The Shape of Algebra in the Mirrors of Mathematics
The Shape of Algebra in the Mirrors of Mathematics

Author: Gabriel Katz

Publisher: World Scientific

Published: 2012

Total Pages: 632

ISBN-13: 9814313599

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The Shape of Algebra is the authors' attempt to share their mathematical experiences with readers who have more than a passing interest in mathematics, but have only a traditional exposure to elementary algebra. Secondary school and college teachers and students who want to expand their horizons in the field will find a fresh presentation of familiar concepts and some unexpected results. This book serves as a text for an "appreciation" course in modern mathematics designed for non-mathematics majors or for first-year students who are considering the possibility of studying mathematics or related disciplines. It can also serve as a source of computer-supported activities that could supplement traditional courses in algebra, multivariable calculus, and complex variable. This book gives the reader a sense of the visual nature of mathematics. Mathematical experiments with universal mapping software VisuMatica, designed by Vladimir Nodel'man, form the very core of the book. Readers are encouraged to reproduce, play with, and expand on these experiments. Numerous problems are interspersed throughout the text to guide the reader. Our treatment of standard algebra is visual and computational. By introducing visual computational environments like VisuMatica, our book promotes this geometric approach to algebra and makes it accessible to readers a great deal earlier. The book will enable our readers to approach its content on three levels: the first one which requires only some fluency with elementary algebraic manipulations; the second one which also presumes familiarity with the notions of derivatives and tangent lines to plane curves, and the third one which uses some basic concepts of multivariable calculus. All three levels are clearly marked in the text, and will allow for a smooth reading and virtual experiments, regardless of the level that our readers will find comfortable.

The Best Writing on Mathematics 2012
The Best Writing on Mathematics 2012

Author: Mircea Pitici

Publisher: Princeton University Press

Published: 2013

Total Pages: 321

ISBN-13: 0691156557

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Collects essays on mathematics, from the mathematical aspects of origami and the mathematics of dating to the frequency and distribution of prime numbers and a ball in five dimensions.

Connected Mathematics 2: the Shape of Algebra
Connected Mathematics 2: the Shape of Algebra

Author: Glenda Lappan

Publisher: Pearson Academic

Published: 2007-12-01

Total Pages: 93

ISBN-13: 9780133661569

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Building on the solid foundation established in Connected Mathematics, over 15,000 students and 300 teachers contributed to the revision. Students will learn mathematics through appealing and engaging problems. The three-step Launch, Explore, Summarize approach helps students develop mathematical thinking and reasoning while practicing and maintaining skills. Users have long praised its appealing and engaging problems and the effective three-step Launch, Explore, and Summarize approach to learning. They've experienced first-hand how the investigations and excercises help students develop mathematical thinking and reasoning while practicing and maintaining skills. And, this research-based curriculum for Grades 6-8 has been funded by the National Science Foundation once again-resulting in Connected Mathematics 2. - Publisher.

Mirror Symmetry
Mirror Symmetry

Author: Kentaro Hori

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 954

ISBN-13: 0821829556

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This thorough and detailed exposition is the result of an intensive month-long course on mirror symmetry sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives with the aim of furthering interaction between the two fields. The material will be particularly useful for mathematicians and physicists who wish to advance their understanding across both disciplines. Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differential equations obtained from a ``mirror'' geometry. The inclusion of D-brane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar-Vafa invariants. This book gives a single, cohesive treatment of mirror symmetry. Parts 1 and 2 develop the necessary mathematical and physical background from ``scratch''. The treatment is focused, developing only the material most necessary for the task. In Parts 3 and 4 the physical and mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory can be described geometrically, and thus mirror symmetry gives rise to a ``pairing'' of geometries. The proof involves applying $R\leftrightarrow 1/R$ circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten theory in the algebraic setting, beginning with the moduli spaces of curves and maps, and uses localization techniques to show that certain hypergeometric functions encode the Gromov-Witten invariants in genus zero, as is predicted by mirror symmetry. Part 5 is devoted to advanced topi This one-of-a-kind book is suitable for graduate students and research mathematicians interested in mathematics and mathematical and theoretical physics.

Mirrors and Reflections
Mirrors and Reflections

Author: Alexandre V. Borovik

Publisher: Springer Science & Business Media

Published: 2009-11-07

Total Pages: 172

ISBN-13: 0387790667

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This graduate/advanced undergraduate textbook contains a systematic and elementary treatment of finite groups generated by reflections. The approach is based on fundamental geometric considerations in Coxeter complexes, and emphasizes the intuitive geometric aspects of the theory of reflection groups. Key features include: many important concepts in the proofs are illustrated in simple drawings, which give easy access to the theory; a large number of exercises at various levels of difficulty; some Euclidean geometry is included along with the theory of convex polyhedra; no prerequisites are necessary beyond the basic concepts of linear algebra and group theory; and a good index and bibliography The exposition is directed at advanced undergraduates and first-year graduate students.

Tropical Geometry and Mirror Symmetry
Tropical Geometry and Mirror Symmetry

Author: Mark Gross

Publisher: American Mathematical Soc.

Published: 2011-01-20

Total Pages: 338

ISBN-13: 0821852329

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Tropical geometry provides an explanation for the remarkable power of mirror symmetry to connect complex and symplectic geometry. The main theme of this book is the interplay between tropical geometry and mirror symmetry, culminating in a description of the recent work of Gross and Siebert using log geometry to understand how the tropical world relates the A- and B-models in mirror symmetry. The text starts with a detailed introduction to the notions of tropical curves and manifolds, and then gives a thorough description of both sides of mirror symmetry for projective space, bringing together material which so far can only be found scattered throughout the literature. Next follows an introduction to the log geometry of Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof of Mikhalkin's tropical curve counting formulas. This latter proof is given in the fourth chapter. The fifth chapter considers the mirror, B-model side, giving recent results of the author showing how tropical geometry can be used to evaluate the oscillatory integrals appearing. The final chapter surveys reconstruction results of the author and Siebert for ``integral tropical manifolds.'' A complete version of the argument is given in two dimensions.

Zachary Zormer
Zachary Zormer

Author: Joanne A. Reisberg

Publisher: Charlesbridge Publishing

Published: 2006

Total Pages: 35

ISBN-13: 1570918759

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Zack, who has it all covered--length, width, perimeter, and area, introduces a Moebius strip, an expanding frame, and how to light up a big area with a small beam of light.