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Covering topics in graph theory, L-functions, p-adic geometry,
Galois representations, elliptic fibrations, genus 3 curves and bad
reduction, harmonic analysis, symplectic groups and mould
combinatorics, this volume presents a collection of papers covering
a wide swath of number theory emerging from the third iteration of
the international Women in Numbers conference, "Women in Numbers -
Europe" (WINE), held on October 14-18, 2013 at the CIRM-Luminy
mathematical conference center in France. While containing
contributions covering a wide range of cutting-edge topics in
number theory, the volume emphasizes those concrete approaches that
make it possible for graduate students and postdocs to begin work
immediately on research problems even in highly complex subjects.
Since the early 2000s, a growing body of scientific studies in
neuropathology, neurology, neurosurgery, biomechanics, statistics,
criminology and psychology has cast doubt on the forensic
reliability of medical determinations of Shaken Baby Syndrome
(SBS), more recently termed Abusive Head Trauma (AHT). Studies have
increasingly documented that accidental short falls and a wide
range of medical conditions, can cause the same symptoms and
findings associated with this syndrome. Nevertheless, inaccurate
diagnoses, unrealistic confidence expression, and wrongful
convictions continue to this day. Bringing together contributions
from a multidisciplinary expert panel of 32 professionals across 8
countries in 16 different specialties, this landmark book tackles
the highly controversial topic of SBS, which lies at the
intersection of medicine, science, and law. With comprehensive
coverage across multiple disciplines, it explains the scientific
evidence challenging SBS and advances efforts to evaluate how
deaths and serious brain injuries in infants should be analysed and
investigated.
In the wrong hands, math can be deadly. Even the simplest numbers
can become powerful forces when manipulated by politicians or the
media, but in the case of the law, your liberty--and your life--can
depend on the right calculation.
In "Math on Trial," mathematicians Leila Schneps and Coralie Colmez
describe ten trials spanning from the nineteenth century to today,
in which mathematical arguments were used--and disastrously
misused--as evidence. They tell the stories of Sally Clark, who was
accused of murdering her children by a doctor with a faulty sense
of calculation; of nineteenth-century tycoon Hetty Green, whose
dispute over her aunt's will became a signal case in the forensic
use of mathematics; and of the case of Amanda Knox, in which a
judge's misunderstanding of probability led him to discount
critical evidence--which might have kept her in jail. Offering a
fresh angle on cases from the nineteenth-century Dreyfus affair to
the murder trial of Dutch nurse Lucia de Berk, Schneps and Colmez
show how the improper application of mathematical concepts can mean
the difference between walking free and life in prison.
A colorful narrative of mathematical abuse, "Math on Trial" blends
courtroom drama, history, and math to show that legal expertise
isn't always enough to prove a person innocent.
Mathematics students and researchers often react to Alexandre
Grothendieck's legendary fame in the world of mathematics by asking
just what the man did to earn him so brilliant a reputation. But as
legitimate as it is, the question is difficult to answer, because
of the particularly abstruse nature of his mathematics and the
wealth of notions he introduced, some of which have become so
natural and so familiar to mathematicians that it is easy to forget
that there was ever an actual individual who first brought them out
into the light, whereas others are so abstract that even experts
may take years to grasp them. It is not merely a matter of stating
some powerful, striking result that he proved, although there are
many of these. But more deeply, Grothendieck's work is based on a
whole system of recasting old ideas in new ways, and it transformed
the entire area of algebraic geometry essentially beyond
recognition. This book attempts to provide a reasonable explanation
of what made Grothendieck the mathematician that he was. Thirteen
articles written by people who knew him personally-some who even
studied or collaborated with him over a period of many
years-portray Grothendieck at work, explaining the nature of his
thought through descriptions of his discoveries and contributions
to various subjects, and with impressions, memories, anecdotes, and
some biographical elements. Seeing him through the eyes of those
who knew him well, the reader will come away with a better
understanding of what made Grothendieck unique.
The 2003 second volume of this account of Kaehlerian geometry and
Hodge theory starts with the topology of families of algebraic
varieties. Proofs of the Lefschetz theorem on hyperplane sections,
the Picard-Lefschetz study of Lefschetz pencils, and Deligne
theorems on the degeneration of the Leray spectral sequence and the
global invariant cycles follow. The main results of the second part
are the generalized Noether-Lefschetz theorems, the generic
triviality of the Abel-Jacobi maps, and most importantly Nori's
connectivity theorem, which generalizes the above. The last part of
the book is devoted to the relationships between Hodge theory and
algebraic cycles. The book concludes with the example of cycles on
abelian varieties, where some results of Bloch and Beauville, for
example, are expounded. The text is complemented by exercises
giving useful results in complex algebraic geometry. It will be
welcomed by researchers in both algebraic and differential
geometry.
This book contains eight expository articles by well-known authors
of the theory of Galois groups and fundamental groups. They focus
on presenting developments, avoiding classical aspects which have
already been described at length in the standard literature. The
volume grew from the special semester held at the MSRI in Berkeley
in 1999 and many of the results are due to work accomplished during
that program. Among the subjects covered are elliptic surfaces,
Grothendieck's anabelian conjecture, fundamental groups of curves
and differential Galois theory in positive characteristic. Although
the articles contain fresh results, the authors have striven to
make them as introductory as possible, making them accessible to
graduate students as well as researchers in algebraic geometry and
number theory. The volume also contains a lengthy overview by Leila
Schneps that sets the individual articles into the broader context
of contemporary research in Galois groups.
The decomposition of the space L2(G(Q)\G(A)), where G is a
reductive group defined over Q and A is the ring of adeles of Q, is
a deep problem at the intersection of number and group theory.
Langlands reduced this decomposition to that of the (smaller)
spaces of cuspidal automorphic forms for certain subgroups of G.
This book describes this proof in detail. The starting point is the
theory of automorphic forms, which can also serve as a first step
towards understanding the Arthur-Selberg trace formula. To make the
book reasonably self-contained, the authors also provide essential
background in subjects such as: automorphic forms; Eisenstein
series; Eisenstein pseudo-series, and their properties. It is thus
also an introduction, suitable for graduate students, to the theory
of automorphic forms, the first written using contemporary
terminology. It will be welcomed by number theorists,
representation theorists and all whose work involves the Langlands
program.
The first of two volumes offering a modern introduction to
Kaehlerian geometry and Hodge structure. The book starts with basic
material on complex variables, complex manifolds, holomorphic
vector bundles, sheaves and cohomology theory, the latter being
treated in a more theoretical way than is usual in geometry. The
author then proves the Kaehler identities, which leads to the hard
Lefschetz theorem and the Hodge index theorem. The book culminates
with the Hodge decomposition theorem. The meanings of these results
are investigated in several directions. Completely self-contained,
the book is ideal for students, while its content gives an account
of Hodge theory and complex algebraic geometry as has been
developed by P. Griffiths and his school, by P. Deligne, and by S.
Bloch. The text is complemented by exercises which provide useful
results in complex algebraic geometry.
Eight expository articles by well-known authors of the theory of Galois groups and fundamental groups focus on recent developments, avoiding classical aspects which have already been described at length in the standard literature. The volume grew from the special semester held at the MSRI in Berkeley in 1999 and many of the new results are due to work accomplished during that program. Among the subjects covered are elliptic surfaces, Grothendieck's anabelian conjecture, fundamental groups of curves and differential Galois theory in positive characteristic. Although the articles contain original results, the authors have striven to make them as introductory as possible, making them accessible to graduate students as well as researchers in algebraic geometry and number theory. The volume also contains a lengthy overview by Leila Schneps that sets the individual articles into the broader context of contemporary research in Galois groups.
The second volume of this modern account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. The main results are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly, Nori's connectivity theorem, which generalizes the above. The last part deals with the relationships between Hodge theory and algebraic cycles. The text is complemented by exercises offering useful results in complex algebraic geometry. Also available: Volume I 0-521-80260-1 Hardback $60.00 C
This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions.
This book surveys progress in the domains described in the hitherto
unpublished manuscript 'Esquisse d'un Programme' (Sketch of a
Program) by Alexander Grothendieck. It will be of wide interest
amongst workers in algebraic geometry, number theory, algebra and
topology.
The first of two companion volumes on anabelian algebraic geometry,
this book contains the famous, but hitherto unpublished manuscript
'Esquisse d'un Programme' (Sketch of a Program) by Alexander
Grothendieck. This work, written in 1984, fourteen years after his
retirement from public life in mathematics, together with the
closely connected letter to Gerd Faltings, dating from 1983 and
also published for the first time in this volume, describe a
powerful program of future mathematics, unifying aspects of
geometry and arithmetic via the central point of moduli spaces of
curves; it is written in an artistic and informal style. The book
also contains several articles on subjects directly related to the
ideas explored in the manuscripts; these are surveys of mathematics
due to Grothendieck, explanations of points raised in the Esquisse,
and surveys on progress in the domains described there.
The decomposition of the space L2(G(Q)\G(A)), where G is a
reductive group defined over Q and A is the ring of adeles of Q, is
a deep problem at the intersection of number and group theory.
Langlands reduced this decomposition to that of the (smaller)
spaces of cuspidal automorphic forms for certain subgroups of G.
This book describes this proof in detail. The starting point is the
theory of automorphic forms, which can also serve as a first step
towards understanding the Arthur-Selberg trace formula. To make the
book reasonably self-contained, the authors also provide essential
background in subjects such as: automorphic forms; Eisenstein
series; Eisenstein pseudo-series, and their properties. It is thus
also an introduction, suitable for graduate students, to the theory
of automorphic forms, the first written using contemporary
terminology. It will be welcomed by number theorists,
representation theorists and all whose work involves the Langlands
program.
The focus of this book is on combinatorial objects, dessins d'enfants, which are drawings with vertices and edges on topological surfaces. Their interest lies in their relation with the set of algebraic curves defined over the closure of the rationals, and the corresponding action of the absolute Galois group on them. The articles contained here unite basic elements of the subject with recent advances. Topics covered include: the explicity association of algebraic curves to dessins, the study of the action of the Galois group on the dessins, computation and combinatorics, relations with modular forms, geometry, generating functions, TeichmÜller and moduli spaces. Researchers in number theory, algebraic geometry or related areas of group theory will find much of interest in this book.
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