Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties ...(展开全部)

Finally, a comprehensive fish identification guide covering the fish-rich reefs of the Pacific. It contains 2,500 underwater photographs of 2,000 species from four of the best marine life authors/photographers in the business. Their collaboration makes it possible for underwater naturalists to identify fishes from Thailand to Tahiti with a single, compact, easy-to-use, no-nonse...(展开全部)

The first edition of this book presented simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut), using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in a...(展开全部)