Part of London Mathematical Society Lecture Note Series
PUBLICATION PLANNED FOR: June 2023
available from June 202
3FORMAT: Paperback
ISBN: 9781009193399
Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that
can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 irreducible deformation
families. This book determines whether the general element of each family admits a Kahler?Einstein metric (and for many
families, for all elements), addressing a question going back to Calabi 70 years ago. The book's solution exploits the relation
between these metrics and the algebraic notion of K-stability. Moreover, the book presents many different techniques to
prove the existence of a Kahler?Einstein metric, containing many additional relevant results such as the classification of
all Kahler?Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all
smooth del Pezzo surfaces. This book will be essential reading for researchers and graduate students working on algebraic
geometry and complex geometry.
'The notion of K-stability for Fano manifold has origins in differential
geometry and geometric analysis but is now also of fundamental importance
in algebraic geometry, with recent developments in moduli theory. This
monograph gives an
account of a large body of research results from the last decade, studying in depth the case of Fano threefolds. The
wealth of material combines in a most attractive way sophisticated modern theory and the detailed study of examples, with
a classical flavour. The authors obtain complete results on the K-stability of generic elements of each of the 105
deformation classes. The concluding chapter contains some fascinating conjectures about the 34 families which may
contain both stable and unstable manifolds, which will surely be the scene for much further work. The book will be an
essential reference for many years to come.' Sir Simon Donaldson, F.R.S., Imperial College London
Introduction
1. K-stability
2. Warm up: smooth del Pezzo surfaces
3. Proof of main theorem: known cases
4. Proof of main theorem: special cases
5. Proof of main theorem: remaining cases
6. The big table
7. Conclusion
Appendix. Technical results used in the proof of main theorem
References
Index.
Part of Cambridge Studies in Advanced Mathematics
available from June 2023
FORMAT: Hardback ISBN: 9781009280006
Cubic hypersurfaces are described by almost the simplest possible polynomial equations, yet their behaviour is rich enough
to demonstrate many of the central challenges in algebraic geometry. With
exercises and detailed references to the wider
literature, this thorough text introduces cubic hypersurfaces and all the techniques needed to study them. The book
starts by laying the foundations for the study of cubic hypersurfaces and of many other algebraic varieties, covering
cohomology and Hodge theory of hypersurfaces, moduli spaces of those and Fano varieties of linear subspaces contained
in hypersurfaces. The next three chapters examine the general machinery applied to cubic hypersurfaces of dimension
two, three, and four. Finally, the author looks at cubic hypersurfaces from a categorical point of view and describes motivic
features. Based on the author's lecture courses, this is an ideal text for graduate students as well as an invaluable
reference for researchers in algebraic geometry.
1. Basic facts
2. Fano varieties of lines
3. Moduli spaces
4. Cubic surfaces
5. Cubic threefolds
6. Cubic fourfolds
7. Derived categories of cubic hypersurfaces
References
Index of notation
Subject index.
*
Part of Cambridge Studies in Advanced Mathematics
available from June 2023
FORMAT: Hardback ISBN: 9780521113618
The first edition of this book provided the first systematic exposition of the arithmetic theory of algebraic groups. This
revised second edition, now published in two volumes, retains the same goals, while incorporating corrections and
improvements, as well as new material covering more recent developments. Volume I begins with chapters covering
background material on number theory, algebraic groups, and cohomology (both abelian and non-abelian), and then turns to
algebraic groups over locally compact fields. The remaining two chapters provide a detailed treatment of arithmetic
subgroups and reduction theory in both the real and adelic settings. Volume II includes new material on groups with
bounded generation and abstract arithmetic groups. With minimal prerequisites and complete proofs given whenever
possible, this book is suitable for self-study for graduate students wishing to learn the subject as well as a reference for
researchers in number theory, algebraic geometry, and related areas.
Provides extended, self-contained accounts of necessary background from number theory and the theory of algebraic
groups, making the book accessible to graduate students and researchers in a variety of areas
Contains an exposition of the classification of classical groups using techniques of non-abelian cohomology, which enables
readers to become familiar with these ideas in the context of very concrete problems
Provides a detailed account of arithmetic groups and reduction theory, allowing readers to see complete proofs of
important results that are widely applied in number theory and related areas
1. Algebraic number theory
2. Algebraic groups
3. Algebraic groups over locally compact fields
4. Arithmetic groups and reduction theory
5. Adeles
References
Index.
available from July 2023
FORMAT: Paperback ISBN: 9781009179911
Written by Sheldon Ross and Erol Pekoz, this text familiarises you with advanced topics in probability while keeping the
mathematical prerequisites to a minimum. Topics covered include measure theory, limit theorems, bounding probabilities
and expectations, coupling and Stein's method, martingales, Markov chains, renewal theory, and Brownian motion. No other
text covers all these topics rigorously but at such an accessible level - all you need is an undergraduate-level
understanding of calculus and probability. New to this edition are sections on the gambler's ruin problem, Stein's method as
applied to exponential approximations, and applications of the martingale stopping theorem. Extra end-of-chapter exercises
have also been added, with selected solutions available.This is an ideal textbook for students taking an advanced
undergraduate or graduate course in probability. It also represents a useful resource for professionals in relevant
application domains, from finance to machine learning.
Provides an intuitive and modern approach to theoretical probability, highlighting a range of its potential applications
Tests and develops understanding with exercises at the end of each chapter, with selected solutions available
New edition includes additional coverage of Stein's method, the martingale stopping theorem, the gambler's ruin problem,
and coupling applications to renewal and other stochastic processes
Features proofs that emphasise intuition rather than more formal ones wherever possible
Preface
1. Measure Theory and Laws of Large Numbers
2. Stein's Method and Central Limit Theorems
3. Conditional Expectation and Martingales
4. Bounding Probabilities and Expectations
5. Markov Chains
6. Renewal Theory
7. Brownian Motion
References
Index.
Part of London Mathematical Society Lecture Note Series
available from August 2023
FORMAT: Paperback ISBN: 9781009097352
Expanding upon the material delivered during the LMS Autumn Algebra School 2020, this volume reflects the fruitful
connections between different aspects of representation theory. Each survey article addresses a specific subject from a
modern angle, beginning with an exploration of the representation theory of associative algebras. The next four chapters
cover important developments of Lie theory in the past two decades, before the final sections introduce the reader to
three strikingly different aspects of group theory. Written at a level suitable for graduate students and researchers in
related fields, this book provides pure mathematicians with a springboard into the vast and growing literature in each area.
Covers a vast array of subjects in algebra and representation theory
Accessible to graduate students, as well as to pure mathematicians working in algebra and its applications
Contains many worked examples and applications, helping the reader to understand
abstract concepts
Introduction David Jordan, Nadia Mazza and Sibylle Schroll
1. Auslander-Reiten theory of finite dimensional algebras Raquel Coelho Simoes
2. Ą-Tilting theory ? an introduction Hipolito Treffinger
3. From frieze patterns to cluster categories Matthew Pressland
4. Infinite-dimensional representations of algebras Rosanna Laking
5. The Springer correspondence Sam Gunningham
6. An introduction to diagrammatic Soergel bimodules Amit Hazi
7. A companion to quantum groups Bart Vlaar
8. Infinite-dimensional Lie algebras and their multivariable generalizations Brian Williams
9. An introduction to crowns in finite groups Gareth Tracey
10. An introduction to totally disconnected locally compact groups and their finiteness conditions Ilaria Castellano
11. Locally analytic representations of p-adic groups Andreas Bode and Nicolas Dupre.