پلی‌توپ‌ها و هندسه گسسته: جلسه ویژه AMS درباره پلی‌توپ‌ها و هندسه گسسته، ۲۱-۲۲ آوریل ۲۰۱۸، دانشگاه شمال شرقی، بوستون، MA

The papers showcase the breadth of discrete geometry through many new methods and results in a variety of topics. Also included are survey articles on some important areas of active research. This volume is aimed at researchers in discrete and convex geometry and researchers who work with abstract polytopes or string C C-groups. It is also aimed at early career mathematicians, including graduate students and postdoctoral fellows, to give them a glimpse of the variety and beauty of these research areas. Topics covered in this volume include: the combinatorics, geometry, and symmetries of convex polytopes; tilings; discrete point sets; the combinatorics of Eulerian posets and interval posets; symmetries of surfaces and maps on surfaces; self-dual polytopes; string C C-groups; hypertopes; and graph coloring. Cover Title page Contents Preface The cd-index: A survey 1. Early history 2. Basic definitions 3. Inequalities 4. Computing the cd-index 5. Bruhat order 6. Algebras 7. Related parameters 8. Conclusion Acknowledgments References d-dimensional self-dual polytopes and Meissner polytopes 1. Introduction and basic terminology 2. Prismoidal constructions of self-dual polytopes 3. Examples of P_{km}, for d≥3 4. Metric embeddings of self-dual polytopes 5. The boundary points of the Reuleaux 4-simplex References On the ranks of string C-group representations for symplectic and orthogonal groups 1. Introduction 2. Orthogonal groups and their geometries 3. The rank 5 construction 4. Proof of Theorem 1.1 5. Concluding remarks Acknowledgments References Perfect colorings of regular graphs 1. Introduction 2. Characterization of color adjacency matrices 3. Counting lemmas 4. Implementation 5. Color adjacency matrices of k-regular graphs 6. Perfect colorings of Platonic graphs 7. Further questions Appendix A: All color adjacency matrices for 3-colorings Appendix B: Perfect colorings of the Platonic graphs Acknowledgments References Tverberg theorems over discrete sets of points Introduction 1. Tverberg numbers over discrete subsets of R2: Proof of Theorem 1 2. Tverberg numbers over \Z3: Proof of Theorem 2 3. Tverberg numbers over S’×\R^{k}: Proof of Theorem 3 4. A generalized fraction selection lemma: Proof of Theorem 4 Acknowledgments References The vertices of primitive zonotopes 1. Introduction 2. Geometric and combinatorial properties 3. An asymptotic estimate for the number of vertices of H1+(d,2) 4. Lower bounds on the number of vertices of H_{∞}(d,1) and H_{∞}+(d,1) 5. Upper bounds on the number of vertices of H_{∞}(d,1) and H_{∞}+(d,1) Acknowledgments References Barycenters of points in polytope skeleta 1. Introduction 2. Inhomogeneous skeleta 3. Unbalanced weights 4. Final remarks Acknowledgments References Two families of locally toroidal regular 4-hypertopes arising from toroids 1. Introduction 2. Preliminaries 3. A family of hypertopes arising from {4,3,4}_{(s,s,0)} 4. A family of hypertopes arising from {3,3,4,3}_{(s,0,0,0)} 5. Final remarks Acknowledgments References Self-polar polytopes 1. Introduction 2. Definitions and preliminaries 3. Self-polarity 4. In low dimensions 5. In higher and lower dimensions 6. The number of vertices of negatively self-polar polytopes 7. Modifications in the same dimension 8. Conclusions and further questions References Isomorphisms of maps on the sphere 1. Introduction 2. Preliminaries 3. Overview of the algorithm 4. From maps to homogeneous maps 5. Homogeneous maps 6. Complexity 7. Concluding remarks References Some enumeration relating to intervals in posets 1. Introduction and background 2. Interval posets and Ehrhart polynomials 3. Chains References String C-group representations of almost simple groups: A survey 1. Introduction 2. Preliminaries 3. Simple groups and rank three string C-group representations 4. Symmetric and alternating groups 5. Projective linear groups 6. Suzuki groups 7. Small Ree groups 8. Orthogonal and symplectic groups 9. Sporadic groups 10. Collateral results 11. C-groups Acknowledgments References Orientation-reversing symmetry of closed surfaces immersed in euclidean 3-space 1. The orbifold and Riemann–Hurwitz equations for G+ 2. The nine cases for G 3. The realization problem: Standard and Klein bottle models 4. The theorems 5. Chirality 6. Questions for further study Acknowledgments References Realizations of the 120-cell 1. Introduction 2. Realizations in general 3. Induced cosine vectors 4. The 600-cell 5. Coordinates and layer vector 6. The quotient H8 7. The quotient H5 8. Small dimensions 9. Dimension 16 –small simplex 10. Dimension 16 –staurotope 11. Dimension 24 12. Dimension 25 13. Dimensions 30 and 40 14. Dimension 48 15. Dimension 36 16. The classification Acknowledgments References Prescribing symmetries and automorphisms for polytopes 1. Introduction 2. Basic notions 3. Preassigning symmetry groups 4. Prescribing involutions as central symmetries 5. Some open problems Acknowledgments References The rhombic triacontahedron and crystallography 1. The overlap puzzle 2. Polyhedra and paralellohedra 3. The RT and non-convex parallelohedra 4. The RT and complex crystals 5. Next steps Acknowledgments References Tilings with congruent edge coronae 1. Introduction 2. Preliminaries 3. Properties of a normal tiling with congruent edge coronae 4. The main result 5. Conclusion and recommendations Acknowledgments References Back Cover This volume contains the proceedings of the International Conference on Algebra and Related Topics, held from July 2-5, 2018, at Mohammed V University, Rabat, Morocco. Linear reserver problems demand the characterization of linear maps between algebras that leave invariant certain properties or certain subsets or relations. One of the most intractable unsolved problems is Kaplansky's conjecture: every surjective unital invertibility preserving linear map between two semisimple Banach algebras is a Jordan homomorphism. Recently, there has been an upsurge of interest in nonlinear preservers, where the maps studied are no longer assumed linear but instead a weak algebraic condition is somehow involved through the preserving property. This volume contains several articles on various aspects of preservers, including such topics as Jordan isomorphisms, Aluthge transform, joint numerical radius on ▫$C^\ast$▫-algebras, advertible complete algebras, and Gelfand-Mazur algebras. The volume also contains a survey on recent progress on local spectrum-preserving maps. Several articles in the volume present results about weighted spaces and algebras of holomorphic or harmonic functions, including biduality in weighted spaces of analytic functions, interpolation in the analytic Wiener algebra, and weighted composition operators on non-locally convex weighted spaces This volume contains the proceedings of the AMS Special Session on Polytopes and Discrete Geometry, held from April 21–22, 2018, at Northeastern University, Boston, Massachusetts. The papers showcase the breadth of discrete geometry through many new methods and results in a variety of topics. Also included are survey articles on some important areas of active research. This volume is aimed at researchers in discrete and convex geometry and researchers who work with abstract polytopes or string $C$-groups. It is also aimed at early career mathematicians, including graduate students and postdoctoral fellows, to give them a glimpse of the variety and beauty of these research areas. Topics covered in this volume include: the combinatorics, geometry, and symmetries of convex polytopes; tilings; discrete point sets; the combinatorics of Eulerian posets and interval posets; symmetries of surfaces and maps on surfaces; self-dual polytopes; string $C$-groups; hypertopes; and graph coloring. Contains the proceedings of the AMS Special Session on Polytopes and Discrete Geometry, held in April 2018, at Northeastern University. The papers showcase the breadth of discrete geometry through many new methods and results in a variety of topics. Also included are survey articles on some important areas of active research.