Softcover
Softcover ISBN: 978-1-4704-7176-7
Product Code: MBK/150
Expected availability date: September 21, 2023
Aspiring and Inspiring is a collection of essays from successful women and gender minority mathematicians on what it takes to build a career in mathematics. The individual essays are intended to advise, encourage, and inspire mathematicians throughout different stages of their careers. Themes emerge as these prominent individuals describe how they managed to persist and rise to positions of leadership in a field which can still be forbidding to many. We read, repeatedly, that individual mentorship matters, that networks of support can be critical, and that finding fulfillment can mean formulating one's own definition of success. Those who aspire to leadership in the field will find much useful advice here.
The cumulative power of the collection carries a strong impact. The glass ceiling is very real in mathematics and is the result of cultural and sociological factors at work in our community. The book makes clear that we won't achieve equality of opportunity merely by exhorting those who are often excluded to change their behaviors and their goals. The need for systemic cultural change is vividly, at times painfully, evident in these stories. As Dr. Erica Graham says in her powerful and moving essay, we need “a different kind of academy”, and we'll only get it by working for it. We can start by reading this book and recognizing the kind of academy we currently have.
Graduate students and researchers interested in professional advancement.
Contents
Preface
Chapter 1. A path to success: Enabled by guardians, connectors and taking adventurous steps (Jacqueline M. Dewar)
Chapter 2. Trust yourself! (Alicia Dickenstein)
Chapter 3. The fight within (Maria Emelianenko)
Chapter 4. Intersectionality as impetus and impediment (Erica J. Graham)
Chapter 5. Find your passion, get organized & cultivate support systems (Allison Henrich)
Chapter 6. Tenure: The rules of the game (Rhonda J. Hughes)
Chapter 7. Make yourself valuable (Jacqueline Jensen-Vallin)
Chapter 8. It’s a wonderful life! - Reflections on the career of a mathematician (Luise-Charlotte Kappe)
Chapter 9. Success is a relative term (Kathryn Leonard)
Chapter 10. Multifaceting: Shattering your own glass ceiling (Perla Myers)
Chapter 11. Positioning yourself to crack the glass ceiling (Catherine A. Roberts)
Chapter 12. I am not your typical role model (or do not follow my steps) (Ivelisse M. Rubio)
Chapter 13. How I became a department head (Irena Swanson)
Chapter 14. Around the glass ceiling (Karen Uhlenbeck)
Chapter 15. The professional advancement of women mathematicians (Shanise Walker)
Chapter 16. Branch cuts: Writing, editing, and ramified complexities (Ursula Whitcher)
Chapter 17. Reach as you climb: Searching for purpose and meaning in academia (Cynthia Wyels)
Credits
Contemporary Mathematics, Volume: 788; 2023
Softcover ISBN: 978-1-4704-7147-7
Expected availability date: August 07, 2023
This volume contains the proceedings of the Alexandre Vinogradov Memorial Conference on Diffieties, Cohomological Physics, and Other Animals, held from December 13?17, 2021, at the Independent University of Moscow and Moscow State University, Moscow, Russia.
The papers are devoted to various interrelations of nonlinear PDEs with geometry and integrable systems. The topics discussed are: gravitational and electromagnetic fields in General Relativity, nonlocal geometry of PDEs, Legendre foliated cocycles on contact manifolds, presymplectic gauge PDEs and Lagrangian BV formalism, jet geometry and high-order phase transitions, bi-Hamiltonian structures of KdV type, bundles of Weyl structures, Lax representations via twisted extensions of Lie algebras, energy functionals and normal forms of knots, and differential invariants of inviscid flows.
The companion volume (Contemporary Mathematics, Volume 789) is devoted to Algebraic and Cohomological Aspects of PDEs.
Graduate students and research mathematicians interested in modern theory of partial differential equation and its relations with differential geometry, integrable systems, fluid mechanics, and field theory.
Articles
Fatemeh Ahangari - Geometric analysis of metric Legendre foliated cocycles on contact manifolds via SODE structure
Dmitri V. Alekseevsky, Alessio Marrani and Andrea Spiro - Special Vinberg cones, invariant admissible cubics and special real manifolds
Andreas ?ap and Jan Slovak - Bundles of Weyl structures and invariant calculus for parabolic geometries
Vladimir N. Chetverikov - Coverings and pseudosymmetries of differential equations
Anna Duyunova - Differential invariants of inviscid flows in pipes
Maxim Grigoriev - Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles
G. F. Helminck and E. A. Panasenko - Minimal realizations of the KP hierarchy, its strict version and their reductions
Josef Jany?ka and Marco Modugno - Minimal coupling of gravitational and electromagnetic fields in General Relativity
Paolo Lorenzoni and Raffaele Vitolo - Projective-geometric aspects of bi-Hamiltonian structures of KdV type
Valentin V. Lychagin - Measurement of random operators, jet geometry and high-order phase transitions
Gianni Manno, Jan Schumm and Andreas Vollmer - Metrics admitting projective and c-projective vector fields
Oleg I. Morozov - Lax representations via twisted extensions of infinite-dimensional Lie algebras: some new results
A. B. Sossinsky - Energy functionals and normal forms of knots and plane curves
Softcover ISBN: 978-1-4704-7051-7
Expected availability date: September 06, 2023
This book is a collection of essays written by a distinguished mathematician with a very long and successful career as a researcher and educator working in many areas of pure and applied mathematics. The author writes about everything he found exciting about math, its history, and its connections with art, and about how to explain it when so many smart people (and children) are turned off by it. The three longest essays touch upon the foundations of mathematics, upon quantum mechanics and Schrodinger's cat phenomena, and upon whether robots will ever have consciousness. Each of these essays includes some unpublished material. The author also touches upon his involvement with and feelings about issues in the larger world. The author's main goal when preparing the book was to convey how much he loves math and its sister fields.
Undergraduate students interested in mathematics in science and society
The Math Behind the Cover
Preface: Confessions of a Polymath
Part 1. Opening More Eyes to Mathematics
Chapter 1. How to Get Middle School Students to Love Formulas & Triangles
i. Algebra
ii. Geometry
Chapter 2. Explaining Grothendieck to Non-mathematicians
i. Nature magazine vs. rings & schemes
ii. A geologist vs. ??? & topoi
Chapter 3. Are Mathematical Formulas Beautiful?
i. Equations as art
ii. Equations reflected in MRI scans and mathematical tribes
Part 2. The History of Mathematics
Chapter 4. Pythagoras’s Rule
i. Its discovery
ii. How did it spread and was it rediscovered?
Chapter 5. The Checkered History of Algebra
i. Babylon
ii. Greece
iii. China
iv. India
v. Early modern Europe
vi. Today
Chapter 6. Multi-cultural Math History in Five Slides
Chapter 7. “Modern” Art/“Modern” Math and the Zeitgeist
i. Beauty and power through randomness
ii. When did abstract, non-figurative art & math start?
iii. Brave new worlds
iv. Full-blown abstraction
Interlude: Intelligent Design in Orion?
Part 3. AI, Neuroscience, and Consciousness
Chapter 8. Parse Trees Are Ubiquitous in Thinking
i. Language
ii. Vision
iii. Actions and plans
iv. The big picture
Chapter 9. Linking Deep Learning and Cortical Functions
i. Neural nets
ii. Tokens vs. distributed data
iii. Transformers and context
iv. Context in the brain
v. What is missing?
Chapter 10. Does/Can Human Consciousness Exist in Animals and Robots?
i. What do neuroscientists say about consciousness?
ii. Consciousness in animals
iii. We need emotions #$@*&!
iv. What do physicists say about consciousness?
v. The philosopher and the sage
Part 4. And Now, Some Bits of Real Math
Chapter 11. Finding the Rhythms of the Primes
Chapter 12. Spaces of Shapes and Rogue Waves
i. Nonlinear gravity waves
ii. Shape spaces
iii. Zakharov’s Hamiltonian
Chapter 13. An Applied Mathematician’s Foundations of Math
i. A warm-up: Arithmetic
ii. Being conservative with second-order arithmetic
iii. The standard foundation: ZFC
iv. The applied perspective
Part 5. Coming to Terms with the Quantum
Chapter 14. Quantum Theory and the Mysterious Collapse
i. Background: Measurements and ‘Copenhagen’
ii. AMU sets
iii. Constraints on macroscopic variables
iv. Molecules
v. Fields
vi. DNA
vii. Bohr bubbles and speculations
Chapter 15. Path Integrals and Quantum Computing
Part 6. Nothing Is Simple in the Real World
Chapter 16. Wake Up!
i. Springer and Klaus Peters
ii. The impact of the internet
Chapter 17. One World or Many?
i. My own experiences
ii. Russia and Shafarevich
iii. India and castes
Chapter 18. Spinoza: Euclid, Ethics, Time
i. Spinoza and substances
ii. A short history of dualism and substances
iii. Spinoza’s Ethics
iv. Relations to various religions and to modern science
Chapter 19. Thoughts on the Future
i. The population explosion
ii. The consequences of this explosion
iii. A safety valve?
iv. Love those robots
v. Playing God with the genome
vi. Unknowns
Figure Credits
Author’s Bibliography
Bibliography
Contemporary Mathematics,Volume: 789; 2023
Softcover ISBN: 978-1-4704-7355-6
Expected availability date: September 07, 2023
This volume contains the proceedings of the Alexandre Vinogradov Memorial Conference on Diffieties, Cohomological Physics, and Other Animals, held from December 13?17, 2021, at Independent University of Moscow and Moscow State University, Moscow, Russia.
The papers reflect the modern interplay between partial differential equations and various aspects of algebra and computer science. The topics discussed are: relations between integrability and differential rings, supermanifolds, differential calculus over graded algebras, noncommutative generalizations of PDEs, quantum vector fields, generalized Nijenhuis torsion, cohomological approach to the geometry of differential equations, the argument shift method, Frolicher structures in the formal Kadomtsev?Petviashvili hierarchy, and computer-based determination of optimal systems of Lie subalgebras.
The companion volume (Contemporary Mathematics, Volume 788) is devoted to Geometry and Mathematical Physics.
Graduate students and research mathematicians interested in the modern theory of partial differential equation in the wide context of commutative and noncommutative algebra and cohomology theories.
Luca Amata and Francesco Oliveri - Automatic Determination of Optimal Systems of Lie Subalgebras: the Package SymboLie
Orest D. Artemovych, Denis L. Blackmore, Rados?aw A. Kycia and Anatolij K. Prykarpatski - New Dubrovin-type integrability theory applications of differential rings
I. A. Bobrova and V. V. Sokolov - Non-abelian Painleve systems with generalized Okamoto integral
Alexei Bocharov - Mathematical Etudes on Quantum Computation
Dimitry Gurevich and Pavel Saponov - Quantum vector fields via quantum doubles and their applications
Hovhannes M. Khudaverdian - Non-linear homomorphisms of algebras of functions are induced by thick morphisms
H. M. Khudaverdian and Th. Th. Voronov - On the Buchstaber?Rees theory of “Frobenius n
-homomorphisms” and its generalization
Jacob Kryczka - Differential Calculus over Graded Commutative Algebras and Vector Bundles with Inner Structures
Fabrizio Pugliese, Giovanni Sparano and Luca Vitagliano - Vinogradov’s cohomological geometry of partial differential equations
Jean-Pierre Magnot, Enrique G. Reyes and Vladimir Rubtsov - Frolicher structures, diffieties, and a formal KP hierarchy
G. Sharygin - Quasi-derivations on Ugln
and the argument shift method
Piergiulio Tempesta and Giorgio Tondo - Polarization of generalized Nijenhuis torsions
Softcover ISBN: 978-1-4704-7247-4
Expected availability date: October 06, 2023
The Mathematical Neighborhoods of School Mathematics visits regions beyond, but proximal to and accessible from school mathematics. Its aim is to give readers a glimpse of not just the rich diversity and adaptability of mathematics, but, most importantly, its interconnections and overall coherence, a perspective not easily available from the school curriculum. This aim entailed assembling a variety of substantial mathematical domains that do not typically cohabit the same volume. The book begins with an in-depth treatment of topics in the school curriculum, often with novel approaches and connections. A unifying thread is the group theoretic study of addition and multiplication in the various number systems of school mathematics. The exposition is mathematically rigorous, including proofs of many fundamental theorems not otherwise easily available in mathematically accessible form.
The Mathematical Neighborhoods of School Mathematics is intended to be a conceptual contribution to mathematics education. It will be a valuable resource in professional development of mathematics teachers, and in mathematical enrichment programs, for both students and teachers. In this regard, many of the chapters are relatively self-contained. It could also serve as a text for undergraduate mathematics majors with an interest in teaching. The exceptional Chapter 11 presents some novel instructional designs for problem-solving activities meant to cultivate “connection-oriented mathematical thinking.”
Hyman Bass is the Samuel Eilenberg Distinguished University Professor of Mathematics and Mathematics Education at the University of Michigan. He is a member of the National Academy of Sciences and of the National Academy of Education.
Jason Brasel, a former high school mathematics teacher, is a mathematics educator and researcher in secondary mathematics, who works at TeachingWorks, University of Michigan.
Pre-service and in-service secondary teachers (and faculty who teach them) interested in expanding their knowledge of the deeper mathematics underlying what they teach.
Contents
Preface
Introduction
Acknowledgments
Chapter 0. Basic Notation and Background
0.1. Sets and set notation
0.2. The basic numbers systems
0.3. Logical Notation and Relations
0.4. Mathematical induction
0.5. Functions
0.6. Orbit Structure
0.7. Counting sets: combinatorics
0.8. Geometric Series
0.9. Exponentials
Chapter 1. Division with Remainder, Place Value, and Order of Magnitude
1.1. Division with Remainder (DwR)
1.2. The Euclidean Algorithm
1.3. Base-10 place value representation
1.4. Base-?? representations of real numbers
1.5. Order of magnitude and significant digits
1.6. Exercises
Chapter 2. Modular congruence and base-?? expansions of rational numbers
2.1. Modular congruence
2.2. Modular congruence of arithmetic expressions
2.3. Modular congruence and base-??: A fast track to ???(??)
2.4. The base-?? representation of a rational number
2.5. Exercises
Chapter 3. Discrete Real Additive Groups
3.1. Metric Spaces
3.2. Additive groups and translation-invariant metrics
3.3. Discrete real additive groups
3.4. Discrete subgroups of \rn
3.5. Commensurability, gcd, and lcm
3.6. Some basic properties of gcd and \lcm
3.7. The group \thereals^{?} and its discrete subgroups
3.8. Appendix: Frobenius numbers
3.9. Appendix: Compactness and connectedness
3.10. Exercises
Chapter 4. Primes and Factorization
4.1. Primes and prime factorization
4.2. Prime power factorization
4.3. Prime-valued polynomials
4.4. Cryptography
4.5. Exercises
Chapter 5. Group Theory and Euclidean Isometry Groups
5.1. Binary operations
5.2. Semigroups, groups, and subgroups
5.3. Homomorphisms, isomorphisms, automorphisms
5.4. G-sets
5.5. Quotient groups ??/??
5.6. Torsion: elements of finite order
5.7. Finite cyclic groups
5.8. Isometries of \thereals? (???3)
5.9. The Euclidean isometry groups ??(??)=????????(\thereals?)
5.10. Appendix: Platonic solids and finite isometry groups of \thereals3
5.11. Appendix: ??-groups and Sylow Theorems
5.12. Exercises
Chapter 6. Rings and Fields
6.1. Rings and fields
6.2. Quotient rings
6.3. Zero-divisors, nilpotents, idempotents, units, and primes
6.4. Chinese remainder theorem
6.5. The multiplicative group (\ints/\ints??)^{?}
6.6. Quadratic Reciprocity
6.7. Polynomial Rings
6.8. Unique Factorization
6.9. Unique Factorization Domains (UFDs)
6.10. Partial Fractions
6.11. Appendix: The ABC Conjecture
Polynomial ABC; Mason’s Theorem
The (Integer) ABC Conjecture (ABC-C)
Relations to Fermat’s Last Theorem (FLT)
6.12. Wieferich Primes; Primitive ??-roots mod ???
6.13. Gaps between integer powers; Catalan and Pillai Conjectures
6.14. Integers with the same prime divisors; The Erd?s-Woods Conjecture
6.15. Waring’s Problem
6.16. Exercises
Chapter 7. Polynomials
7.1. Polynomial division with remainder (PDwR) and Roots
7.2. Polynomial interpolation
7.3. Binomial Theorem and Multinomial Theorem
7.4. The Inclusion-Exclusion Formula (I-EF)
7.5. Derangements
7.6. Symmetric functions
7.7. Multi-linearity and symmetry
7.8. Integer polynomial dynamics
7.9. Complex polynomial dynamics
7.10. The landscape of polynomial equations
7.11. Exercises
Chapter 8. Combinatorics
8.1. What is combinatorics?
8.2. Binomial coefficients and the set of injections [??]→[??]
8.3. The Frobenius Endomorphism and Binomial congruences
8.4. Graphs and trees. ??-??+??=2
8.5. Multinomial coefficients and surjections [??]→[??]
8.6. Partitions
8.7. Mobius inversion
8.8. Sperner’s theorem
8.9. Binary sequences and Catalan numbers ??(??)
8.10. Some of the many incarnations of ??(??)
8.11. Discrete Probability
8.12. Exercises
Chapter 9. Discrete Calculus
9.1. Sequences as functions ??:\nats→\thereals; Δ,??:\RN→\RN
9.2. The Fundamental Theorem of Discrete Calculus
9.3. The Binomial Polynomials ??_{??}(??)= \binom{??}??
9.4. The Binomial Taylor Expansion
9.5. Binomial representations of ??^{??}
9.6. Sums of ??^{???} powers: ??_{??}(??+1)=1^{??}+2^{??}+…+??^{??}
9.7. Geometric representations of ??_{??}(??+1)
9.8. ??_{??} as a polynomial in ???
9.9. Exercises
Chapter 10. Complex numbers
10.1. Definition and models of \comps
10.2. Discrete additive subgroups and subrings of \comps
10.3. The complex exponential function exp:\comps→\comps
10.4. The multiplicative group structure of \comps
10.5. Roots of unity
10.6. Discrete subgroups of \comps^{?}
10.7. The Fundamental Theorem of Algebra
10.8. \comps and Euclidean Transformations
10.9. Pythagorean Theorem, Pythagorean Triples, and Gaussian Integers
10.10. Quaternions
10.11. Exercises
Chapter 11. Mathematical Connections
11.1. Connection-oriented mathematical thinking
11.2. Connection-making in solving a problem
11.3. Connections between mathematical problems
11.4. Different problems reducible to a common model
11.5. Problems related to the Euclidean Algorithm
11.6. The magical marriage of two games
Index
Pure and Applied Undergraduate Texts, Volume: 63
Softcover ISBN: 978-1-4704-7204-7
Expected availability date: November 19, 2023
This textbook bridges the gap between lower-division mathematics courses and advanced mathematical thinking. Featuring clear writing and appealing topics, the book introduces techniques for writing proofs in the context of discrete mathematics. By illuminating the concepts behind techniques, the authors create opportunities for readers to sharpen critical thinking skills and develop mathematical maturity.
Beginning with an introduction to sets and logic, the book goes on to establish the basics of proof techniques. From here, chapters explore proofs in the context of number theory, combinatorics, functions and cardinality, and graph theory. A selection of extension topics concludes the book, including continued fractions, infinite arithmetic, and the interplay among Fibonacci numbers, Pascal's triangle, and the golden ratio.
A Discrete Transition to Advanced Mathematics is suitable for an introduction to proof course or a course in discrete mathematics. Abundant examples and exercises invite readers to get involved, and the wealth of topics allows for course customization and further reading. This new edition has been expanded and modernized throughout, featuring:
A new chapter on combinatorial geometry;
An expanded treatment of the combinatorics of indistinguishable objects;
New sections on the inclusion-exclusion principle and circular permutations;
Over 365 new exercises.
Undergraduate students who need a strong conceptual foundation for higher mathematical thinking.
Contents
Preface
Preface to the Second Edition
Chapter 1. Sets and Logic
1.1. Sets
1.2. Set Operations
1.3. Partitions
1.4. Logic and Truth Tables
1.5. Quantifiers
1.6. Implications
Chapter 2. Proofs
2.1. Proof Techniques
2.2. Mathematical Induction
2.3. The Pigeonhole Principle
Chapter 3. Number Theory
3.1. Divisibility
3.2. The Euclidean Algorithm
3.3. The Fundamental Theorem of Arithmetic
3.4. Divisibility Tests
3.5. Number Patterns
Chapter 4. Combinatorics
4.1. Getting from Point A to Point B
4.2. The Fundamental Principle of Counting
4.3. A Formula for the Binomial Coefficients
4.4. Permutations with Indistinguishable Objects
4.5. Combinations with Indistinguishable Objects
4.6. The Inclusion-Exclusion Principle
4.7. Circular Permutations
4.8. Probability
Chapter 5. Relations
5.1. Relations
5.2. Equivalence Relations
5.3. Partial Orders
5.4. Quotient Spaces
Chapter 6. Functions and Cardinality
6.1. Functions
6.2. Inverse Relations and Inverse Functions
6.3. Cardinality of Infinite Sets
6.4. An Order Relation for Cardinal Numbers
Chapter 7. Graph Theory
7.1. Graphs
7.2. Matrices, Digraphs, and Relations
7.3. Shortest Paths in Weighted Graphs
7.4. Trees
Chapter 8. Sequences
8.1. Sequences
8.2. Finite Differences
8.3. Limits of Sequences of Real Numbers
8.4. Some Convergence Properties
8.5. Infinite Arithmetic
8.6. Recurrence Relations
Chapter 9. Fibonacci Numbers and Pascal’s Triangle
9.1. Pascal’s Triangle
9.2. The Fibonacci Numbers
9.3. The Golden Ratio
9.4. Fibonacci Numbers and the Golden Ratio
9.5. Pascal’s Triangle and the Fibonacci Numbers
Chapter 10. Combinatorial Geometry in the Plane
10.1. Polygons and Convex Sets
10.2. Pick’s Theorem
10.3. Irrational Approximations of ??
10.4. Cotes’s Theorem (optional)
10.5. Tiling and Visibility
10.6. Covering Properties and Geometry of Point Sets
10.7. Linear Algebra and Packing the Plane
10.8. Helly’s Theorem
Chapter 11. Continued Fractions
11.1. Finite Continued Fractions
11.2. Convergents of a Continued Fraction
11.3. Infinite Continued Fractions
11.4. Applications of Continued Fractions
Answers or Hints for Selected Exercises
Bibliography
Index