Hardcover ISBN: 978-1-4704-7411-9
Product Code: CRMM/39
Expected availability date: November 05, 2023
CRM Monograph Series Volume: 39
2023; 127 pp
MSC: Primary 11; 57; 14; 32;
Note: This book is in French.
Ce livre constitue un expose detaille de la serie de cours donnes en 2020 par le Prof. Nicolas Bergeron, titulaire de la Chaire Aisenstadt au CRM de Montreal.
L'objet de ce texte est une ample generalisation d'une famille d'identites classiques, notamment la formule d'addition de la fonction cotangente ou celle des series d'Eisenstein. Le livre relie ces identites a la cohomologie de certains sous-groupes arithmetiques du groupe lineaire general. Il rend explicite ces relations au moyen de la theorie des symboles modulaires de rang superieur, devoilant finalement un lien concret entre des objets de nature topologique et algebrique.
This book provides a detailed exposition of the material presented in a series of lectures given in 2020 by Prof. Nicolas Bergeron while he held the Aisenstadt Chair at the CRM in Montreal.
The topic is a broad generalization of certain classical identities such as the addition formulas for the cotangent function and for Eisenstein series. The book relates these identities to the cohomology of arithmetic subgroups of the general linear group. It shows that the relations can be made explicit using the theory of higher rank modular symbols, ultimately unveiling a concrete link between topological and algebraic objects.
Graduate students and researchers interested in modular forms and/or special values of L
-functions.
Construction de cocycles : aspects topologiques
Enonces des principaux resultats : cocycles explicites
Cohomologie dfarrangements dfhyerplans : representants canoniques
Formes differentielles sur lfespace symetrique associe a SLn(C)
Compactifications de Satake, de Tits et symboles modulaires
Cocycles de GLn(C)
explicites
Series dfEisenstein associees a Õ
Cocycle multicatif du groupe rationnel GLn(Q)+
Cocycle elliptique du groupe rationnel GLn(Q)+
Annexe A. Cohomologie equivariante et complexe de de Rham simplicial
Annexe B. Classe dfEisenstein affine et theorie de lfobstruction
Softcover ISBN: 978-1-4704-7112-5
Product Code: SURV/275
Expected availability date: December 03, 2023
Mathematical Surveys and Monographs Volume: 275
2023; 533 pp
MSC: Primary 13; 14; 32; 42;
Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundle-valued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briancon?Skoda theorem and Hilbert's Nullstellensatz, to arithmetic intersection theory and to tropical geometry.
This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the one-dimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra.
Graduate students and researchers interested in residue theory.
Softcover ISBN: 978-1-4704-6992-4
Product Code: CONM/790
Contemporary Mathematics Volume: 790
2023; 157 pp
This volume contains the proceedings of the Conference on Compactifications, Configurations, and Cohomology, held from October 22?24, 2021, at Northeastern University, Boston, MA.
Some of the most active and fruitful mathematical research occurs at the interface of algebraic geometry, representation theory, and topology. Noteworthy examples include the study of compactifications in three specific settings?algebraic group actions, configuration spaces, and hyperplane arrangements. These three types of compactifications enjoy common structural features, including relations to root systems, combinatorial descriptions of cohomology rings, the appearance of iterated blow-ups, the geometry of normal crossing divisors, and connections to mirror symmetry in physics. On the other hand, these compactifications are often studied independently of one another.
The articles focus on new and existing connections between the aforementioned three types of compactifications, thereby setting the stage for further research. It draws on the discipline-specific expertise of all contributors, and at the same time gives a unified, self-contained reference for compactifications and related constructions in different contexts.
Graduate students and research mathematicians interested in compactification problems in algebraic geometry, algebraic topology, and Lie theory.
Ana B?libanu - A quasi-Poisson structure on the multiplicative Grothendieck?Springer resolution
Patrick Brosnan - Volumes of definable sets in o-minimal expansions and affine GAGA theorems
Peter Crooks and Markus Roser - Hessenberg varieties and Poisson slices
Graham Denham and Avi Steiner - Geometry of logarithmic derivations of hyperplane arrangements
Iva Halacheva - Shift of argument algebras and de Concini?Procesi spaces
Ben Knudsen - Projection spaces and twisted Lie algebras
Alexandru I. Suciu - Cohomology, Bocksteins, and resonance varieties in characteristic 2
Part of Elements in Non-local Data Interactions: Foundations and Applications
AVAILABILITY: Not yet published - available from October 2023
FORMAT: Paperback
ISBN: 9781009331005
Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. This Element provides a unified framework to handle these approaches via Markov chains. The authors consider stochastic normalizing flows as a pair of Markov chains fulfilling some properties, and show how many state-of-the-art models for data generation fit into this framework. Indeed numerical simulations show that including stochastic layers improves the expressivity of the network and allows for generating multimodal distributions from unimodal ones. The Markov chains point of view enables the coupling of both deterministic layers as invertible neural networks and stochastic layers as Metropolis-Hasting layers, Langevin layers, variational autoencoders and diffusion normalizing flows in a mathematically sound way. The authors' framework establishes a useful mathematical tool to combine the various approaches.
1. Introduction
2. Preliminaries
3. Normalizing Flows
4. Stochastic Normalizing Flows
5. Stochastic Layers
6. Conditional Generative Modeling
7. Numerical Results
8. Conclusions and Open Questions
References.
Part of Cambridge Monographs on Applied and Computational Mathematics
Not yet published - available from January 2024
FORMAT: Hardback
ISBN: 9781009298780
Many materials can be modeled either as discrete systems or as continua, depending on the scale. At intermediate scales it is necessary to understand the transition from discrete to continuous models and variational methods have proved successful in this task, especially for systems, both stochastic and deterministic, that depend on lattice energies. This is the first systematic and unified presentation of research in the area over the last 20 years. The authors begin with a very general and flexible compactness and representation result, complemented by a thorough exploration of problems for ferromagnetic energies with applications ranging from optimal design to quasicrystals and percolation. This leads to a treatment of frustrated systems, and infinite-dimensional systems with diffuse interfaces. Each topic is presented with examples, proofs and applications. Written by leading experts, it is suitable as a graduate course text as well as being an invaluable reference for researchers.
Illustrates the pros and cons of a variety of general demonstration techniques and results, allowing readers to easily choose the technique to suit their own problems
Features a number of prototypical examples with open problems to inspire future research
Presents each topic independently with examples, theoretical analyses and applications, suitable for graduate courses at different levels
1. Introduction
2. Preliminaries
3. Homogenization of pairwise systems with positive coefficients
4. Compactness and integral representation
5. Random lattices
6. Extensions
7. Frustrated systems
8. Perspectives towards dense graphs
A. Multiscale analysis
B. Spin systems as limits of elastic interactions
References
Index.
Part of Cambridge Studies in Advanced Mathematics
Not yet published - available from February 2024
FORMAT: Hardback
ISBN: 9781009257817
This book introduces convex polytopes and their graphs, alongside the results and methodology required to study them. It guides the reader from the basics to current research, presenting many open problems to facilitate the transition. The book includes results not found in a book before such as: the edge connectivity and linkedness of graphs of polytopes; the characterisation of their cycle space; the Minkowski decomposition of polytopes from the perspective of geometric graphs; the very recent lower bound theorem of Xue on the number of faces of polytopes with small number of vertices; and Kalai's rigidity proof of the Lower bound theorem for simplicial polytopes. This accessible introduction covers prerequisites from linear algebra, graph theory, and polytope theory, and ends each chapter with exercises of varying difficulty to help the reader engage with new concepts, making it ideal for students and researchers new to the field.
Helps readers build intuition with examples and descriptive illustrations
Guides readers to the edge of the field, compiling over 20 significant open problems
Covers prerequisite knowledge from linear algebra, graph theory, and polytope theory
Preface
1. Introduction
2. Polytopes
3. Polytopal graphs
4. Connectivity
5. Reconstruction
6. Decomposition
7. Diameter
8. Faces
A. Open problems
B. Topology
C. Graphs
References
Glossary
Index.