Feynman Amplitudes, Periods and Motives

Author: Luis Álvarez-Cónsul,José Ignacio Burgos-Gil,Kurusch Ebrahimi-Fard

Publisher: American Mathematical Soc.

ISBN: 1470422476

Category: Mathematical physics

Page: 289

View: 9121

This volume contains the proceedings of the International Research Workshop on Periods and Motives--A Modern Perspective on Renormalization, held from July 2-6, 2012, at the Instituto de Ciencias Matemáticas, Madrid, Spain. Feynman amplitudes are integrals attached to Feynman diagrams by means of Feynman rules. They form a central part of perturbative quantum field theory, where they appear as coefficients of power series expansions of probability amplitudes for physical processes. The efficient computation of Feynman amplitudes is pivotal for theoretical predictions in particle physics. Periods are numbers computed as integrals of algebraic differential forms over topological cycles on algebraic varieties. The term originated from the period of a periodic elliptic function, which can be computed as an elliptic integral. Motives emerged from Grothendieck's "universal cohomology theory", where they describe an intermediate step between algebraic varieties and their linear invariants (cohomology). The theory of motives provides a conceptual framework for the study of periods. In recent work, a beautiful relation between Feynman amplitudes, motives and periods has emerged. The articles provide an exciting panoramic view on recent developments in this fascinating and fruitful interaction between pure mathematics and modern theoretical physics.

Geometry of Moduli Spaces and Representation Theory

Author: Roman Bezrukavnikov,Alexander Braverman,Zhiwei Yun

Publisher: American Mathematical Soc.

ISBN: 1470435748

Category: Algebraic varieties

Page: 436

View: 6509

This book is based on lectures given at the Graduate Summer School of the 2015 Park City Mathematics Institute program “Geometry of moduli spaces and representation theory”, and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory. Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan–Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry. Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections. The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.

Algebraic Groups

The Theory of Group Schemes of Finite Type over a Field

Author: J. S. Milne

Publisher: Cambridge University Press

ISBN: 1316739155

Category: Mathematics

Page: N.A

View: 7982

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

Albanese and Picard 1-motives

Author: Luca Barbieri-Viale,V. Srinivas

Publisher: Societe Mathematique De France


Category: Mathematics

Page: 104

View: 4426

Algebraic Groups and Arithmetic

Author: S. G. Dani,Gopal Prasad

Publisher: Narosa Publishing House


Category: Mathematics

Page: 570

View: 7037

Algebraic Groups and Arithmetic is an area in which major advances have been made in recent decades. The School of Mathematics of the Tata Institute of Fundamental Research has been one of the significant contributors to the progress, under the leadership of Professor M.S. Raghunathan. The Tata Institute organised a conference on the theme during 17-22 December 2001, on the occasion of Professor Raghunathan turning sixty. The conference received enthusiastic response, and there were lectures by several experts on forefront topics in the theme, including group-theoretic aspects, diophantine approximation, modular forms, representation theory, interactions with topology and geometry, dynamics on homogeneous spaces. This volume is a collection of papers emerging from the Conference. In addition to original papers by several leading mathematicians in the area, it also includes two expository papers on the work of Professor M.S. Raghunathan, by the late Professor Armand Borel and Professor Gopal Prasad, which had also been presented at the Conference.

Modular Forms and Special Cycles on Shimura Curves. (AM-161)

Author: Stephen S. Kudla,Michael Rapoport,Tonghai Yang

Publisher: Princeton University Press

ISBN: 9780691125510

Category: Mathematics

Page: 373

View: 8876

Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soulé arithmetic Chow groups of "M". The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations. The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.

Automorphic Forms, Representation Theory and Arithmetic

Papers presented at the Bombay Colloquium 1979

Author: S. Gelbart,G. Harder,K. Iwasawa,H. Jaquet,N.M. Katz,I. Piatetski-Shapiro,S. Raghavan,T. Shintani,H.M. Stark,D. Zagier

Publisher: Springer

ISBN: 3662007347

Category: Mathematics

Page: 355

View: 3714

International Colloquium an Automorphic Forms, Representation Theory and Arithmetic. Published for the Tata Institute of Fundamental Research, Bombay

On Certain L-functions

Conference on Certain L-functions in Honor of Freydoon Shahidi, July 23-27, 2007, Purdue University, West Lafayette, Indiana

Author: James Arthur

Publisher: American Mathematical Soc.

ISBN: 0821852043

Category: Mathematics

Page: 647

View: 8155

This volume constitutes the proceedings of a conference, ``On Certain $L$-functions'', held July 23-27, 2007 at Purdue University, West Lafayette, Indiana. The conference was organized in honor of the 60th birthday of Freydoon Shahidi, widely recognized as having made groundbreaking contributions to the Langlands program. The articles in this volume represent a snapshot of the state of the field from several viewpoints. Contributions illuminate various areas of the study of geometric, analytic, and number theoretic aspects of automorphic forms and their $L$-functions, and both local and global theory are addressed. Topics discussed in the articles include Langlands functoriality, the Rankin-Selberg method, the Langlands-Shahidi method, motivic Galois groups, Shimura varieties, orbital integrals, representations of $p$-adic groups, Plancherel formula and its consequences, the Gross-Prasad conjecture, and more. The volume also includes an expository article on Shahidi's contributions to the field, which serves as an introduction to the subject. Experts will find this book a useful reference, and beginning researchers will be able to use it to survey major results in the Langlands program.

The Geometry of Algebraic Cycles

Proceedings of the Conference on Algebraic Cycles, Columbus, Ohio, March 25-29, 2008

Author: Reza Akhtar,Patrick Brosnan,Roy Joshua

Publisher: American Mathematical Soc.

ISBN: 0821851918

Category: Mathematics

Page: 187

View: 5083

The subject of algebraic cycles has its roots in the study of divisors, extending as far back as the nineteenth century. Since then, and in particular in recent years, algebraic cycles have made a significant impact on many fields of mathematics, among them number theory, algebraic geometry, and mathematical physics. The present volume contains articles on all of the above aspects of algebraic cycles. It also contains a mixture of both research papers and expository articles, so that it would be of interest to both experts and beginners in the field.

Proceedings of the CIMPA-TIFR School on Probability Measures on Groups

Recent Directions and Trends

Author: S. G. Dani

Publisher: Narosa Publishing House

ISBN: 9788173197031

Category: Mathematics

Page: 363

View: 1280

Many aspects of the classical probability theory based on vector spaces were generalised in the second half of the 20th century to measures on groups, especially Lie Groups. Courses were well-received, and this volume represents improved, edited and refereed versions of the notes.

Guide to Psychological Assessment with Asians

Author: Lorraine T. Benuto,Nicholas Thaler,Brian D. Leany

Publisher: Springer

ISBN: 1493907964

Category: Psychology

Page: 474

View: 2785

To effectively serve minority clients, clinicians require a double understanding: of both evidence-based practice and the cultures involved. This particularly holds true when working with Asian-Americans, a diverse and growing population. The Guide to Psychological Assessment with Asians synthesizes real-world challenges, empirical findings, clinical knowledge and common-sense advice to create a comprehensive framework for practice. This informed resource is geared toward evaluation of first-generation Asian Americans and recent immigrants across assessment methods (self-report measures, projective tests), settings (school, forensic) and classes of disorders (eating, substance, sexual). While the Guide details cross-cultural considerations for working with Chinese-, Japanese-, Korean and Indian-American clients, best practices are also included for assessing members of less populous groups without underestimating, overstating or stereotyping the role of ethnicity in the findings. In addition, contributors discuss diversity of presentation within groups and identify ways that language may present obstacles to accurate evaluation. Among the areas covered in this up-to-date reference: Structured and semi-structured clinical interviews. Assessment of acculturation, enculturation and culture. IQ testing. Personality disorders. Cognitive decline and dementia. Mood disorders and suicidality. Neuropsychological assessment of children, adolescents and adults. Culture-bound syndromes. Designed for practitioners new to working with Asian clients as well as those familiar with the population, the Guide to Psychological Assessment with Asians is exceedingly useful to neuropsychologists, clinical psychologists, health psychologists and clinical social workers.

Mathematical Lives

Protagonists of the Twentieth Century From Hilbert to Wiles

Author: CLAUDIO BARTOCCI,Renato Betti,Angelo Guerraggio,Roberto Lucchetti

Publisher: Springer Science & Business Media

ISBN: 9783642136061

Category: Mathematics

Page: 238

View: 1713

Steps forward in mathematics often reverberate in other scientific disciplines, and give rise to innovative conceptual developments or find surprising technological applications. This volume brings to the forefront some of the proponents of the mathematics of the twentieth century, who have put at our disposal new and powerful instruments for investigating the reality around us. The portraits present people who have impressive charisma and wide-ranging cultural interests, who are passionate about defending the importance of their own research, are sensitive to beauty, and attentive to the social and political problems of their times. What we have sought to document is mathematics’ central position in the culture of our day. Space has been made not only for the great mathematicians but also for literary texts, including contributions by two apparent interlopers, Robert Musil and Raymond Queneau, for whom mathematical concepts represented a valuable tool for resolving the struggle between ‘soul and precision.’

Proceedings of the International Colloquium on Lie Groups and Ergodic Theory, Mumbai, 1996

Author: S. G. Dani,Tata Institute of Fundamental Research

Publisher: Narosa Publishing House

ISBN: 9788173192357

Category: Mathematics

Page: 386

View: 6916

This volume is a collection of papers related to lectures delivered in an international colloquium held at the Tata Institute of Fundamental Research, Mumbai, in January 1996. The colloquium, which was designated a Golden Jubilee event of the Institute, was aimed at bringing into focus various recent developments in ergodic theory, related to Lie groups and discrete subgroups. Experts from all over the world spoke at the meeting, on different aspects of the topic.

Periods and Nori Motives

Author: Annette Huber,Stefan Müller-Stach

Publisher: Springer

ISBN: 3319509268

Category: Mathematics

Page: 372

View: 8369

This book casts the theory of periods of algebraic varieties in the natural setting of Madhav Nori’s abelian category of mixed motives. It develops Nori’s approach to mixed motives from scratch, thereby filling an important gap in the literature, and then explains the connection of mixed motives to periods, including a detailed account of the theory of period numbers in the sense of Kontsevich-Zagier and their structural properties. Period numbers are central to number theory and algebraic geometry, and also play an important role in other fields such as mathematical physics. There are long-standing conjectures about their transcendence properties, best understood in the language of cohomology of algebraic varieties or, more generally, motives. Readers of this book will discover that Nori’s unconditional construction of an abelian category of motives (over fields embeddable into the complex numbers) is particularly well suited for this purpose. Notably, Kontsevich's formal period algebra represents a torsor under the motivic Galois group in Nori's sense, and the period conjecture of Kontsevich and Zagier can be recast in this setting. Periods and Nori Motives is highly informative and will appeal to graduate students interested in algebraic geometry and number theory as well as researchers working in related fields. Containing relevant background material on topics such as singular cohomology, algebraic de Rham cohomology, diagram categories and rigid tensor categories, as well as many interesting examples, the overall presentation of this book is self-contained.

The Mind of an Engineer

Author: Purnendu Ghosh,Baldev Raj

Publisher: Springer

ISBN: 9811001197

Category: Science

Page: 458

View: 4117

The Indian National Academy of Engineering (INAE) promotes the endeavour of the practitioners of engineering and technology and related sciences to solve the problems of national importance. The book is an initiative of the INAE and a reflection of the experiences of some of the Fellows of the INAE in the fields of science, technology and engineering. The book is about the reminiscences, eureka moments, inspirations, challenges and opportunities in the journey the professionals took toward self-realisation and the goals they achieved. The book contains 58 articles on diverse topics that truly reflects the way the meaningful mind of an engineer works.

Recurrence Sequences

Author: Graham Everest, Alf van der Poorten,Igor Shparlinski,Thomas Ward

Publisher: American Mathematical Soc.

ISBN: 1470423154


Page: 318

View: 9463

Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.

Data Gathering, Analysis and Protection of Privacy through Randomized Response Techniques: Qualitative and Quantitative Human Traits

Author: N.A

Publisher: Elsevier

ISBN: 0444635718

Category: Mathematics

Page: 544

View: 1596

Data Gathering, Analysis and Protection of Privacy through Randomized Response Techniques: Qualitative and Quantitative Human Traits tackles how to gather and analyze data relating to stigmatizing human traits. S.L. Warner invented RRT and published it in JASA, 1965. In the 50 years since, the subject has grown tremendously, with continued growth. This book comprehensively consolidates the literature to commemorate the inception of RR. Brings together all relevant aspects of randomized response and indirect questioning Tackles how to gather and analyze data relating to stigmatizing human traits Gives an encyclopedic coverage of the topic Covers recent developments and extrapolates to future trends

Elliptic Curves, Modular Forms and Cryptography

Proceedings of the Advanced Instructional Workshop on Algebraic Number Theory

Author: Ashwani K. Bhandari,D.S. Nagaraj,B. Ramakrishnan,T.N. Venkataramana

Publisher: Springer

ISBN: 9386279150

Category: Mathematics

Page: 360

View: 2278