Series: Encyclopaedia of Mathematical Sciences , Vol. 137
2008, Approx. 270 p., Hardcover
ISBN: 978-3-540-76756-5
Due: January 16, 2008
About this book
Schubert varieties lie at the cross roads of algebraic geometry, combinatorics, commutative algebra, and representation theory. They are an important class of subvarieties of flag varieties, interesting in their own right, and providing an inductive tool for studying flag varieties. The literature on them is vast, for they are ubiquitous?they have been intensively studied over the last fifty years, from many different points of view and by many different authors.
This book is mainly a detailed account of a particularly interesting instance of their occurrence: namely, in relation to classical invariant theory. More precisely, it is about the connection between the first and second fundamental theorems of classical invariant theory on the one hand and standard monomial theory for Schubert varieties in certain special flag varieties - the ordinary, orthogonal, and symplectic Grassmannians - on the other. Historically, this connection was the prime motivation for the development of standard monomial theory. Determinantal varieties and basic concepts of geometric invariant theory arise naturally in establishing the connection.
The book also treats, in the last chapter, some other applications of standard monomial theory, e.g., to the study of certain naturally occurring affine algebraic varieties that, like determinantal varieties, can be realized as open parts of Schubert varieties.
Written for:
Researchers in algebra and algebraic geometry
Keywords
Classical invariant theory
Determinantal varieties
Grassmannians
Schubert varieties
standard monomial theory
Series: Lecture Notes in Mathematics , Vol. 1693
1st edition, 1998, LNM 1693: Minimax and Monotonicity, by Springer Heidelberg
2nd exp. ed., 2008, Hardcover
ISBN: 978-1-4020-6918-5
About this book
In this new edition of LNM 1693 the essential idea is to reduce questions on monotone multifunctions to questions on convex functions. However, rather than using a “big convexification” of the graph of the multifunction and the “minimax technique”for proving the existence of linear functionals satisfying certain conditions, the Fitzpatrick function is used. The journey begins with a generalization of the Hahn-Banach theorem uniting classical functional analysis, minimax theory, Lagrange multiplier theory and convex analysis and culminates in a survey of current results on monotone multifunctions on a Banach space.
The first two chapters are aimed at students interested in the development of the basic theorems of functional analysis, which leads painlessly to the theory of minimax theorems, convex Lagrange multiplier theory and convex analysis. The remaining five chapters are useful for those who wish to learn about the current research on monotone multifunctions on (possibly non reflexive) Banach space.
Table of contents
Preface.- Introduction.- I.The Hahn-Banach-Lagrange Theorem and Some Consequences.- II.Fenchel Duality.- III.Multifunctions, SSD Spaces, Monotonicity and Fitzpatrick Functions.- V.Monotone Multifunctions on Reflexive Banach Spaces.- VI.Special Maximally Monotone Multifunctions.- VII.The Sum Problem for General Banach Spaces.- VIII.Open Problems.- IX.Glossary of Classes of Multifunctions.- X.A Selection of Results.- References.- Subject index.- Symbol index.
Series: Lecture Notes in Mathematics , Vol. 1930
2008, VI, 136 p., Softcover
ISBN: 978-3-540-75931-7
Due: February 2008
About this book
This set of lectures, which had its origin in a mini course delivered at the Summer Program of IMPA (Rio de Janeiro), is an introduction to intrinsic scaling, a powerful method in the analysis of degenerate and singular PDEs.
In the first part, the theory is presented from scratch for the model case of the degenerate p-Laplace equation. This approach brings to light what is really essential in the method, leaving aside technical refinements needed to deal with more general equations, and is entirely self-contained.
The second part deals with three applications of the theory to relevant models arising from flows in porous media and phase transitions. The aim is to convince the reader of the strength of the method as a systematic approach to regularity for this important class of equations.
Written for:
Researchers and graduate students in partial differential equations
Keywords:
Degenerate and singular PDEs
MSC(2000): 35D10, 35K65
intrinsic scaling
p-Laplacian
regularity
Series: Lecture Notes in Mathematics , Vol. 1931
2008, Approx. 390 p., Softcover
ISBN: 978-3-540-76891-3
Due: January 2008
About this book
Six leading experts lecture on a wide spectrum of recent results on the subject of the title, providing both a solid reference and deep insights on current research activity. Michael Cowling presents a survey of various interactions between representation theory and harmonic analysis on semisimple groups and symmetric spaces. Alain Valette recalls the concept of amenability and shows how it is used in the proof of rigidity results for lattices of semisimple Lie groups. Edward Frenkel describes the geometric Langlands correspondence for complex algebraic curves, concentrating on the ramified case where a finite number of regular singular points is allowed. Masaki Kashiwara studies the relationship between the representation theory of real semisimple Lie groups and the geometry of the flag manifolds associated with the corresponding complex algebraic groups. David Vogan deals with the problem of getting unitary representations out of those arising from complex analysis, such as minimal globalizations realized on Dolbeault cohomology with compact support. Nolan Wallach illustrates how representation theory is related to quantum computing, focusing on the study of qubit entanglement.
Table of contents
Preface by E. C. Tarabusi, A. D’Agnolo, M. Picardello.- M. Cowling: Applications of representation theory to harmonic analysis of Lie groups (and vice versa).- E. Frenkel: Ramifications of the geometric Langlands Program.- M. Kashiwara: Equivariant derived category and representation of real semisimple Lie groups.- A.Valette: Amenability and Margulis super-rigidity.- D. A. Vogan, Jr: Unitary Representations and Complex Analysis.- N. R. Wallach: Quantum computing and entanglement for mathematicians.
Series: Developments in Mathematics , Vol. 16
2008, Approx. 430 p., Hardcover
ISBN: 978-3-211-74279-2
Due: February 2008
About this book
This volume contains 21 research and survey papers on recent developments in the field of diophantine approximation. This includes contributions to Wolfgang Schmidt's subspace theorem and its applications to diophantine equations and to the study of linear recurring sequences. The articles are either in the spirit of more classical diophantine analysis or of a geometric or combinatorial flavor. Several articles deal with estimates for the number of solutions of diophantine equations as well as with congruences and polynomials. In addition, the volume contains transcendence results for special functions and contributions to metric diophantine approximation and discrepancy theory. The articles are based on lectures given at a conference at the Erwin Schrodinger-Institute (Vienna, 2003), where many leading experts in the field of diophantine approximation participated.
Table of contents
-Dedication to Wolfgang Tichy.-Schaffer´s Determinant Argument.-Arithmetic progressions and Tic-Tac-Toe games.-Metric discrepancy results for sequences {NkX } and Diophantine equations.-Mahler´s classification of numbers compared with Kosma´s, II.-Rational approximations to a q-analogue of p and some other q-series.-Orthogonality and digit shifts in the classical Mean Squares problem in irregularities of point distribution.-Applications of the Subspace Theorem to certain Diophantine problems.-A generalization of the Subspace Theorem with polynomials of higher degree.-On the Diophantine equation Gn (x) = Gm (y) with Q(x,y) = 0.-A criterion for polynomials to divide infinitely many k-nomials.-Approximants de Pade des q-Polylogarithmes.-The set of solutions of some equation for linear recurrence sequences.-Counting algebraic numbers with large height I.-Class number conditions for the diagonal case of the equation of Nagell-Ljunggren.-Construction of approximations to zeta-values.-Quelques aspects Diophantiens des varietes Toriques Projectives.-Une inegalite de Lojasiewicz arithmetique.-On the continued fraction expansion of a class of numbers.-The number of solutions of a linear homogeneous congruence.-A note on Lyapunov theory for Brun algorithm.-Orbit sums and modular vector invariants.-New irrationality results for dologarithms of rational numbers.