When thinking about this book, three questions come to mind: What are motives? What is motivic cohomology? How does this book fit into these frameworks? Let me begin with a very brief answer to these questions. The theory of motives is a branch of algebraic geometry dealing with algebraic varieties over a fixed field k. The basic idea is simple in its audacity: enlarge the category of varieties...

The term modular form, generically, is very broad. These objects most naturally exist within the scope of number theory and play central roles in its various branches, yet they also play important roles in other fields of mathematics. For example, modular forms might conjure up such diverse thoughts as Fermat’s Last Theorem, the Langlands program, the Riemann Hypothesis, arithmetic applications...

Some complete and some partial characterizations of continua are obtained in terms of the existence of an e-selection on one or another of their hyperspaces for each e > 0 .

In a first course in complex analysis, students learn a theorem that states that if an analytic function is zero on a non-discrete set inside a region in the complex plane, then the function must be identically zero. In particular, the values that an analytic function takes in the neighborhood of a single point completely determine the function in the whole region. This, of course, is very usef...

The book contains a collection of articles by participants of the Working Week on Resolution of Singularities held at Obergurgl in Tirol, September 7-14, 1997. It is dedicated to Oscar Zariski, the founder of the school of algebraic geometry in the United States. During his long career as a mathematician he obtained groundbreaking results in algebra and algebraic geometry. Many years of his car...

The recent turmoil in financial markets has been partly caused by insufficient attention to rigorous financial modeling. Among the causes of this failing is the relative shortage of mathematically well trained professionals in the financial services industry. Shreve is a co-founder of one of the oldest and most successful masters degree programs in financial engineering, established at Carnegie...

In his fundamental 1985 paper [6], Drinfeld attached a certain Hopf algebra, which he called a Yangian, to each finite dimensional simple Lie algebra over the ground field C. These Hopf algebras can be regarded as a tool for producing rational solutions of the quantum Yang-Baxter equation and are one of the main families of examples in Drinfeld's seminal ICM address [8] which marked the beginni...

Maximum principles are among the most powerful and widely used analytic tools in the study of second-order linear and nonlinear elliptic and parabolic equations. They enable us to obtain valuable information about (real valued) solutions of differential equations and inequalities (such as a priori pointwise estimates, and uniqueness and stability results) without the need to know in advance the...

Real algebraic geometry studies real algebraic sets, i.e., real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular, real polynomial mappings). The history of real algebraic geometry goes back to ancient Greece. In the third century BCE, Archimedes and Apollonius systematically studied problems on conic sections [17], and also introduc...

Differential Algebra and the Analytic Case Author(s): A. Seidenberg Source: Proceedings of the American Mathematical Society, Vol. 9, No. 1 (Feb., 1958), pp. 159164 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2033416 Accessed: 17/03/2010 14:56 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http...