Due 2019-02-08
1st ed. 2018, XVI, 413 p.
150 illus.
Hardcover
ISBN 978-3-030-05212-6
CMS Books in Mathematics
Mathematics : Manifolds and Cell Complexes (incl. Diff. Topology)
Introduces key concepts and constructions both diagrammatic and algebraic
in the field
Exemplifies aspects of problem solving approaches inherent in mathematics
Demonstrates a range of mathematical concepts tangibly through
instantiations in context
Exposes reader to foundations and applications of mathematical constructions
Provides exercises throughout text
This book provides an accessible introduction to knot theory, focussing on Vassiliev invariants,
quantum knot invariants constructed via representations of quantum groups, and how these
two apparently distinct theories come together through the Kontsevich invariant. Consisting of
four parts, the book opens with an introduction to the fundamentals of knot theory, and to
knot invariants such as the Jones polynomial. The second part introduces quantum invariants
of knots, working constructively from first principles towards the construction of Reshetikhin-
Turaev invariants and a description of how these arise through Drinfeld and Jimbo's quantum
groups. Its third part offers an introduction to Vassiliev invariants, providing a careful account
of how chord diagrams and Jacobi diagrams arise in the theory, and the role that Lie algebras
play. The final part of the book introduces the Konstevich invariant. This is a universal quantum
invariant and a universal Vassiliev invariant, and brings together these two seemingly different
families of knot invariants. The book provides a detailed account of the construction of the
Jones polynomial via the quantum groups attached to sl(2), the Vassiliev weight system arising
from sl(2), and how these invariants come together through the Kontsevich invariant.
Due 2019-03-14
1st ed. 2019, Approx. 150 p.
Hardcover
ISBN 978-981-13-3986-8
Mathematics : Mathematical Physics
Offers a unique and unified overview of symplectic geometry
Highlights the differential properties of symplectic manifolds
Great interest for the emerging field of "Geometric Science of Information
This introductory book offers a unique and unified overview of symplectic geometry,
highlighting the differential properties of symplectic manifolds. It consists of six chapters:
Some Algebra Basics, Symplectic Manifolds, Cotangent Bundles, Symplectic G-spaces, Poisson
Manifolds, and A Graded Case, concluding with a discussion of the differential properties of
graded symplectic manifolds of dimensions (0,n). It is a useful reference resource for students
and researchers interested in geometry, group theory, analysis and differential equations.
1st ed. 2019, X, 280 p.
Hardcover
ISBN 978-981-13-3800-7
Probability Theory and Stochastic Modelling
Mathematics : Probability Theory and Stochastic Processes
Provides systematic treatment of the Malliavin calculus on the Wiener?
Poisson space, introducing Sobolev norms
Uses the flow property of the solution of stochastic differential equations and
application to dual jump-diffusions
Is a study of fundamental solutions through stochastic analysis without the
aid of partial differential equations
This monograph presents a modern treatment of (1) stochastic differential equations and (2)
diffusion and jump-diffusion processes. The simultaneous treatment of diffusion processes and
jump processes in this book is unique: Each chapter starts from continuous processes and
then proceeds to processes with jumps. In the first part of the book, it is shown that solutions
of stochastic differential equations define stochastic flows of diffeomorphisms. Then, the
relation between stochastic flows and heat equations is discussed. The latter part investigates
fundamental solutions of these heat equations (heat kernels) through the study of the
Malliavin calculus. The author obtains smooth densities for transition functions of various types
of diffusions and jump-diffusions and shows that these density functions are fundamental
solutions for various types of heat equations and backward heat equations. Thus, in this book
fundamental solutions for heat equations and backward heat equations are constructed
independently of the theory of partial differential equations. Researchers and graduate student
in probability theory will find this book very useful.
Due 2019-06-12
1st ed. 2019, Approx. 350 p.
Hardcover
ISBN 978-3-030-05673-5
Algebra and Applications
Mathematics : Category Theory, Homological Algebra
Postulates a model of weak n-categories using structures (called n-fold
categories) with strictly associative compositions
Encompasses intuitive introductions to new concepts, which would otherwise
remain very technical
Provides diagrammatic summaries and road-maps to guide the reader
Offers a very thorough introduction to multi-simplicial techniques, including
figures illustrating geometric interpretations in low dimensions
This monograph presents a new model of mathematical structures called weak n-categories.
These structures find their motivation in a wide range of fields, from algebraic topology to
mathematical physics, algebraic geometry and mathematical logic. While strict n-categories are
easily defined in terms associative and unital composition operations they are of limited use in
applications, which often call for weakened variants of these laws. The author proposes a new
approach to this weakening, whose generality arises not from a weakening of such laws but
from the very geometric structure of its cells; a geometry dubbed weak globularity. The new
model, called weakly globular n-fold categories, is one of the simplest known algebraic
structures yielding a model of weak n-categories. The central result is the equivalence of this
model to one of the existing models, due to Tamsamani and further studied by Simpson. This
theory has intended applications to homotopy theory, mathematical physics and to longstanding
open questions in category theory. As the theory is described in elementary terms and
the book is largely self-contained, it is accessible to beginning graduate students and to
mathematicians from a wide range of disciplines well beyond higher category theory. The new
model makes a transparent connection between higher category theory and homotopy theory,
rendering it particularly suitable for category theorists and algebraic topologists. Although the
results are complex, readers are guided with an intuitive explanation before each concept is
introduced, and with diagrams showing the interconnections between the main ideas and
results.
The Langlands Programme is one of the most important areas in modern pure mathematics. The importance of this volume lies in its potential to recast many aspects of the programme in an entirely new context. For example, the morphisms in the monomial category of a locally p-adic Lie group have a distributional description, due to Bruhat in his thesis. Admissible representations in the programme are often treated via convolution algebras of distributions and representations of Hecke algebras. The monomial embedding, introduced in this book, elegantly fits together these two uses of distribution theory. The author follows up this application by giving the monomial category treatment of the Bernstein Centre, classified by Deligne?Bernstein?Zelevinsky.
This book gives a new categorical setting in which to approach well-known topics. Therefore, the context used to explain examples is often the more generally accessible case of representations of finite general linear groups. For example, Galois base-change and epsilon factors for locally p-adic Lie groups are illustrated by the analogous Shintani descent and Kondo?Gauss sums, respectively. General linear groups of local fields are emphasized. However, since the philosophy of this book is essentially that of homotopy theory and algebraic topology, it includes a short appendix showing how the buildings of Bruhat?Tits, sufficient for the general linear group, may be generalised to the tom Dieck spaces (now known as the Baum?Connes spaces) when G is a locally p-adic Lie group.
The purpose of this monograph is to describe a functorial embedding of the category of admissible k-representations of a locally profinite topological group G into the derived category of the additive category of the admissible k-monomial module category. Experts in the Langlands Programme may be interested to learn that when G is a locally p-adic Lie group, the monomial category is closely related to the category of topological modules over a sort of enlarged Hecke algebra with generators corresponding to characters on compact open modulo the centre subgroups of G. Having set up this functorial embedding, how the ingredients of the celebrated Langlands Programme adapt to the context of the derived monomial module category is examined. These include automorphic representations, epsilon factors and L-functions, modular forms, Weil?Deligne representations, Galois base change and Hecke operators.
Finite Modulo the Centre Groups
GL2 of a Local Field
Automorphic Representations
GLnK in General
Monomial Resolutions and Deligne Representations
Kondo Style Invariants
Hecke Operators and Monomial Resolutions
Could Galois Descent be Functorial?
PSH-algebras and the Shintani Correspondence
Appendices:
Galois Descent of Representations
Remarks on a Paper of Guy Henniart
Finite General Linear and Symmetric Groups
Locally p-adic Lie Groups
Graduate students and researchers in automorphic forms reviewing the concepts
from a local-algebraic point of view.