (By marriage theorem) • G ∈ C 0 iff it has 2 -orientation (out-degrees at most 2). • H ≤ s G iff G can be 2 -oriented with no edge from H to G \ H . Corollary C 0 is a class of all finite 2 -orientations D 0 with directions forgotten. D 0 is closed for free amalgamation over successor-closed substructures.

  32. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Ramsey expansions of C 0 and orientations Theorem (Kechris, Pestov, Todorˇ cevi` c, 2005) Let F be a Fraïssé limit, then the following are equivalent. • Automorphism group of F is extremely amenable; • Age ( F ) has the Ramsey property.

  33. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Ramsey expansions of C 0 and orientations Theorem (Kechris, Pestov, Todorˇ cevi` c, 2005) Let F be a Fraïssé limit, then the following are equivalent. • Automorphism group of F is extremely amenable; • Age ( F ) has the Ramsey property. Denote by M 0 the generalised Fraïssé limit of C 0 . Theorem (Evans 2015) If M + 0 is a Ramsey expansion of M 0 , then Aut ( M + 0 ) fixes a 2 -orientation.

  34. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Ramsey expansions of C 0 and orientations Theorem (Kechris, Pestov, Todorˇ cevi` c, 2005) Let F be a Fraïssé limit, then the following are equivalent. • Automorphism group of F is extremely amenable; • Age ( F ) has the Ramsey property. Denote by M 0 the generalised Fraïssé limit of C 0 . Theorem (Evans 2015) If M + 0 is a Ramsey expansion of M 0 , then Aut ( M + 0 ) fixes a 2 -orientation. Proof. • Consider G acting on the space X ( M 0 ) of 2-orientations of M 0 (a G -flow). • As Aut ( M + 0 ) is extremely amenable, there is some S ∈ X ( M 0 ) which is fixed by Aut ( M + 0 ) . • Aut ( M + 0 ) is a subgroup of Aut ( S ) .

  35. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) .

  36. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) . • Let ( C + 0 , ⊑ ) be a Ramsey expansion of ( C 0 , ≤ s ) , then every A ∈ C 0 has infinitely many expansions in ( C + 0 ; ⊑ ) .

  37. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) . • Let ( C + 0 , ⊑ ) be a Ramsey expansion of ( C 0 , ≤ s ) , then every A ∈ C 0 has infinitely many expansions in ( C + 0 ; ⊑ ) . • Given two 2-orientations A ⊆ B , we write A ⊑ s B if there is no edge from A to B \ A . • ⊑ is coarser than ⊑ s for 2-orientation fixed by ( C + 0 , ⊑ ) .

  38. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) . • Let ( C + 0 , ⊑ ) be a Ramsey expansion of ( C 0 , ≤ s ) , then every A ∈ C 0 has infinitely many expansions in ( C + 0 ; ⊑ ) . • Given two 2-orientations A ⊆ B , we write A ⊑ s B if there is no edge from A to B \ A . • ⊑ is coarser than ⊑ s for 2-orientation fixed by ( C + 0 , ⊑ ) .

  39. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F No precompact Ramsey expansions of C 0 Theorem (Evans 2016) There is no precompact Ramsey expansion of ( C 0 ; ≤ s ) . • Let ( C + 0 , ⊑ ) be a Ramsey expansion of ( C 0 , ≤ s ) , then every A ∈ C 0 has infinitely many expansions in ( C + 0 ; ⊑ ) . • Given two 2-orientations A ⊆ B , we write A ⊑ s B if there is no edge from A to B \ A . • ⊑ is coarser than ⊑ s for 2-orientation fixed by ( C + 0 , ⊑ ) . Proof. • Every vertex v ∈ M + 0 has out-degree at most 2, but infinite in-degree. • Oriented path v 1 → v 2 → v 2 · · · v n always extended by a vertex v 0 to v 0 → v 1 → v 2 → v 2 · · · v n .

  40. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey Denote by D ≺ 0 the class of all finite ordered 2-orientations. Theorem (H., Evans, Nešetˇ ril, 2015+) D ≺ 0 is a Ramsey class.

  41. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey Denote by D ≺ 0 the class of all finite ordered 2-orientations. Theorem (H., Evans, Nešetˇ ril, 2015+) D ≺ 0 is a Ramsey class. Proof. → ( | B | ) | A | • Given A , B ∈ D ≺ 0 put N − 2 . • Extend language by unary predicates R 1 , R 2 , . . . R N . • Given | B | tuple � b = ( b 1 , b 2 , . . . b | B | ) , denote by B � b expansion of B where i -th vertex is in relation R b i . � n � • P 0 is a disjoint union of B � v , v ∈ . | B | • Put u ∼ v if successor-closure of u is isomorphic to v . → ( B ) A • C = P 0 / ∼ . C − 2 .

  42. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey B A

  43. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey B A 1 2 3 4 5

  44. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey B A 1 2 3 4 5

  45. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F D ≺ 0 is Ramsey B A 1 2 3 4 5

  46. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Optimality of Ramsey expansion Question: (Tsankov) Is ( D ≺ 0 ; ⊑ s ) any better than the trivial Ramsey expansion?

  47. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Optimality of Ramsey expansion Question: (Tsankov) Is ( D ≺ 0 ; ⊑ s ) any better than the trivial Ramsey expansion? Theorem (H., Evans, Nešetˇ ril, 2016+) There exists G 0 ⊂ D ≺ 0 such that • ( G 0 ; ⊑ s ) is strong expansion of ( C 0 ; ≤ s ) , • ( G 0 ; ⊑ s ) is Ramsey classes, • N G 0 , the group of automorphisms of Fraïssé limit of ( G 0 ; ⊑ s ) is maximal amongst extremely amenable subgroups of Aut ( M 0 ) . • Class of all self-sufficient substructures of G 0 has an Expansion Property with respect to C 0 and thus give a minimal Aut ( M 0 ) flow.

  48. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ .

  49. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) .

  50. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) . Theorem For every A + ∈ D 1 there exists B ∈ C 1 such that every expansion B + ∈ D 1 contains A + as a self-sufficient substructure.

  51. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) . Theorem For every A + ∈ D 1 there exists B ∈ C 1 such that every expansion B + ∈ D 1 contains A + as a self-sufficient substructure. Proof by induction on | A + | . v

  52. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) . Theorem For every A + ∈ D 1 there exists B ∈ C 1 such that every expansion B + ∈ D 1 contains A + as a self-sufficient substructure. Proof by induction on | A + | . v • Every A ∈ D 1 has vertex v of in-degree 0. • A 0 = A \ { v } .

  53. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Expasion property of non-precompactness Definition K ′ has expansion property wrt K if for every A ∈ K there exists B ∈ K such that every expansion of B in K ′ contains every expansion of A in K ′ . Denote by ( D 1 ; ⊑ s ) the class of all finite acyclic orientations. Denote by ( C 1 ; ⊑ s ) unoriented reduct of ( D 1 ; ⊑ s ) . Theorem For every A + ∈ D 1 there exists B ∈ C 1 such that every expansion B + ∈ D 1 contains A + as a self-sufficient substructure. Proof by induction on | A + | . v • Every A ∈ D 1 has vertex v of in-degree 0. • A 0 = A \ { v } . • Construct B 0 by induction hypothesis. • Extend every copy of A 0 in B 0 to A by 5 copies of v .

  54. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Extension property of non-precompact expansion Definition Suppose A ∈ D 1 we put A ∈ E 1 iff: 1 If l ( a ) ≺ l ( b ) . 2 If l ( a ) = l ( b ) then order is defined lexicographically by descending chains of their successors l ( a ) denote the level of vertex a .

  55. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Extension property of non-precompact expansion Definition Suppose A ∈ D 1 we put A ∈ E 1 iff: 1 If l ( a ) ≺ l ( b ) . 2 If l ( a ) = l ( b ) then order is defined lexicographically by descending chains of their successors l ( a ) denote the level of vertex a . Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure.

  56. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure. Proof.

  57. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure. Proof.

  58. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure. Proof.

  59. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Theorem (H., Evans, Nešetˇ ril, 2016+) For every A + ∈ E 1 there exists B ∈ C 1 such that every expansion B + ∈ E 1 contains A as self-sufficient substructure. Proof.

  60. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has no Hrushovski property Given strong class ( C ; ≤ ) , a strong partial automorphism of A ∈ C is an isomorphism f : D → E for some D , E ≤ A .

  61. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has no Hrushovski property Given strong class ( C ; ≤ ) , a strong partial automorphism of A ∈ C is an isomorphism f : D → E for some D , E ≤ A . Definition ( C , ≤ ) has the extension property for strong partial automorphisms (EPPA) if ∀ A ∈C ∃ B ∈C , A ≤ B such that every strong partial automorphism of A extends to an automorphism of B .

  62. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has no Hrushovski property Given strong class ( C ; ≤ ) , a strong partial automorphism of A ∈ C is an isomorphism f : D → E for some D , E ≤ A . Definition ( C , ≤ ) has the extension property for strong partial automorphisms (EPPA) if ∀ A ∈C ∃ B ∈C , A ≤ B such that every strong partial automorphism of A extends to an automorphism of B . Theorem (Evans, 2016, easier argument by Tsankov) Aut ( M 0 ) is not amenable and thus ( C 0 ; ≤ s ) has no EPPA. Explicit example given by Zaniar Ghadernezhad.

  63. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has no Hrushovski property Given strong class ( C ; ≤ ) , a strong partial automorphism of A ∈ C is an isomorphism f : D → E for some D , E ≤ A . Definition ( C , ≤ ) has the extension property for strong partial automorphisms (EPPA) if ∀ A ∈C ∃ B ∈C , A ≤ B such that every strong partial automorphism of A extends to an automorphism of B . Theorem (Evans, 2016, easier argument by Tsankov) Aut ( M 0 ) is not amenable and thus ( C 0 ; ≤ s ) has no EPPA. Explicit example given by Zaniar Ghadernezhad. Theorem (H., Evans, Nešetˇ ril, 2017+) The class of all finite 2 -orientations has EPPA.

  64. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has Hrushovski expansion Theorem (H., Evans, Nešetˇ ril, 2017+) Let D be a class of finite 2 -orientations closed for free amalgamation over successor-closed substructures. Then D has EPPA.

  65. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has Hrushovski expansion Theorem (H., Evans, Nešetˇ ril, 2017+) Let D be a class of finite 2 -orientations closed for free amalgamation over successor-closed substructures. Then D has EPPA. Proof. • Given A ∈ D construct B 0 ∈ D as follows: 1 Vertices of B 0 are pairs ( v , f ) where v ∈ A and f ∈ Sym ( B ) . 2 ( v , f ) → ( v ′ , f ′ ) iff f = f ′ and f ( v ) → f ( v ′ ) is edge of A .

  66. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has Hrushovski expansion Theorem (H., Evans, Nešetˇ ril, 2017+) Let D be a class of finite 2 -orientations closed for free amalgamation over successor-closed substructures. Then D has EPPA. Proof. • Given A ∈ D construct B 0 ∈ D as follows: 1 Vertices of B 0 are pairs ( v , f ) where v ∈ A and f ∈ Sym ( B ) . 2 ( v , f ) → ( v ′ , f ′ ) iff f = f ′ and f ( v ) → f ( v ′ ) is edge of A . • Put ( v , f ) ∼ ( v , f ′ ) iff there is isomorphism of successor-closures α of ( v , f ) and ( v , f ′ ) such that α ( u , h ) = ( h , h ′ ) . • B = B 0 / ∼ .

  67. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Hrushovski construction has Hrushovski expansion Theorem (H., Evans, Nešetˇ ril, 2017+) Let D be a class of finite 2 -orientations closed for free amalgamation over successor-closed substructures. Then D has EPPA. Proof. • Given A ∈ D construct B 0 ∈ D as follows: 1 Vertices of B 0 are pairs ( v , f ) where v ∈ A and f ∈ Sym ( B ) . 2 ( v , f ) → ( v ′ , f ′ ) iff f = f ′ and f ( v ) → f ( v ′ ) is edge of A . • Put ( v , f ) ∼ ( v , f ′ ) iff there is isomorphism of successor-closures α of ( v , f ) and ( v , f ′ ) such that α ( u , h ) = ( h , h ′ ) . • B = B 0 / ∼ . Along with Herwig-Lascar theorem this also shows EPPA for unary Cherlin-Shelah-Shi classes and more. Can be also combined with Hodkinson-Otto construction to obtain irreducible-structure faithful EPPA.

  68. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Summary 1 ( C 1 ; ≤ s ) (reducts of acyclic 2-orientations) ¬ Ramsey, ¬ EPPA, AP 2 ( D 1 ; ⊑ s ) (acyclic 2-orientations) EP wrt C 1 , ¬ Ramsey, EPPA, Minimal flow, AP

  69. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Summary 1 ( C 1 ; ≤ s ) (reducts of acyclic 2-orientations) ¬ Ramsey, ¬ EPPA, AP 2 ( D 1 ; ⊑ s ) (acyclic 2-orientations) EP wrt C 1 , ¬ Ramsey, EPPA, Minimal flow, AP 3 ( D ≺ 1 ; ⊑ s ) (ordered acyclic 2-orientations) ¬ EP wrt C 1 nor D 1 , Ramsey, ¬ EPPA, ¬ Minimal flow, AP

  70. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Summary 1 ( C 1 ; ≤ s ) (reducts of acyclic 2-orientations) ¬ Ramsey, ¬ EPPA, AP 2 ( D 1 ; ⊑ s ) (acyclic 2-orientations) EP wrt C 1 , ¬ Ramsey, EPPA, Minimal flow, AP 3 ( D ≺ 1 ; ⊑ s ) (ordered acyclic 2-orientations) ¬ EP wrt C 1 nor D 1 , Ramsey, ¬ EPPA, ¬ Minimal flow, AP 4 ( E 1 ; ⊑ s ) (admisively ordered acyclic 2-orientations) EP wrt C 1 and D 1 Ramsey, ¬ EPPA, ¬ Minimal flow, AP

  71. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Summary 1 ( C 1 ; ≤ s ) (reducts of acyclic 2-orientations) ¬ Ramsey, ¬ EPPA, AP 2 ( D 1 ; ⊑ s ) (acyclic 2-orientations) EP wrt C 1 , ¬ Ramsey, EPPA, Minimal flow, AP 3 ( D ≺ 1 ; ⊑ s ) (ordered acyclic 2-orientations) ¬ EP wrt C 1 nor D 1 , Ramsey, ¬ EPPA, ¬ Minimal flow, AP 4 ( E 1 ; ⊑ s ) (admisively ordered acyclic 2-orientations) EP wrt C 1 and D 1 Ramsey, ¬ EPPA, ¬ Minimal flow, AP 5 ( E ′ 1 ; ≤ s ) (All self sufficient substructures of E 1 ) EP wrt C 1 and D 1 ¬ Ramsey, ¬ EPPA, Minimal flow, ¬ AP

  72. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Summary 1 ( C 0 ; ≤ s ) (reducts of 2-orientations) ¬ Ramsey, ¬ EPPA, AP 2 ( D − 0 ; ⊑ s ) (2-orientations with strongly connected components unoriented) EP wrt C 0 ¬ Ramsey, EPPA, Minimal flow, AP 3 ( D 0 ; ⊑ s ) (2-orientations) ¬ EP wrt C 0 , ¬ Ramsey, EPPA, ¬ Minimal flow, AP 4 ( D ≺ 0 ; ⊑ s ) (ordered 2-orientations) ¬ EP wrt C 0 nor D 0 , Ramsey, ¬ EPPA, ¬ Minimal flow, AP

  73. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F Summary 1 ( C 0 ; ≤ s ) (reducts of 2-orientations) ¬ Ramsey, ¬ EPPA, AP 2 ( D − 0 ; ⊑ s ) (2-orientations with strongly connected components unoriented) EP wrt C 0 ¬ Ramsey, EPPA, Minimal flow, AP 3 ( D 0 ; ⊑ s ) (2-orientations) ¬ EP wrt C 0 , ¬ Ramsey, EPPA, ¬ Minimal flow, AP 4 ( D ≺ 0 ; ⊑ s ) (ordered 2-orientations) ¬ EP wrt C 0 nor D 0 , Ramsey, ¬ EPPA, ¬ Minimal flow, AP 5 ( E 0 ; ⊑ s ) (admissive orderings and 2-orientations) EP wrt C 0 but no D 0 , Ramsey, ¬ EPPA, ¬ Minimal flow, AP 6 ( E ′ 0 ; ≤ s ) (All self sufficient substructures of E 0 ) EP wrt C 0 but no D 0 , ¬ Ramsey, ¬ EPPA, Minimal flow, ¬ AP 7 ( D ′ 0 ; ≤ s ) (reducts E ′ 0 ) EP wrt C 0 but no D 0 , ¬ Ramsey, ¬ EPPA, Minimal flow, ¬ AP

  74. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F The ω -categorical case • F : R ≥ 0 → R ≥ 0

  75. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F The ω -categorical case • F : R ≥ 0 → R ≥ 0 • C 0 = { B : δ ( A ) ≥ 0 for all A ⊆ B } . C F = { B : δ ( A ) ≥ F ( | A | ) for all A ⊆ B } .

  76. Structural Ramsey Hrushovski construction Orientations Ramsey property Expansion property EPPA C F The ω -categorical case • F : R ≥ 0 → R ≥ 0 • C 0 = { B : δ ( A ) ≥ 0 for all A ⊆ B } . C F = { B : δ ( A ) ≥ F ( | A | ) for all A ⊆ B } . • A ≤ s B iff δ ( A ) ≤ δ ( B ′ ) for all finite B ′ with A ⊂ B ′ ⊆ B . A ≤ d B iff δ ( A ) < δ ( B ′ ) for all finite B ′ with A ⊂ B ′ ⊆ B .

Recommend

Automorphism groups and Ramsey properties of sparse graphs. David Evans Dept. of Mathematics,

Automorphism groups and Ramsey properties of sparse graphs. David Evans Dept. of Mathematics,

Automorphism groups and Ramsey properties of sparse graphs. David Evans Dept. of Mathematics, Imperial College London. 1 / 19 Joint work with Jan Hubi cka and Jaroslav Neet ril T HEMES : Automorphism groups of nice model-theoretic

777 views • 65 slides
On p -groups with automorphism groups of prescribed properties Luke Morgan The Centre for the

On p -groups with automorphism groups of prescribed properties Luke Morgan The Centre for the

Introduction Maximally symmetric p -groups The groups On p -groups with automorphism groups of prescribed properties Luke Morgan The Centre for the Mathematics of Symmetry and Computation The University of Western Australia Joint work with:

983 views • 48 slides
STABILIZERS OF VERTICES OF GRAPHS WITH PRIMITIVE AUTOMORPHISM GROUPS AND A STRONG VERSION OF THE

STABILIZERS OF VERTICES OF GRAPHS WITH PRIMITIVE AUTOMORPHISM GROUPS AND A STRONG VERSION OF THE

STABILIZERS OF VERTICES OF GRAPHS WITH PRIMITIVE AUTOMORPHISM GROUPS AND A STRONG VERSION OF THE SIMS CONJECTURE Anatoly S. Kondratev N.N. Krasovskii Institute of Mathematics and Mechanics of UB RAS and Ural Federal University,

396 views • 26 slides
Automorphism Groups of Compact Quantum Groups joint work with Alex Chirvasitu Issan Patri

Automorphism Groups of Compact Quantum Groups joint work with Alex Chirvasitu Issan Patri

Automorphism Groups of Compact Quantum Groups joint work with Alex Chirvasitu Issan Patri December 19, 2016 Chennai Mathematical Institute (CMI), Chennai Table of contents 1. Automorphism Groups of CQGs 2. Automorphism Groups of CQGs

448 views • 22 slides
Ramsey properties of random graphs Rajko Nenadov Monash University 29th August 2016

Ramsey properties of random graphs Rajko Nenadov Monash University 29th August 2016

Ramsey properties of random graphs Rajko Nenadov Monash University 29th August 2016 Introduction A graph G is Ramsey for H G H if every red/blue colouring of the edges of G contains a monochromatic copy of H Introduction A graph G is

1.48k views • 81 slides
Automorphism groups of Schur rings over cyclic groups of prime-power order Reinhard Pschel

Automorphism groups of Schur rings over cyclic groups of prime-power order Reinhard Pschel

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Automorphism groups of Schur rings over cyclic groups of prime-power order Reinhard Pschel joint work with Mikhail Klin Institut fr Algebra Technische

1.47k views • 98 slides
Symmetric cohomology of groups Mahender Singh Knots, Braids and Automorphism Groups Novosibirsk

Symmetric cohomology of groups Mahender Singh Knots, Braids and Automorphism Groups Novosibirsk

Symmetric cohomology of groups Mahender Singh Knots, Braids and Automorphism Groups Novosibirsk July 2014 Mahender Singh IISER Mohali Introduction Cohomology of groups is a contravariant functor turning groups and modules over groups into

809 views • 34 slides
Open Problems Concerning Automorphism Groups of Projective Planes G. Eric Moorhouse Department

Open Problems Concerning Automorphism Groups of Projective Planes G. Eric Moorhouse Department

Projective Planes Subplanes Orbits Open Problems Concerning Automorphism Groups of Projective Planes G. Eric Moorhouse Department of Mathematics University of Wyoming BIRS 25 April 2011 G. Eric Moorhouse Automorphism Groups of Projective

959 views • 39 slides
Automorphism breaking in locally finite graphs Florian Lehner Graz University of Technology

Automorphism breaking in locally finite graphs Florian Lehner Graz University of Technology

Automorphism breaking in locally finite graphs Florian Lehner Graz University of Technology CanaDAM Memorial University of Newfoundland June 10, 2013 Distinguishing Graphs Florian Lehner Automorphism breaking in locally finite graphs

613 views • 36 slides
Vertex-Minimal ertex-Minimal Planar Planar Graphs Graphs with with a Prescrib Prescribed ed

Vertex-Minimal ertex-Minimal Planar Planar Graphs Graphs with with a Prescrib Prescribed ed

Vertex-Minimal ertex-Minimal Planar Planar Graphs Graphs with with a Prescrib Prescribed ed Automorphism utomorphism Group Group C.J. Jones, Sarah E. Lubow, and Carlie J. Triplitt University of Texas at Tyler Automorphism Group of a

435 views • 27 slides
Automorphism orbit membership for right-angled Artin groups Matthew B. Day University of

Automorphism orbit membership for right-angled Artin groups Matthew B. Day University of

Automorphism orbit membership for right-angled Artin groups Matthew B. Day University of Arkansas matthewd@uark.edu GAGTA7@CCNY, May 30, 2013 Ar iv reference: 1211:0078, Full-featured peak reduction in right-angled Artin groups Matthew Day

874 views • 75 slides
Topological dynamics and ergodic theory of automorphism groups Alexander S. Kechris Harvard;

Topological dynamics and ergodic theory of automorphism groups Alexander S. Kechris Harvard;

Topological dynamics and ergodic theory of automorphism groups Alexander S. Kechris Harvard; November 18, 2013 Topological dynamics and ergodic theory of automorphism groups Introduction I will discuss some aspects of the topological dynamics

1.32k views • 109 slides
Computing automorphism groups and testing groups for isomorphism Derek Holt University of

Computing automorphism groups and testing groups for isomorphism Derek Holt University of

Computing automorphism groups and testing groups for isomorphism Derek Holt University of Warwick Questions, Algorithms, and Computations in Abstract Group Theory, Braunschweig, May 2013 Derek Holt (University of Warwick) Computing

685 views • 47 slides
Automorphism Groups of Homogeneous Structures Andrs Villaveces - Universidad Nacional de

Automorphism Groups of Homogeneous Structures Andrs Villaveces - Universidad Nacional de

Reconstructing models The Small Index Property SIP beyond first order Examples: quasiminimal classes, the Zilber field, j-invariants Automorphism Groups of Homogeneous Structures Andrs Villaveces - Universidad Nacional de Colombia - Bogot

947 views • 52 slides
Genericity, Infinitary Interpretations, and Automorphism Groups of Structures Russell Miller

Genericity, Infinitary Interpretations, and Automorphism Groups of Structures Russell Miller

Genericity, Infinitary Interpretations, and Automorphism Groups of Structures Russell Miller Queens College &amp; CUNY Graduate Center Southeastern Logic Symposium University of Florida 5 March 2017 (Joint work with Matthew Harrison-Trainor,

428 views • 22 slides
Topological simplicity of automorphism groups of some generic ab-initio structures Zaniar

Topological simplicity of automorphism groups of some generic ab-initio structures Zaniar

Review of Generic structures Review of automorphism group of a countable structure Topological simplicity Topological simplicity of automorphism groups of some generic ab-initio structures Zaniar Ghadernezhad Institut f ur Mathematische

339 views • 20 slides
Properties of the automorphism group and a probabilistic construction of a class of countable

Properties of the automorphism group and a probabilistic construction of a class of countable

Properties of the automorphism group and a probabilistic construction of a class of countable labeled structures Dragan Ma sulovi c and Igor Dolinka Department of Mathematics and Informatics University of Novi Sad, Serbia AAA 83 Novi

479 views • 24 slides
Interlacing Families I: Bipartite Ramanujan Graphs of all Degrees Nikhil Srivastava (Microsoft

Interlacing Families I: Bipartite Ramanujan Graphs of all Degrees Nikhil Srivastava (Microsoft

Interlacing Families I: Bipartite Ramanujan Graphs of all Degrees Nikhil Srivastava (Microsoft Research) Adam Marcus (Yale), Daniel Spielman (Yale) Expander Graphs Sparse regular well-connected graphs with many properties of random graphs.

1.31k views • 126 slides
Gallai-Ramsey Number of Graphs Yaping Mao School of Mathematics and Statistics Qinghai Normal

Gallai-Ramsey Number of Graphs Yaping Mao School of Mathematics and Statistics Qinghai Normal

Gallai-Ramsey Number of Graphs Yaping Mao School of Mathematics and Statistics Qinghai Normal University, Xining, Qinghai, China May 2019 Introduction Main results Outline 1 Introduction Classical Ramsey number Gallai Ramsey number under K 3

1.19k views • 105 slides
Automorphism Groups of Graphical Models and Lifted Variational Inference Hung Hai Bui 1 Tuyen N.

Automorphism Groups of Graphical Models and Lifted Variational Inference Hung Hai Bui 1 Tuyen N.

Automorphism Groups of Graphical Models and Lifted Variational Inference Hung Hai Bui 1 Tuyen N. Huynh 2 Sebastian Riedel 3 1 Laboratory for Natural Language Understanding, Nuance Communications 2 AI Center, SRI International 3 Department of

501 views • 26 slides
Automorphism groups of spaces with many symmetries Aleksandra Kwiatkowska University of Bonn

Automorphism groups of spaces with many symmetries Aleksandra Kwiatkowska University of Bonn

Ultrahomogeneous structures The pseudo-arc and the Lelek fan Projective Fra ss e theory Applications Automorphism groups of spaces with many symmetries Aleksandra Kwiatkowska University of Bonn September 23, 2016 Aleksandra

877 views • 67 slides
Spectral theory of automorphism groups in QFT Wojciech Dybalski (G ottingen) A A A 0 0

Spectral theory of automorphism groups in QFT Wojciech Dybalski (G ottingen) A A A 0 0

Spectral theory of automorphism groups in QFT Wojciech Dybalski (G ottingen) A A A 0 0 0 X 1 Outline 1. Particle content in QM and QFT. 2. Space translations in QM and QFT. 3. Spectral decomposition: A = A pp A pc

347 views • 17 slides
Constructing Small Generating Sets for the Multiplicative Groups of Algebras over Finite Fields

Constructing Small Generating Sets for the Multiplicative Groups of Algebras over Finite Fields

Constructing Small Generating Sets for the Multiplicative Groups of Algebras over Finite Fields Ming-Deh Huang, Lian Liu University of Southern California ISSAC16, July 19-22 Motivation Expander graphs are sparse graphs that are well

366 views • 22 slides
Extremal results for sparse pseudorandom graphs Yufei Zhao Massachusetts Institute of Technology

Extremal results for sparse pseudorandom graphs Yufei Zhao Massachusetts Institute of Technology

Extremal results for sparse pseudorandom graphs Yufei Zhao Massachusetts Institute of Technology Joint work with David Conlon and Jacob Fox Sparse extensions Extending classical results to sparse settings. For example: Sparse extensions

882 views • 55 slides

More recommend