Schubert Calculus and Its Applications in Combinatorics and Representation Theory
Schubert Calculus and Its Applications in Combinatorics and Representation Theory

Author: Jianxun Hu

Publisher: Springer Nature

Published: 2020-10-24

Total Pages: 367

ISBN-13: 9811574510

DOWNLOAD EBOOK

This book gathers research papers and surveys on the latest advances in Schubert Calculus, presented at the International Festival in Schubert Calculus, held in Guangzhou, China on November 6–10, 2017. With roots in enumerative geometry and Hilbert's 15th problem, modern Schubert Calculus studies classical and quantum intersection rings on spaces with symmetries, such as flag manifolds. The presence of symmetries leads to particularly rich structures, and it connects Schubert Calculus to many branches of mathematics, including algebraic geometry, combinatorics, representation theory, and theoretical physics. For instance, the study of the quantum cohomology ring of a Grassmann manifold combines all these areas in an organic way. The book is useful for researchers and graduate students interested in Schubert Calculus, and more generally in the study of flag manifolds in relation to algebraic geometry, combinatorics, representation theory and mathematical physics.

The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics
The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics

Author: James Haglund

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 178

ISBN-13: 0821844113

DOWNLOAD EBOOK

This work contains detailed descriptions of developments in the combinatorics of the space of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics. These developments have led in turn to some surprising discoveries in the combinatorics of Macdonald polynomials.

High-Dimensional Probability
High-Dimensional Probability

Author: Roman Vershynin

Publisher: Cambridge University Press

Published: 2018-09-27

Total Pages: 299

ISBN-13: 1108415199

DOWNLOAD EBOOK

An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.

Foundations of Computational Mathematics
Foundations of Computational Mathematics

Author: Ronald A. DeVore

Publisher: Cambridge University Press

Published: 2001-05-17

Total Pages: 418

ISBN-13: 9780521003490

DOWNLOAD EBOOK

Collection of papers by leading researchers in computational mathematics, suitable for graduate students and researchers.

Finite Frames
Finite Frames

Author: Peter G. Casazza

Publisher: Springer Science & Business Media

Published: 2012-09-14

Total Pages: 492

ISBN-13: 0817683739

DOWNLOAD EBOOK

Hilbert space frames have long served as a valuable tool for signal and image processing due to their resilience to additive noise, quantization, and erasures, as well as their ability to capture valuable signal characteristics. More recently, finite frame theory has grown into an important research topic in its own right, with a myriad of applications to pure and applied mathematics, engineering, computer science, and other areas. The number of research publications, conferences, and workshops on this topic has increased dramatically over the past few years, but no survey paper or monograph has yet appeared on the subject. Edited by two of the leading experts in the field, Finite Frames aims to fill this void in the literature by providing a comprehensive, systematic study of finite frame theory and applications. With carefully selected contributions written by highly experienced researchers, it covers topics including: * Finite Frame Constructions; * Optimal Erasure Resilient Frames; * Quantization of Finite Frames; * Finite Frames and Compressed Sensing; * Group and Gabor Frames; * Fusion Frames. Despite the variety of its chapters' source and content, the book's notation and terminology are unified throughout and provide a definitive picture of the current state of frame theory. With a broad range of applications and a clear, full presentation, this book is a highly valuable resource for graduate students and researchers across disciplines such as applied harmonic analysis, electrical engineering, quantum computing, medicine, and more. It is designed to be used as a supplemental textbook, self-study guide, or reference book.

Differentiable Measures and the Malliavin Calculus
Differentiable Measures and the Malliavin Calculus

Author: Vladimir Igorevich Bogachev

Publisher: American Mathematical Soc.

Published: 2010-07-21

Total Pages: 506

ISBN-13: 082184993X

DOWNLOAD EBOOK

This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject.

Affine Insertion and Pieri Rules for the Affine Grassmannian
Affine Insertion and Pieri Rules for the Affine Grassmannian

Author: Thomas Lam

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 103

ISBN-13: 0821846582

DOWNLOAD EBOOK

The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${\rm Gr}$ associated with $SL(n,\mathbb{C})$.Their main results are: Pieri rules for the Schubert bases of $H^*({\rm Gr})$ and $H_*({\rm Gr})$, which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. A new combinatorial definition for $k$-Schur functions, which represent the Schubert basis of $H_*({\rm Gr})$. A combinatorial interpretation of the pairing $H^*({\rm Gr})\times H_*({\rm Gr}) \rightarrow\mathbb Z$ induced by the cap product.

Mathematics for Physics
Mathematics for Physics

Author: Michael Stone

Publisher: Cambridge University Press

Published: 2009-07-09

Total Pages: 821

ISBN-13: 1139480618

DOWNLOAD EBOOK

An engagingly-written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics used in research in physics. The first half of the book focuses on the traditional mathematical methods of physics – differential and integral equations, Fourier series and the calculus of variations. The second half contains an introduction to more advanced subjects, including differential geometry, topology and complex variables. The authors' exposition avoids excess rigor whilst explaining subtle but important points often glossed over in more elementary texts. The topics are illustrated at every stage by carefully chosen examples, exercises and problems drawn from realistic physics settings. These make it useful both as a textbook in advanced courses and for self-study. Password-protected solutions to the exercises are available to instructors at www.cambridge.org/9780521854030.