A publication of the Societe Mathematique de France
The curved βγ system is a nonlinear σ-model with a Riemann surface as the source and a complex manifold X as the target. Its classical solutions pick out the holomorphic maps from the Riemann surface into X. Physical arguments identify its algebra of operators with a vertex algebra known as the chiral differential operators (CDO) of X. The authors verify these claims mathematically by constructing and quantizing rigorously this system using machinery developed by Kevin Costello and the second author, which combine renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Furthermore, the authors find that the factorization algebra of quantum observables of the curved βγ system encodes the sheaf of chiral differential operators. In this sense our approach provides deformation quantization for vertex algebras. As in many approaches to deformation quantization, a key role is played by Gelfand-Kazhdan formal geometry.
The authors begin by constructing a quantization of the βγ system with an n-dimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields Wn on the target disk, and it is naturally identified with the first Pontryagin class in Gelfand-Fuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of Wn by Ω^2cl, the closed 2-forms on the disk. By machinery mentioned above, the authors then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal βγ vertex algebra. Next, the authors introduce a version of Gelfand-Kazhdan formal geometry suitable for factorization algebras, and verify that for a complex manifold X with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs of X.
Graduate students and research mathematicians.
stAcrisque, Volume: 419;
2020; 210 pp; Softcover
MSC: Primary 81; 17; 55;
Print ISBN: 978-2-85629-920-3
Product Code: AST/419
The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalized, converges. The model examples are the Kasner solutions. The main result of the article is optimal energy estimates. However, the author also derives asymptotics and demonstrate that the leading order asymptotics can be specified (also in situations where the asymptotics are not convergent).
It is sometimes argued that if the factors multiplying the spatial derivatives decay exponentially (for a system of wave equations), then the spatial derivatives can be ignored. This line of reasoning is incorrect; the author gives examples of equations such that: (1) the factors multiplying the spatial derivatives decay exponentially, (2) the factors multiplying the time derivatives are constants, (3) the energies of individual modes of solutions asymptotically decay exponentially, and (4) the energies of generic solutions grow as exp[exp(t)]ast→∞.
When the factors multiplying the spatial derivatives grow exponentially, the Fourier modes of solutions oscillate with a frequency that grows exponentially. To obtain asymptotics, the author fixes a mode and considers the net evolution over one period. Moreover, he replaces the evolution (over one period) with a matrix multiplication. He cannot calculate the matrices explicitly; he approximates them. To obtain the asymptotics theb author needs to calculate a matrix product where there is no bound on the number of factors, and where each factor can only be approximated. Nevertheless, he obtains detailed asymptotics. In fact, it is possible to isolate an overall behaviour (growth/decay) from the (increasingly violent) oscillatory behaviour. Moreover, then author is also in a position to specify the leading order asymptotics.
Graduate students and researchers.
AstAcrisque
Volume: 420; 2020; 512 pp; Softcover
MSC: Primary 35; 53; 58; 83;
Print ISBN: 978-2-85629-926-5
Product Code: AST/420
This book is the second volume of two books aiming to present results in Complex Geometry from a firmly geometric point of view. It begins with the construction of the space of compact cycles of a given complex space. The chapters cover a wide variety of topics: (1) Intersection theory; (2) From Douady morphism to cycles; (3) Chow variety; (4) Convexity in spaces of cycles; and (5) Cycles and Kahlerian varieties.
Cours Specialises
Volume: 27; 2020; 567 pp; Hardcover
MSC: Primary 32;
Print ISBN: 978-2-85629-907-4
Product Code: COSP/27
The goal of this book is to develop a theory of join and slices for strict ∞-categories. To any pair of strict ∞-categories, the authors associate a third one that they call their join. This operation is compatible with the usual join of categories up to truncation. The authors show that the join defines a monoidal category structure on the category of strict ∞-categories and that it respects connected inductive limits in each variable. In particular, the authors obtain the existence of some right adjoints; these adjoints define ∞-categorical slices, in a generalized sense. They state some conjectures about the functoriality of the join and the slices with respect to higher lax and oplax transformations and they prove some first results in this direction. These results are used in another paper to establish a Quillen Theorem A for strict ∞-categories. Finally, in an appendix, the authors revisit the Gray tensor product of strict ∞categories. One of the main tools used in this paper is Steiner's theory of augmented directed complexes.
Memoires de la Societe Mathematique de France, Volume: 165
2020; 213 pp; Softcover
MSC: Primary 18; 55;
Print ISBN: 978-2-85629-921-0
Product Code: SMFMEM/165
To Be Published: 30 November 2020
Publisher: International Press of Boston, Inc.
Paperback
2020 Paperback (ISBN 9781571463951)
104 pages
This is the fifteenth issue (Vol. 8, No. 1, July 2020) of the Notices of the International Congress of Chinese Mathematicians (or ICCM Notices, for short), the official periodical of the ICCM organization.
Published semi-annually, the Notices bring news, research, and presentation of various perspectives, relevant to Chinese mathematics development and education.
Readers of the Notices will find research papers on various topics by prominent experts from around the world, interesting and timely articles on current applications and trends, biographical and historical essays, profiles of important institutions of research and learning, and more.
Provides a self-contained graduate introduction to the spectral theory of
ODEs, with emphasis on Sturm?Liouville equations
First book covering both the right and left-definite cases, in both regular and
singular settings
Includes both forward and inverse spectral problems, as well as inverse
scattering theory
Few prerequisites with background material covered in the appendices
This graduate textbook offers an introduction to the spectral theory of ordinary differential
equations, focusing on Sturm?Liouville equations. Sturm?Liouville theory has applications in
partial differential equations and mathematical physics. Examples include classical PDEs such
as the heat and wave equations. Written by leading experts, this book provides a modern,
systematic treatment of the theory. The main topics are the spectral theory and eigenfunction
expansions for Sturm?Liouville equations, as well as scattering theory and inverse spectral
theory. It is the first book offering a complete account of the left-definite theory for Sturm?
Liouville equations. The modest prerequisites for this book are basic one-variable real analysis,
linear algebra, as well as an introductory course in complex analysis. More advanced
background required in some parts of the book is completely covered in the appendices. With
exercises in each chapter, the book is suitable for advanced undergraduate and graduate
courses, either as an introduction to spectral theory in Hilbert space, or to the spectral theory
of ordinary differential equations. Advanced topics such as the left-definite theory and the
Camassa?Holm equation, as well as bibliographical notes, make the book a valuable reference
for experts.
1st ed. 2020, IX, 379 p.
Softcover
ISBN 978-3-030-59087-1
Product category : Graduate/advanced undergraduate textbook
Series : Universitext
Mathematics : Analysis
Provides an essential overview of modern topics in Clifford analysis
Dedicated to Prof. Wolfgang Sprosig
Quaternionic and Clifford analysis are an extension of complex analysis into higher dimensions.
The unique starting point of Wolfgang Sprosig’s work was the application of quaternionic
analysis to elliptic differential equations and boundary value problems. Over the years, Clifford
analysis has become a broad-based theory with a variety of applications both inside and
outside of mathematics, such as higher-dimensional function theory, algebraic structures,
generalized polynomials, applications of elliptic boundary value problems, wavelets, image
processing, numerical and discrete analysis. The aim of this volume is to provide an essential
overview of modern topics in Clifford analysis, presented by specialists in the field, and to
honor the valued contributions to Clifford analysis made by Wolfgang Sprosig throughout his
career.
1st ed. 2019, XXI, 503 p. 22
illus., 10 illus. in color.
Hardcover
ISBN 978-3-030-23853-7
Product category : Contributed volume
Series : Trends in Mathematics
Softcover
ISBN 978-3-030-23856-8
Mathematics : Functions of a Complex Variable