نمایش‌های جبر، هندسه و فیزیک: سخنرانی‌های ممتاز موریس آسلندر و کنفرانس بین‌المللی، ۲۵-۳۰ آوریل ۲۰۱۸، وودز هول، ماساچوست (ریاضیات معاصر)

This volume contains selected expository lectures delivered at the 2018 Maurice Auslander Distinguished Lectures and International Conference, held April 25–30, 2018, at the Woods Hole Oceanographic Institute, Woods Hole, MA. Reflecting recent developments in modern representation theory of algebras, the selected topics include an introduction to a new class of quiver algebras on surfaces, called “geodesic ghor algebras”, a detailed presentation of Feynman categories from a representation-theoretic viewpoint, connections between representations of quivers and the structure theory of Coxeter groups, powerful new applications of approximable triangulated categories, new results on the heart of a $t$-structure, and an introduction to methods of constructive category theory. Cover Title page Contents Preface Examples of geodesic ghor algebras on hyperbolic surfaces 1. Introduction 2. Ghor algebras: Background and main results 3. Examples References Feynman categories and representation theory Introduction 1. Representations from a categorical viewpoint 2. Feynman categories 3. Constructions and examples 4. Modules and enriched Feynman categories 5. Bar, co–bar, Feynman transforms, & master equations 6. W-construction and cubical structures 7. Outlook Appendix A. Graph glossary and graphical Feynman categories Appendix B. Graph description of F+, F^{+gcp} and F^{hyp} Appendix C. Double categories, 2–categories and monoidal categories Appendix D. Model structures Acknowledgments References Preprojective roots and graph monoids of Coxeter groups Introduction 1. Graph monoid 2. Preprojective roots 3. Preprojective roots and finite Coxeter groups 4. Reduced c-admissible words References Approximable triangulated categories 1. Introduction 2. Background 3. Approximability—the intuition, which comes from D(R) 4. Measuring the complexity of an object 5. The formal definition of approximability 6. The main theorems 7. More about the strong generation of \dperfX and D^{b}_{coh}(X) 8. More about finite R–linear functors H:\big[\dperfX\big]^{op}⟶\ModR and ̃H:D^{b}_{coh}(X)⟶\ModR 9. The categories \dperfX and D^{b}_{coh}(X) determine each other Appendix A. Some dumb maps in D_{}C^{C’}(A), and the proof that the third map of the triangle is a cochain map Appendix B. The assumption that the short exact sequences of cochain complexes are degreewise split is harmless Appendix C. Translating the approach to derived categories we presented here to the more standard one in the literature Acknowledgments References Methods of constructive category theory Introduction 1. Category constructors 1.1. Computable categories 1.2. \Ab-categories 1.3. Additive closure 1.4. Homomorphism structures 1.5. Freyd category 1.6. Computing with Freyd categories 1.6.1. Equality of morphisms 1.6.2. Cokernels 1.6.3. Kernels 1.6.4. The abelian case 1.6.5. Homomorphisms 1.7. Computing natural transformations 2. Constructive diagram chases 2.1. Additive relations 2.2. Category of generalized morphisms 2.3. Computation rules 2.4. Cohomology 2.5. Snake lemma 2.6. Generalized homomorphism theorem 2.7. Computing spectral sequences References The HRS tilting process and Grothendieck hearts of t-structures 1. Introduction 2. Preliminaries 3. Projective and injective objects in the heart. Quasi-(co)tilting torsion pairs 4. When is the heart of a torsion pair a Grothendieck category? 5. Beyond the HRS case: Some recent results Acknowledgments References Back Cover This Volume Contains Selected Expository Lectures Delivered At The 2018 Maurice Auslander Distinguished Lectures And International Conference, Held April 25–30, 2018, At The Woods Hole Oceanographic Institute, Woods Hole, Ma. Reflecting Recent Developments In Modern Representation Theory Of Algebras, The Selected Topics Include An Introduction To A New Class Of Quiver Algebras On Surfaces, Called “geodesic Ghor Algebras”, A Detailed Presentation Of Feynman Categories From A Representation-theoretic Viewpoint, Connections Between Representations Of Quivers And The Structure Theory Of Coxeter Groups, Powerful New Applications Of Approximable Triangulated Categories, New Results On The Heart Of A T T-structure, And An Introduction To Methods Of Constructive Category Theory. Presents selected expository lectures delivered at the 2018 Maurice Auslander Distinguished Lectures and International Conference, held in April 2018. Topics include an introduction to a new class of quiver algebras on surfaces called 'geodesic ghor algebras', and a presentation of Feynman categories from a representation-theoretic viewpoint.