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Can artificial intelligence learn mathematics? The question is at
the heart of this original monograph bringing together theoretical
physics, modern geometry, and data science. The study of Calabi-Yau
manifolds lies at an exciting intersection between physics and
mathematics. Recently, there has been much activity in applying
machine learning to solve otherwise intractable problems, to
conjecture new formulae, or to understand the underlying structure
of mathematics. In this book, insights from string and quantum
field theory are combined with powerful techniques from complex and
algebraic geometry, then translated into algorithms with the
ultimate aim of deriving new information about Calabi-Yau
manifolds. While the motivation comes from mathematical physics,
the techniques are purely mathematical and the theme is that of
explicit calculations. The reader is guided through the theory and
provided with explicit computer code in standard software such as
SageMath, Python and Mathematica to gain hands-on experience in
applications of artificial intelligence to geometry. Driven by data
and written in an informal style, The Calabi-Yau Landscape makes
cutting-edge topics in mathematical physics, geometry and machine
learning readily accessible to graduate students and beyond. The
overriding ambition is to introduce some modern mathematics to the
physicist, some modern physics to the mathematician, and machine
learning to both.
'The book is an engaging and influential collection of significant
contributions from an assembly of world expert leaders and pioneers
from different fields, working at the interface between topology
and physics or applications of topology to physical systems ... The
book explores many interesting and novel topics that lie at the
intersection between gravity, quantum fields, condensed matter,
physical cosmology and topology ... A rich, well-organized, and
comprehensive overview of remarkable and insightful connections
between physics and topology is here made available to the physics
reader.'Contemporary PhysicsSince its birth in Poincare's seminal
1894 'Analysis Situs', topology has become a cornerstone of
mathematics. As with all beautiful mathematical concepts, topology
inevitably - resonating with that Wignerian principle of the
effectiveness of mathematics in the natural sciences - finds its
prominent role in physics. From Chern-Simons theory to topological
quantum field theory, from knot invariants to Calabi-Yau
compactification in string theory, from spacetime topology in
cosmology to the recent Nobel Prize winning work on topological
insulators, the interactions between topology and physics have been
a triumph over the past few decades.In this eponymous volume, we
are honoured to have contributions from an assembly of grand
masters of the field, guiding us with their world-renowned
expertise on the subject of the interplay between 'Topology' and
'Physics'. Beginning with a preface by Chen Ning Yang on his
recollections of the early days, we proceed to a novel view of
nuclei from the perspective of complex geometry by Sir Michael
Atiyah and Nick Manton, followed by an entree toward recent
developments in two-dimensional gravity and intersection theory on
the moduli space of Riemann surfaces by Robbert Dijkgraaf and
Edward Witten; a study of Majorana fermions and relations to the
Braid group by Louis H Kauffman; a pioneering investigation on
arithmetic gauge theory by Minhyong Kim; an anecdote-enriched
review of singularity theorems in black-hole physics by Sir Roger
Penrose; an adventure beyond anyons by Zhenghan Wang; an apercu on
topological insulators from first-principle calculations by Haijun
Zhang and Shou-Cheng Zhang; finishing with synopsis on quantum
information theory as one of the four revolutions in physics and
the second quantum revolution by Xiao-Gang Wen. We hope that this
book will serve to inspire the research community.
'The book is an engaging and influential collection of significant
contributions from an assembly of world expert leaders and pioneers
from different fields, working at the interface between topology
and physics or applications of topology to physical systems ... The
book explores many interesting and novel topics that lie at the
intersection between gravity, quantum fields, condensed matter,
physical cosmology and topology ... A rich, well-organized, and
comprehensive overview of remarkable and insightful connections
between physics and topology is here made available to the physics
reader.'Contemporary PhysicsSince its birth in Poincare's seminal
1894 'Analysis Situs', topology has become a cornerstone of
mathematics. As with all beautiful mathematical concepts, topology
inevitably - resonating with that Wignerian principle of the
effectiveness of mathematics in the natural sciences - finds its
prominent role in physics. From Chern-Simons theory to topological
quantum field theory, from knot invariants to Calabi-Yau
compactification in string theory, from spacetime topology in
cosmology to the recent Nobel Prize winning work on topological
insulators, the interactions between topology and physics have been
a triumph over the past few decades.In this eponymous volume, we
are honoured to have contributions from an assembly of grand
masters of the field, guiding us with their world-renowned
expertise on the subject of the interplay between 'Topology' and
'Physics'. Beginning with a preface by Chen Ning Yang on his
recollections of the early days, we proceed to a novel view of
nuclei from the perspective of complex geometry by Sir Michael
Atiyah and Nick Manton, followed by an entree toward recent
developments in two-dimensional gravity and intersection theory on
the moduli space of Riemann surfaces by Robbert Dijkgraaf and
Edward Witten; a study of Majorana fermions and relations to the
Braid group by Louis H Kauffman; a pioneering investigation on
arithmetic gauge theory by Minhyong Kim; an anecdote-enriched
review of singularity theorems in black-hole physics by Sir Roger
Penrose; an adventure beyond anyons by Zhenghan Wang; an apercu on
topological insulators from first-principle calculations by Haijun
Zhang and Shou-Cheng Zhang; finishing with synopsis on quantum
information theory as one of the four revolutions in physics and
the second quantum revolution by Xiao-Gang Wen. We hope that this
book will serve to inspire the research community.
The juxtaposition of 'machine learning' and 'pure mathematics and
theoretical physics' may first appear as contradictory in terms.
The rigours of proofs and derivations in the latter seem to reside
in a different world from the randomness of data and statistics in
the former. Yet, an often under-appreciated component of
mathematical discovery, typically not presented in a final draft,
is experimentation: both with ideas and with mathematical data.
Think of the teenage Gauss, who conjectured the Prime Number
Theorem by plotting the prime-counting function, many decades
before complex analysis was formalized to offer a proof.Can modern
technology in part mimic Gauss's intuition? The past five years saw
an explosion of activity in using AI to assist the human mind in
uncovering new mathematics: finding patterns, accelerating
computations, and raising conjectures via the machine learning of
pure, noiseless data. The aim of this book, a first of its kind, is
to collect research and survey articles from experts in this
emerging dialogue between theoretical mathematics and machine
learning. It does not dwell on the well-known multitude of
mathematical techniques in deep learning, but focuses on the
reverse relationship: how machine learning helps with mathematics.
Taking a panoramic approach, the topics range from combinatorics to
number theory, and from geometry to quantum field theory and string
theory. Aimed at PhD students as well as seasoned researchers, each
self-contained chapter offers a glimpse of an exciting future of
this symbiosis.
Live performance of Verdi's opera, recorded in 2013 on the 100th
anniversary of the Verona Arena Festival. The production stars Hui
He, Fabio Sartori and Giovanna Casolla with Omer Meir Wellber
conducting the Orchestra and Chorus of the Arena di Verona.
The increasing demand for materials which enable a more efficient
energy conversion - materials allowing higher operating
temperatures and a lower weight of the components with preferably
better corrosion resistance- has put intermetallic-based alloys
back into focus. For example, the recent use of light weight,
TiAl-based alloys for automotive and aerospace applications has
spurred research on other intermetallic-based alloys. In addition
to structural intermetallics for extreme environments encountered
in advanced energy systems, this book also focuses on fundamental
and interdisciplinary aspects of novel intermetallic-metal systems
(e.g., Co-based superalloys) and functional intermetallics that can
store energy, generate power and enhance reliability. Topics
include: intermetallics for hydrogen storage and thermoelectric
applications; iron aluminides - physical metallurgy; titanium
aluminides - physical metallurgy; titanium aluminides - structure,
properties and coatings; iron aluminides, titanium aluminides,
nickel aluminides and silicides; nickel/cobalt superalloys and
nickel aluminides; niobium and molybdenum silicide-based alloys;
laves phases and fundamental aspects of intermetallics - phase
stability, defects and theory.
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