Contemporary Mathematics, Volume: 689
2017; 189 pp; Softcover
Print ISBN: 978-1-4704-2847-1
This volume contains the proceedings of the AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory and the AMS Special Session on Spatial Graphs, both held from October 24?25, 2015, at California State University, Fullerton, CA.
Included in this volume are articles that draw on techniques from geometry and algebra to address topological problems about knot theory and spatial graph theory, and their combinatorial generalizations to equivalence classes of diagrams that are preserved under a set of Reidemeister-type moves.
The interconnections of these areas and their connections within the broader field of topology are illustrated by articles about knots and links in spatial graphs and symmetries of spatial graphs in S3 and other 3-manifolds.
Graduate students and research mathematicians interested in knot theory.
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Contemporary Mathematics, Volume: 690
2017; 322 pp; Softcover
Print ISBN: 978-1-4704-2256-1
This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27?29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s.
The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters.
This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas.
Graduate students and research mathematicians interested in set theory, foundations, and related areas.
Part of Australian Mathematical Society Lecture Series
Publication planned for: August 2017
availability: Not yet published - available from August 2017
format: Hardback
isbn: 9781108417365
format: Paperback
isbn: 9781108404952
Enumerative combinatorics, in its algebraic and analytic forms, is vital to many areas of mathematics, from model theory to statistical mechanics. This book, which stems from many years' experience of teaching, invites students into the subject and prepares them for more advanced texts. It is suitable as a class text or for individual study. The author provides proofs for many of the theorems to show the range of techniques available, and uses examples to link enumerative combinatorics to other areas of study. The main section of the book introduces the key tools of the subject (generating functions and recurrence relations), which are then used to study the most important combinatorial objects, namely subsets, partitions, and permutations of a set. Later chapters deal with more specialised topics, including permanents, SDRs, group actions and the Redfield?Polya theory of cycle indices, Mobius inversion, the Tutte polynomial, and species.
1. Introduction
2. Formal power series
3. Subsets, partitions and permutations
4. Recurrence relations
5. The permanent
6. q-analogues
7. Group actions and cycle index
8. Mobius inversion
9. The Tutte polynomial
10. Species
11. Analytic methods: a first look
12. Further topics
13. Bibliography and further directions
Index.
Part of Cambridge Studies in Advanced Mathematics
Publication planned for: September 2017
availability: Not yet published - available from September 2017
format: Paperback
isbn: 9781316639566
This up-to-date introduction to Griffiths' theory of period maps and period domains focusses on algebraic, group-theoretic and differential geometric aspects. Starting with an explanation of Griffiths' basic theory, the authors go on to introduce spectral sequences and Koszul complexes that are used to derive results about cycles on higher-dimensional algebraic varieties such as the Noether?Lefschetz theorem and Nori's theorem. They explain differential geometric methods, leading up to proofs of Arakelov-type theorems, the theorem of the fixed part and the rigidity theorem. They also use Higgs bundles and harmonic maps to prove the striking result that not all compact quotients of period domains are Kahler. This thoroughly revised second edition includes a new third part covering important recent developments, in which the group-theoretic approach to Hodge structures is explained, leading to Mumford?Tate groups and their associated domains, the Mumford?Tate varieties and generalizations of Shimura varieties.
Review of previous edition: 'This book, dedicated to Philip Griffiths, provides an excellent introduction to the study of periods of algebraic integrals and their applications to complex algebraic geometry. In addition to the clarity of the presentation and the wealth of information, this book also contains numerous problems which render it ideal for use in a graduate course in Hodge theory.' Mathematical Reviews
Part I. Basic Theory:
1. Introductory examples
2. Cohomology of compact Kahler manifolds
3. Holomorphic invariants and cohomology
4. Cohomology of manifolds varying in a family
5. Period maps looked at infinitesimally
Part II. Algebraic Methods:
6. Spectral sequences
7. Koszul complexes and some applications
8. Torelli theorems
9. Normal functions and their applications
10. Applications to algebraic cycles: Nori's theorem
Part III. Differential Geometric Aspects:
11. Further differential geometric tools
12. Structure of period domains
13. Curvature estimates and applications
14. Harmonic maps and Hodge theory
Part IV. Additional Topics:
15. Hodge structures and algebraic groups
16. Mumford?Tate domains
17. Hodge loci and special subvarieties
Appendix A. Projective varieties and complex manifolds
Appendix B. Homology and cohomology
Appendix C. Vector bundles and Chern classes
Appendix D. Lie groups and algebraic groups
References
Index.
Part of Cambridge Studies in Advanced Mathematics
Publication planned for: September 2017
availability: Not yet published - available from September 2017
format: Hardback
isbn: 9781107156371
format: Paperback
isbn: 9781316609880
The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev?Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert, and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi?Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev?Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch?Gabber?Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.
1. Quaternion algebras
2. Central simple algebras and Galois descent
3. Techniques from group cohomology
4. The cohomological Brauer group
5. Severi?Brauer varieties
6. Residue maps
7. Milnor K-theory
8. The Merkurjev-Suslin theorem
9. Symbols in positive characteristic
Appendix. A breviary of algebraic geometry
Bibliography
Index.