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In recent years, number theory and arithmetic geometry have been
enriched by new techniques from noncommutative geometry, operator
algebras, dynamical systems, and K-Theory. This volume collects and
presents up-to-date research topics in arithmetic and
noncommutative geometry and ideas from physics that point to
possible new connections between the fields of number theory,
algebraic geometry and noncommutative geometry. The articles
collected in this volume present new noncommutative geometry
perspectives on classical topics of number theory and arithmetic
such as modular forms, class field theory, the theory of reductive
p-adic groups, Shimura varieties, the local L-factors of arithmetic
varieties. They also show how arithmetic appears naturally in
noncommutative geometry and in physics, in the residues of Feynman
graphs, in the properties of noncommutative tori, and in the
quantum Hall effect.
The first instances of deformation theory were given by Kodaira and
Spencer for complex structures and by Gerstenhaber for associative
algebras. Since then, deformation theory has been applied as a
useful tool in the study of many other mathematical structures, and
even today it plays an important role in many developments of
modern mathematics. This volume collects a few self-contained and
peer-reviewed papers by experts which present up-to-date research
topics in algebraic and motivic topology, quantum field theory,
algebraic geometry, noncommutative geometry and the deformation
theory of Poisson algebras. They originate from activities at the
Max-Planck-Institute for Mathematics and the Hausdorff Center for
Mathematics in Bonn.
This book is aimed at presenting different methods and perspectives
in the theory of Quantum Groups, bridging between the algebraic,
representation theoretic, analytic, and differential-geometric
approaches. It also covers recent developments in Noncommutative
Geometry, which have close relations to quantization and quantum
group symmetries. The volume collects surveys by experts which
originate from an acitvity at the Max-Planck-Institute for
Mathematics in Bonn.
Quantum cohomology, the theory of Frobenius manifolds and the
relations to integrable systems are flourishing areas since the
early 90's.
An activity was organized at the Max-Planck-Institute for
Mathematics in Bonn, with the purpose of bringing together the main
experts in these areas. This volume originates from this activity
and presents the state of the art in the subject.
This book is aimed at presenting different methods and perspectives
in the theory of Quantum Groups, bridging between the algebraic,
representation theoretic, analytic, and differential-geometric
approaches. It also covers recent developments in Noncommutative
Geometry, which have close relations to quantization and quantum
group symmetries. The volume collects surveys by experts which
originate from an acitvity at the Max-Planck-Institute for
Mathematics in Bonn.
This volume comprises both research and survey articles originating
from the conference on Arithmetic and Geometry around Quantization
held in Istanbul in 2006. A wide range of topics related to
quantization are covered, thus aiming to give a glimpse of a broad
subject in very different perspectives.
The first instances of deformation theory were given by Kodaira and
Spencer for complex structures and by Gerstenhaber for associative
algebras. Since then, deformation theory has been applied as a
useful tool in the study of many other mathematical structures, and
even today it plays an important role in many developments of
modern mathematics.
This volume collects a few self-contained and peer-reviewed papers
by experts which present up-to-date research topics in algebraic
and motivic topology, quantum field theory, algebraic geometry,
noncommutative geometry and the deformation theory of Poisson
algebras. They originate from activities at the
Max-Planck-Institute for Mathematics and the Hausdorff Center for
Mathematics in Bonn.
Exploring common themes in modern art, mathematics, and science,
including the concept of space, the notion of randomness, and the
shape of the cosmos. This is a book about art-and a book about
mathematics and physics. In Lumen Naturae (the title refers to a
purely immanent, non-supernatural form of enlightenment),
mathematical physicist Matilde Marcolli explores common themes in
modern art and modern science-the concept of space, the notion of
randomness, the shape of the cosmos, and other puzzles of the
universe-while mapping convergences with the work of such artists
as Paul Cezanne, Mark Rothko, Sol LeWitt, and Lee Krasner. Her
account, focusing on questions she has investigated in her own
scientific work, is illustrated by more than two hundred color
images of artworks by modern and contemporary artists. Thus
Marcolli finds in still life paintings broad and deep philosophical
reflections on space and time, and connects notions of space in
mathematics to works by Paul Klee, Salvador Dali, and others. She
considers the relation of entropy and art and how notions of
entropy have been expressed by such artists as Hans Arp and Fernand
Leger; and traces the evolution of randomness as a mode of artistic
expression. She analyzes the relation between graphical
illustration and scientific text, and offers her own
watercolor-decorated mathematical notebooks. Throughout, she
balances discussions of science with explorations of art, using one
to inform the other. (She employs some formal notation, which can
easily be skipped by general readers.) Marcolli is not simply
explaining art to scientists and science to artists; she charts
unexpected interdependencies that illuminate the universe.
The unifying theme of this book is the interplay among
noncommutative geometry, physics, and number theory. The two main
objects of investigation are spaces where both the noncommutative
and the motivic aspects come to play a role: space-time, where the
guiding principle is the problem of developing a quantum theory of
gravity, and the space of primes, where one can regard the Riemann
Hypothesis as a long-standing problem motivating the development of
new geometric tools. The book stresses the relevance of
noncommutative geometry in dealing with these two spaces. The first
part of the book deals with quantum field theory and the geometric
structure of renormalization as a Riemann-Hilbert correspondence.
It also presents a model of elementary particle physics based on
noncommutative geometry. The main result is a complete derivation
of the full Standard Model Lagrangian from a very simple
mathematical input. Other topics covered in the first part of the
book are a noncommutative geometry model of dimensional
regularization and its role in anomaly computations, and a brief
introduction to motives and their conjectural relation to quantum
field theory. The second part of the book gives an interpretation
of the Weil explicit formula as a trace formula and a spectral
realization of the zeros of the Riemann zeta function. This is
based on the noncommutative geometry of the adele class space,
which is also described as the space of commensurability classes of
Q-lattices, and is dual to a noncommutative motive (endomotive)
whose cyclic homology provides a general setting for spectral
realizations of zeros of L-functions. The quantum statistical
mechanics of the space of Q-lattices, in one and two dimensions,
exhibits spontaneous symmetry breaking. In the low-temperature
regime, the equilibrium states of the corresponding systems are
related to points of classical moduli spaces and the symmetries to
the class field theory of the field of rational numbers and of
imaginary quadratic fields, as well as to the automorphisms of the
field of modular functions. The book ends with a set of analogies
between the noncommutative geometries underlying the mathematical
formulation of the Standard Model minimally coupled to gravity and
the moduli spaces of Q-lattices used in the study of the zeta
function.
Modified gravity models play an important role in contemporary
theoretical cosmology. The present book proposes a novel approach
to the topic based on techniques from noncommutative geometry,
especially the spectral action functional as a gravity model. The
book discusses applications to early universe models and slow-roll
inflation models, to the problem of cosmic topology, to
non-isotropic cosmologies like mixmaster universes and Bianchi IX
gravitational instantons, and to multifractal structures in
cosmology.Relations between noncommutative and algebro-geometric
methods in cosmology is also discussed, including the occurrence of
motives, periods, and modular forms in spectral models of gravity.
Modified gravity models play an important role in contemporary
theoretical cosmology. The present book proposes a novel approach
to the topic based on techniques from noncommutative geometry,
especially the spectral action functional as a gravity model. The
book discusses applications to early universe models and slow-roll
inflation models, to the problem of cosmic topology, to
non-isotropic cosmologies like mixmaster universes and Bianchi IX
gravitational instantons, and to multifractal structures in
cosmology.Relations between noncommutative and algebro-geometric
methods in cosmology is also discussed, including the occurrence of
motives, periods, and modular forms in spectral models of gravity.
This book presents recent and ongoing research work aimed at
understanding the mysterious relation between the computations of
Feynman integrals in perturbative quantum field theory and the
theory of motives of algebraic varieties and their periods. One of
the main questions in the field is understanding when the residues
of Feynman integrals in perturbative quantum field theory evaluate
to periods of mixed Tate motives. The question originates from the
occurrence of multiple zeta values in Feynman integrals
calculations observed by Broadhurst and Kreimer.Two different
approaches to the subject are described. The first, a "bottom-up"
approach, constructs explicit algebraic varieties and periods from
Feynman graphs and parametric Feynman integrals. This approach,
which grew out of work of Bloch-Esnault-Kreimer and was more
recently developed in joint work of Paolo Aluffi and the author,
leads to algebro-geometric and motivic versions of the Feynman
rules of quantum field theory and concentrates on explicit
constructions of motives and classes in the Grothendieck ring of
varieties associated to Feynman integrals. While the varieties
obtained in this way can be arbitrarily complicated as motives, the
part of the cohomology that is involved in the Feynman integral
computation might still be of the special mixed Tate kind. A
second, "top-down" approach to the problem, developed in the work
of Alain Connes and the author, consists of comparing a Tannakian
category constructed out of the data of renormalization of
perturbative scalar field theories, obtained in the form of a
Riemann-Hilbert correspondence, with Tannakian categories of mixed
Tate motives. The book draws connections between these two
approaches and gives an overview of other ongoing directions of
research in the field, outlining the many connections of
perturbative quantum field theory and renormalization to motives,
singularity theory, Hodge structures, arithmetic geometry,
supermanifolds, algebraic and non-commutative geometry.The text is
aimed at researchers in mathematical physics, high energy physics,
number theory and algebraic geometry. Partly based on lecture notes
for a graduate course given by the author at Caltech in the fall of
2008, it can also be used by graduate students interested in
working in this area.
This book presents recent and ongoing research work aimed at
understanding the mysterious relation between the computations of
Feynman integrals in perturbative quantum field theory and the
theory of motives of algebraic varieties and their periods. One of
the main questions in the field is understanding when the residues
of Feynman integrals in perturbative quantum field theory evaluate
to periods of mixed Tate motives. The question originates from the
occurrence of multiple zeta values in Feynman integrals
calculations observed by Broadhurst and Kreimer.Two different
approaches to the subject are described. The first, a "bottom-up"
approach, constructs explicit algebraic varieties and periods from
Feynman graphs and parametric Feynman integrals. This approach,
which grew out of work of Bloch-Esnault-Kreimer and was more
recently developed in joint work of Paolo Aluffi and the author,
leads to algebro-geometric and motivic versions of the Feynman
rules of quantum field theory and concentrates on explicit
constructions of motives and classes in the Grothendieck ring of
varieties associated to Feynman integrals. While the varieties
obtained in this way can be arbitrarily complicated as motives, the
part of the cohomology that is involved in the Feynman integral
computation might still be of the special mixed Tate kind. A
second, "top-down" approach to the problem, developed in the work
of Alain Connes and the author, consists of comparing a Tannakian
category constructed out of the data of renormalization of
perturbative scalar field theories, obtained in the form of a
Riemann-Hilbert correspondence, with Tannakian categories of mixed
Tate motives. The book draws connections between these two
approaches and gives an overview of other ongoing directions of
research in the field, outlining the many connections of
perturbative quantum field theory and renormalization to motives,
singularity theory, Hodge structures, arithmetic geometry,
supermanifolds, algebraic and non-commutative geometry.The text is
aimed at researchers in mathematical physics, high energy physics,
number theory and algebraic geometry. Partly based on lecture notes
for a graduate course given by the author at Caltech in the fall of
2008, it can also be used by graduate students interested in
working in this area.
Written by a scholar recognized for important and diverse
contributions to mathematical physics, geometry and number theory,
this book is a erudite and brilliantly original exploration of
parallel developments in (mostly modern) art, mathematics, and
physics through the study of topics such as the still-life genre,
physical and artistic visions of nothingness, the mathematical
concept of space, the geometry of prime numbers, particle physics
and cosmology, and artistic and mathematical encounters with
randomness. A final chapter shows how the language of art,
especially surrealist and dadaist art, can help raise awareness and
stimulate debate around some darker aspects of the mathematical
profession and some of the psychological difficulties associated to
the work of mathematical research. While the intended audience does
not necessarily consist of readers with a scientific background,
the book will be organized in such a way that it can be read at two
different levels, with some chapters that require no prior
knowledge of mathematics, and some others that explore more
advanced material. All key mathematical notions will be introduced
and explained.
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