Classical modular forms on the upper half plane, with respect to the modular group SL(2,Z) and its congruence subgroups, have arisen naturally in number theory, complex analysis, topology, mathematical physics, and many other subjects. The closely related automorphic representations are basic notions in the celebrated Langlands program, which was proposed by Langlands in the late 1960s and has since revolutionized the fields of number theory, arithmetic algebraic geometry, and representation theory. This volume consists of expanded lecture notes from a 2007 international conference in Guangzhou, China, at which several leading experts in number theory presented introductions to, and surveys of, many aspects of automorphic forms and the Langlands program.

Automorphic Forms and the Langlands Program (volume 9 of the Advanced Lectures in Mathematics series)

Science & Math

Various, Contributors

Mathematics

International Press of Boston, Incorporated

In a career of nearly sixty years of mathematical research, Guangchang Dong's influence on the development of partial differential equations in China has been immense, at both teaching and research levels. To celebrate Prof. Dong's eightieth birthday, an international conference called Elliptic and Parabolic Equations and Applications was held in August 2008 at Zhejiang University in Hangzhou, China. This volume presents fifteen papers in all -- some drawn from lectures given at the conference, others by his friends and former students.

Trends in Partial Differential Equations (volume 10 of the Advanced Lectures in Mathematics series)

VARIOUS

Bian, Baojun

Li, Shenghong

Wang, Xu-Jia

Cohomology of groups is a fundamental tool in many subjects of modern mathematics. One important generalized cohomology theory is the algebraic K-theory. Indeed, algebraic K-groups of rings are important invariants of the rings and have played important roles in algebra, topology, number theory, etc. This volume consists of expanded lecture notes from a 2007 seminar at Zhejiang University in China, at which several leading experts presented introductions, to and surveys of, many aspects of cohomology of groups and algebraic K-theory, along with their broad applications. Two foundational papers on algebraic K-theory by Daniel Quillen are also included.

Cohomology of Groups and Algebraic K-theory (volume 12 of the Advanced Lectures in Mathematics series)

Contents Hyper-Kähler manifolds (by A. Dancer) Compact Riemannian manifolds with exceptional holonomy (by D. Joyce) Kähler-Einstein manifolds of positive scalar curvature (by G. Tian) Quaternion-Kähler Geometry (by S. Salamon) 3-Sasakian manifolds (by C. Boyer & K. Galicki) Ricci flow and Einstein metrics in low dimensions (by B. Chow) Rigidity and compactness of Einstein metrics (by P. Petersen) Einstein deformations of hyperbolic metrics (by O. Biquard) Four-dimensional Einstein manifolds, and beyond (by C. LeBrun) Einstein metrics from symmetry and bundle constructions (by McKenzie Wang) General relativity (by K. Tod) The stability of Minkowski Space-Time (by D. Christodoulou) Einstein-Weyl Geometry (by D. Calderbank & H. Pedersen)

Surveys in Differential Geometry, Vol. 6: Essays on Einstein manifolds (2010 re-issue)

Contents The geometry of classical particles (by M. Atiyah) Singularities in the work of Friedrich Hirzebruch (by E. Brieskorn) The moduli space of abelian varieties and the singularities of the theta divisor (by C. Gilberto & G. Van der Geer) Holomorphic spheres in loop groups and Bott periodicity (by R.L. Cohen, E. Lupercio & G.B. Segal) Moment map and diffeomorphisms (by S.K. Donaldson) Dirac charge quantization and generalized differential cohomology (by D.S. Freed) The holomorphic kernel of the Rankin-Selberg convolution (by D. Goldfeld & S. Zhang) Equivariant de Rham theory and graphs (by V. Guillemin & C. Zara) Morse theory and Stokes' theorem (by F.R. Harvey & H.B. Lawson, Jr) The Atiyah-Bott-Singer fixed point theorem and number theory (by F. Hirzebruch) The moduli space of complex Lagrangian submanifolds (by N.J. Hitchin) Which Singer is that? (by R.V. Kadison) Curvature and function theory on Riemannian manifolds (by P. Li) Mirror Principle. III (by B.H. Lian, K. Liu & S.-T. Yau) Mirror Principle. IV (by B.H. Lian, K. Liu & S.-T. Yau) Three constructions of Frobenius manifolds: A comparative study (by Yu.I. Manin) On Ricci-flat twistor theory (by R. Penrose) On the geometry of nilpotent orbits (by W. Schmid & K. Vilonen) Seiberg-Witten invariants, self-dual harmonic 2-forms and Hofer-Wysocki-Zehnder formalism (by C.H. Taubes) Unifying themes in topological field theories (by C. Vafa) Noncommutative Yang-Mills theory and string theory (by E. Witten)

Surveys in Differential Geometry, Vol. 7: Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer (2010 re-issue)

Contents Projective planes, Severi varieties and spheres (by M. Atiyah & J. Berndt) Degeneration of Einstein metrics and metrics with special holonomy (by J. Cheeger) The min-max construction of minimal surfaces (by T.H. Colding & C. De Lellis) Universal volume bounds in Riemannian manifolds (by C.B. Croke & M. Katz) A Kawamata-Viehweg vanishing theorem on compact Kähler manifolds (by J.-P. Demailly & T. Peternel) Moment maps in differential geometry (by S.K. Donaldson) Local rigidity for cocycles (by D. Fisher & G.A. Margulis) Einstein metrics, four-manifolds, and differential topology (by C. LeBrun) Topological quantum field theory for Calabi-Yau threefolds and G2-manifold (by N.C. Leung) Geometric results in classical minimal surface theory (by W.H. Meeks III) On global existence of wave maps with critical regularity (by A. Nahmod) Discreteness of minimal models of Kodaira dimension zero and subvarieties of moduli stacks (by E. Viehweg & K. Zuo) Geometry of the Weil-Peterson completion of Teichmüller space (by S.A. Wolpert)

Surveys in Differential Geometry, Vol. 8: Lectures on geometry and topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck (2010 re-issue)

Contents Anomalous diffusion and stability of Harnack inequalities (by M.T. Barlow) From isoperimetric inequalities to heat kernels via symmetrization (by G. Besson) Discrete isoperimetric inequalities (by F. Chung) An excursion into geometric analysis (by T.H. Colding & W.P. Minicozzi II) Eigenvalues of elliptic operators and geometric applications (by A. Grigor'yan, Y. Netrusov & S.-T. Yau) Spectral gap, logarithmic Sobolev constant, and geometric bounds (by M. Ledoux) Discrete analytic functions: An exposition (by L. Lovász) Conformal properties in classical minimal surface theory (by W.H. Meeks III & J. Pérez) Analysis of the cut locus via heat kernel (by R. Neel & D. Strook) Analysis on Riemannian co-compact covers (by L. Saloff-Coste) Functoriality and small eigenvalues of Laplacian on Riemann surfaces (by F. Shahidi) The inverse spectral problem (by S. Zelditch)

Surveys in Differential Geometry, Vol. 9: Eigenvalues of Laplacians and other geometric operators (2010 re-issue)

Surveys in Differential Geometry, Vol. 11: Metric and comparison geometry (2010 re-issue)

Contents On the conformal scalar curvature equation and related problems (by S. Brendle) A survey of the Kähler-Ricci Flow and Yau's Uniformization Conjecture (by A. Chau, L.-F. Tam) Recent developments on the Hamilton's Ricci Flow (by H.-D. Cao, B.-L. Chen, and X.-P. Zhu) Curvature flows in semi-Riemannian manifolds (by C. Gerhardt) Global regularity and singularity development for wave maps (by J. Krieger) Relativistic Teichmüller theory: a Hamilton-Jacobi approach to 2+1-dimensional Einstein gravity (by V. Moncrief) Monotonicity and Li-Yau-Hamilton inequalities (by L. Ni) Singularities of mean curvature flow and flow with surgeries (by C. Sinestrari) Some recent developments in Lagrangian mean curvature flows (by M.-T. Wang)

Surveys in Differential Geometry, Vol. 12: Geometric flows (2010 re-issue)

Traditionally, elementary number theory is a branch of number theory dealing with the integers, but without use of techniques from other mathematical fields. In this concise and elegant book, the author has sought to pare away all material that might be considered extraneous to a three-hour-per-week, twelve-week semester course in elementary number theory. The author presents in natural sequence the basic ideas and results of elementary number theory, laying a strong foundation for later studies in algebraic number theory and analytic number theory. The only background knowledge required of the reader is of some simple properties of the system of integers. Elementary Number Theory begins with a few preliminaries on induction principles, followed by a quick review of division algorithms. Then in the second chapter, the author touches upon the usage of divisors, the greatest (or least) common divisor (multiple), the Euclidean algorithm, and linear indeterminate equations. This foundation supports discussions in the subsequent chapters concerning: prime numbers; congruences; congruent equations; and, finally, three additional topics (comprising cryptography, Diophantine equations and Gaussian integers). Each chapter concludes with exercises that both illustrate the theory and provide practice in the techniques. Answers to even-numbered problems are given at the end of the book.

Elementary Number Theory

Xiao, Jie

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