توپولوژی، هندسه و دینامیک: یادبود V. A. روخلین

From an August 2019 conference in St. Petersburg, Russia, 20 papers discuss aspects of mathematics of particular interest to Russian mathematician Rokhlin (1919-84). The topics include the amenability of groupoids and asymptotic invariance of convolution powers, the convergence of equilibrium measures corresponding to finite subgroups of infinite graphs: new examples, Poincaré polynomials of generic torus orbit closures in Schubert varieties, a geometric description of the Hochschild cohomology of group algebras, and maximally inflected real trigonal curves of Hirzebruch surfaces. Annotation ©2021 Ringgold, Inc., Portland, OR (protoview.com) Cover Title page Contents Preface V. A. Rokhlin (23 August 1919–3 December 1984), materials for the biography Bibliography of V. A. Rokhlin About V. A. Rokhlin Teaching mathematics to non-mathematicians Notes by Oleg Viro Vladimir Abramovich Rokhlin and algebraic topology 1. Introduction 2. Bordism groups 3. The signature and its applications 4. The signature of 4-dimensional manifolds 5. Framed bordism, and the Rokhlin and Milnor–Kervaire theorems 6. Thom spaces and Atiyah duality 7. The theories of complex bordism U_{*}(X) and cobordism U*(X) 8. The loop space of S3 and the coefficients of the Chen–Dold character 9. The signature of partially framed manifolds References Amenability of groupoids and asymptotic invariance of convolution powers Introduction 1. Amenable groupoids 2. Markov chains on groupoids and approximate invariance 3. Amenable actions References Slopes of links and signature formulas 1. Introduction 2. The signature formula 3. The slope 4. Slopes via C-complexes 5. Concordance invariance References B-rigidity of the property to be an almost Pogorelov polytope Introduction 1. Cohomology ring of a moment-angle manifold of a simple 3-polytope 2. B-rigidity of Pogorelov polytopes 3. Cohomological rigidity of the property to be an almost Pogorelov polytope 4. Generalization of the technique to almost Pogorelov polytopes 5. Remark Acknowledgments References The first homology of a real cubic is generated by lines 1. Introduction 2. The case of nodal cubics 3. Passing to nonsingular cubics 4. Concluding remarks Acknowledgments References Circular orders, ultra-homogeneous order structures, and their automorphism groups 1. Introduction 2. Some generalizations of (extreme) amenability 3. Circular order, topology, and inverse limits 4. Ultrahomogeneous actions on circularly ordered sets 5. The Fraïssé class of finite circularly ordered systems and the KPT theory 6. Automatic continuity and Roelcke precompactness 7. Some perspectives and questions 8. Appendix: Large ultrahomogeneous circularly ordered sets Acknowledgments References Convergence of equilibrium measures corresponding to finite subgraphs of infinite graphs: New examples 1. Introduction 2. Preliminary information and statement of the problem 3. Linear graphs 4. Existence of an irregular sequence 5. Existence of a regular sequence 6. Concluding remarks References Anti-symplectic involutions on rational symplectic 4-manifolds 1. Introduction 2. Tools 3. Proof outline References Dolbeault cohomology of complex manifolds with torus action 1. Introduction 2. Preliminaries: holomorphic foliations on complex manifolds 3. Fujiki foliations 4. Basic Dolbeault cohomology of the canonical foliations on complex moment-angle manifolds 5. Manifolds with maximal torus actions 6. Dolbeault cohomology of moment-angle manifolds Acknowledgment References Poincaré polynomials of generic torus orbit closures in Schubert varieties 1. Introduction 2. Backgrounds: Polytopes and projective toric varieties 3. Generic torus orbit closures in Schubert varieties and their Poincaré polynomials 4. Proof of Theorem 3.6 5. Concluding remarks Acknowledgment References Higher order Massey products and applications Introduction 1. Massey products in cohomology 2. Massey products and Lie algebras representations 3. k-step Massey products in Lie algebra cohomology 4. Non-trivial Massey products in Lie algebra cohomology 5. Massey products in Koszul homology of local rings 6. Massey products in Toric Topology and nonformality of polyhedral products Acknowledgments References Discreteness of deformations of cocompact discrete subgroups 1. Introduction 2. Preliminaries 3. Deformations and discreteness Acknowledgments References Topological isotopy and Cochran’s derived invariants 1. Introduction 2. Invariants 3. Realization 4. Rationality Acknowledgment References Geometric description of the Hochschild cohomology of group algebras 1. Introduction 2. The smooth version of Johnson’s problem 3. Hochschild (co)homology 4. Hochschild homology 5. Conclusion 6. Addendum: Comparison of homology and cohomology References A user’s guide to basic knot and link theory 1. Main definitions and results on knots 2. Main definitions and results on links 3. Some basic tools 4. The Gauss linking number modulo 2 via plane diagrams 5. The Arf invariant 6. Appendix: Proper colorings 7. Oriented knots and links and their connected sums 8. The Gauss linking number via plane diagrams 9. The Casson invariant 10. Alexander-Conway polynomial 11. Vassiliev-Goussarov invariants 12. Appendix: Some details Acknowledgments References Group actions: Entropy, mixing, spectra, and generic properties 1. Basic definitions 2. G-actions and spectra of boundary value problems 3. Entropy 4. Generic properties: Definition 5. Approximation of group actions 6. Cardinal-valued invariants of measure-preserving transformations 7. Spectral problems 8. Rokhlin’s multiple mixing problem 9. Linear extensions of dynamical systems: The spectral theory and MET References Rokhlin’s theorem, a problem and a conjecture Maximally inflected real trigonal curves on Hirzebruch surfaces 1. Introduction 2. Trigonal curves and elliptic surfaces 3. Dessins 4. Skeletons 5. A constructive description of maximally inflected trigonal curves 6. Rigid isotopies and week equivalence Acknowledgment References Back Cover V.A. Rokhlin, materials for the biography / A.M. Vershik -- Teaching mathematics to non-mathematicians / V.A. Rokhlin -- Vladimir Abramovich Rokhlin and algebraic topology / V. M. Buchstaber -- Amenability of groupoids and asymptotic invariance of convolution powers / T. Buhler and V.A. Kaimanovich -- Slopes of links and signature formulas / A. Degtyarev, V. Florens, and A.G. Lecuona -- B-rigidity of the property to be an almost Pogorelov polytope / N. Yu. Erokhovets -- The first homology of a real cubic is generated by lines / S. Finashin and V. Kharlamov -- Circular orders, ultra-homogeneous order structures, and their automorphism groups / E. Glasner and M. Megrelishvili -- Convergence of equilibrium measures corresponding to finite subgraphs of infinite graphs : new examples / B.M. Gurevich -- Anti-symplectic involutions on rational symplectic 4-manifolds / V. Kharlamov and V. Shevchishin -- Dolbeault cohomology of complex manifolds with torus action / R. Krutowski and T. Panov -- Poincaré polynomials of generic torus orbit closures in Schubert varieties / E. Lee, M. Masuda, S. Park, and J. Song -- Higher order Massey products and applications / I. Limonchenko and D. Millionshchikov -- Discreteness of deformations of cocompact discrete subgroups / G.A. Margulis and G.A. Soifer -- Topological isotopy and Cochran's derived invariants / S.A. Melikhov -- Geometric description of the Hochschild cohomology of group algebras / A.S. Mishchenko -- A user's guide to basic knot and link theory / A. Skopenkov -- Group actions : entropy, mixing, spectra, and generic properties / A. Stepin and S. Tikhonov -- Rokhlin's theorem, a problem and a conjecture / D. Sullivan -- Maximally inflected real trigonal curves on Hirzebruch surfaces / V.I. Zvonilov Vladimir Abramovich Rokhlin (8/23/1919–12/03/1984) was one of the leading Russian mathematicians of the second part of the twentieth century. His main achievements were in algebraic topology, real algebraic geometry, and ergodic theory. The volume contains the proceedings of the Conference on Topology, Geometry, and Dynamics: V. A. Rokhlin-100, held from August 19–23, 2019, at The Euler International Mathematics Institute and the Steklov Institute of Mathematics, St. Petersburg, Russia. The articles deal with topology of manifolds, theory of cobordisms, knot theory, geometry of real algebraic manifolds and dynamical systems and related topics. The book also contains Rokhlin's biography supplemented with copies of actual very interesting documents. Vladimir Abramovich Rokhlin (1919-1984) was one of the leading Russian mathematicians of the second part of the twentieth century. Topics covered in this volume include topology of manifolds, theory of cobordisms, knot theory, geometry of real algebraic manifolds and dynamical systems and related topics.