گروه‌های جبری: ساختارها و عمل‌ها: سخنرانی‌های کلیفورد ۲۰۱۵ درباره گروه‌های جبری، ساختارها و عمل‌ها، ۲ تا ۵ مارس ۲۰۱۵، دانشگاه تولین، نیواورلئان، لوئیزیانا

This volume contains the proceedings of the 2015 Clifford Lectures on Algebraic Groups: Structures and Actions, held from March 2–5, 2015, at Tulane University, New Orleans, Louisiana. This volume consists of six articles on algebraic groups, including an enhanced exposition of the classical results of Chevalley and Rosenlicht on the structure of algebraic groups; an enhanced survey of the recently developed theory of pseudo-reductive groups; and an exposition of the recently developed operational $K$-theory for singular varieties. In addition, there are three research articles containing previously unpublished foundational results on birational automorphism groups of algebraic varieties; solution of Hermite-Joubert problem over $p$-closed fields; and cohomological invariants and applications to classifying spaces. The old and new results presented in these articles will hopefully become cornerstones for the future development of the theory of algebraic groups and applications. Graduate students and researchers working in the fields of algebraic geometry, number theory, and representation theory will benefit from this unique and broad compilation of fundamental results on algebraic group theory. Cover Title page Contents Preface Computing torus-equivariant K-theory of singular varieties 1. Introduction 2. Some background 3. Bivariant theories 4. Kimura’s exact sequences and the Kan extension property 5. Riemann-Roch theorems 6. Localization theorems 7. Other directions References Algebraic structures of groups of birational transformations 1. Introduction 2. Structures given by families of transformations 3. Flat families and scheme structure References The Hermite-Joubert problem over p-closed fields 1. Introduction 2. Geometry of the hypersurfaces X_{n,p} and Y_{n,p} 3. Proof of Theorem 1.3: (1) ⟹ (2) 4. Proof of Theorem 1.3: (2) ⟹ (3) 5. Proof of Theorem 1.3: (3) ⟹ (1) 6. Proof of Theorem 1.4 7. Density of rational points on hypersurfaces 8. Proof of Assertions (∗) and (∗∗) 9. Remarks on Theorems 1.3 and 1.4 10. The Hermite-Joubert problem for p=2 11. The Hermite-Joubert problem for p=3 12. When are there solutions to (1.1) and (1.2)? 13. Proof of Theorem 1.5 14. Beyond Theorem 1.5 Acknowledgements References Some structure theorems for algebraic groups 1. Introduction 2. Basic notions and results 2.1. Group schemes 2.2. Actions of group schemes 2.3. Linear representations 2.4. The neutral component 2.5. Reduced subschemes 2.6. Torsors 2.7. Homogeneous spaces and quotients 2.8. Exact sequences, isomorphism theorems 2.9. The relative Frobenius morphism 3. Proof of Theorem 1 3.1. Affine algebraic groups 3.2. The affinization theorem 3.3. Anti-affine algebraic groups 4. Proof of Theorem 2 4.1. The Albanese morphism 4.2. Abelian torsors 4.3. Completion of the proof of Theorem 2 5. Some further developments 5.1. The Rosenlicht decomposition 5.2. Equivariant compactification of homogeneous spaces 5.3. Commutative algebraic groups 5.4. Semi-abelian varieties 5.5. Structure of anti-affine groups 5.6. Commutative algebraic groups (continued) 6. The Picard scheme 6.1. Definitions and basic properties 6.2. Structure of Picard varieties 7. The automorphism group scheme 7.1. Basic results and examples 7.2. Blanchard’s lemma 7.3. Varieties with prescribed connected automorphism group References Structure and classification of pseudo-reductive groups 1. Introduction 1.1. Motivation 1.2. Initial definitions and examples 1.3. Terminology and notation 1.4. Simplifications and corrections 2. Standard groups and dynamic methods 2.1. Basic properties of pseudo-reductive groups 2.2. The standard construction 2.3. Dynamic techniques and pseudo-parabolic subgroups 3. Roots, root groups, and root systems 3.1. Root groups 3.2. Pseudo-simplicity and root systems 3.3. Open cell 4. Structure theory 4.1. Bruhat decomposition 4.2. Pseudo-completeness 4.3. Properties of pseudo-parabolic subgroups 5. Refined structure theory 5.1. Further rational conjugacy 5.2. General Bruhat decomposition 5.3. Relative roots 5.4. Applications of refined structure 6. Central extensions and standardness 6.1. Central quotients 6.2. Central extensions 7. Non-standard constructions 7.1. Groups of minimal type 7.2. Rank-1 groups and applications 7.3. A non-standard construction 7.4. Root fields and standardness 7.5. Basic exotic constructions 8. Groups with a non-reduced root system 8.1. Preparations for birational constructions 8.2. Construction via birational group laws 8.3. Properties of birational construction 9. Classification of forms 9.1. Automorphisms and Galois-twisting 9.2. Tits-style classification 10. Structural classification 10.1. Exceptional constructions 10.2. Generalized standard groups Acknowledgements References Index Invariants of algebraic groups and retract rationality of classifying spaces 1. Introduction 2. Galois cohomology 3. Retract rational varieties 4. Retract rational classifying spaces 5. Cohomology of classifying spaces 6. Invariants of algebraic groups 7. Degree 1 invariants with coefficients in Galois module 8. Brauer invariants 9. Invariants of degree 3 with coefficients in \Q/\Z(2) References Back Cover Contains the proceedings of the 2015 Clifford Lectures on Algebraic Groups, held March 2015. The six articles cover an enhanced exposition of the classical results of Chevalley and Rosenlicht on the structure of algebraic groups; an enhanced survey of the recently developed theory of pseudo-reductive groups; and an exposition of the recently developed operational $K$-theory for singular varieties.