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Lectures on Invariant Theory

Author : Igor Dolgachev
Publisher : Cambridge University Press
Release : 2003-08-07
Category : Mathematics
ISBN : 0521525489

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Book Lectures on Invariant Theory Description/Summary:

The primary goal of this 2003 book is to give a brief introduction to the main ideas of algebraic and geometric invariant theory. It assumes only a minimal background in algebraic geometry, algebra and representation theory. Topics covered include the symbolic method for computation of invariants on the space of homogeneous forms, the problem of finite-generatedness of the algebra of invariants, the theory of covariants and constructions of categorical and geometric quotients. Throughout, the emphasis is on concrete examples which originate in classical algebraic geometry. Based on lectures given at University of Michigan, Harvard University and Seoul National University, the book is written in an accessible style and contains many examples and exercises. A novel feature of the book is a discussion of possible linearizations of actions and the variation of quotients under the change of linearization. Also includes the construction of toric varieties as torus quotients of affine spaces.

Lectures on Invariant Theory

Author : Igor V. Dolgachev
Publisher : Unknown
Release : 2014-05-14
Category : MATHEMATICS
ISBN : 1107362261

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Book Lectures on Invariant Theory Description/Summary:

This 2003 book is a brief introduction to algebraic and geometric invariant theory with numerous examples and exercises.

Harmonic Analysis, Group Representations, Automorphic Forms, and Invariant Theory

Author : Jian-Shu Li
Publisher : World Scientific
Release : 2007
Category : Mathematics
ISBN : 9789812770783

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Book Harmonic Analysis, Group Representations, Automorphic Forms, and Invariant Theory Description/Summary:

This volume carries the same title as that of an international conference held at the National University of Singapore, 9-11 January 2006 on the occasion of Roger E. Howe's 60th birthday. Authored by leading members of the Lie theory community, these contributions, expanded from invited lectures given at the conference, are a fitting tribute to the originality, depth and influence of Howe's mathematical work. The range and diversity of the topics will appeal to a broad audience of research mathematicians and graduate students interested in symmetry and its profound applications.

Invariant Theory

Author : T.A. Springer
Publisher : Springer
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540373704

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Book Invariant Theory Description/Summary:

Lectures on Seiberg-Witten Invariants

Author : John D. Moore
Publisher : Springer
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540685920

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Book Lectures on Seiberg-Witten Invariants Description/Summary:

In the fall of 1994, Edward Witten proposed a set of equations which give the main results of Donaldson theory in a far simpler way than had been thought possible. The purpose of these notes is to provide an elementary introduction to the equations that Witten proposed. They are directed towards graduate students who have already taken a basic course in differential geometry and topology.

Geometric Invariant Theory and Decorated Principal Bundles

Author : Alexander H. W. Schmitt
Publisher : European Mathematical Society
Release : 2008
Category : Mathematics
ISBN : 3037190655

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Book Geometric Invariant Theory and Decorated Principal Bundles Description/Summary:

The book starts with an introduction to Geometric Invariant Theory (GIT). The fundamental results of Hilbert and Mumford are exposed as well as more recent topics such as the instability flag, the finiteness of the number of quotients, and the variation of quotients. In the second part, GIT is applied to solve the classification problem of decorated principal bundles on a compact Riemann surface. The solution is a quasi-projective moduli scheme which parameterizes those objects that satisfy a semistability condition originating from gauge theory. The moduli space is equipped with a generalized Hitchin map. Via the universal Kobayashi-Hitchin correspondence, these moduli spaces are related to moduli spaces of solutions of certain vortex type equations. Potential applications include the study of representation spaces of the fundamental group of compact Riemann surfaces. The book concludes with a brief discussion of generalizations of these findings to higher dimensional base varieties, positive characteristic, and parabolic bundles. The text is fairly self-contained (e.g., the necessary background from the theory of principal bundles is included) and features numerous examples and exercises. It addresses students and researchers with a working knowledge of elementary algebraic geometry.

Random Matrix Theory

Author : Percy Deift,Dimitri Gioev
Publisher : American Mathematical Soc.
Release : 2009-01-01
Category : Mathematics
ISBN : 9780821883570

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Book Random Matrix Theory Description/Summary:

"This book features a unified derivation of the mathematical theory of the three classical types of invariant random matrix ensembles-orthogonal, unitary, and symplectic. The authors follow the approach of Tracy and Widom, but the exposition here contains a substantial amount of additional material, in particular, facts from functional analysis and the theory of Pfaffians. The main result in the book is a proof of universality for orthogonal and symplectic ensembles corresponding to generalized Gaussian type weights following the authors' prior work. New, quantitative error estimates are derived." --Book Jacket.

Computational Invariant Theory

Author : Harm Derksen,Gregor Kemper
Publisher : Springer
Release : 2015-12-23
Category : Mathematics
ISBN : 9783662484227

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Book Computational Invariant Theory Description/Summary:

This book is about the computational aspects of invariant theory. Of central interest is the question how the invariant ring of a given group action can be calculated. Algorithms for this purpose form the main pillars around which the book is built. There are two introductory chapters, one on Gröbner basis methods and one on the basic concepts of invariant theory, which prepare the ground for the algorithms. Then algorithms for computing invariants of finite and reductive groups are discussed. Particular emphasis lies on interrelations between structural properties of invariant rings and computational methods. Finally, the book contains a chapter on applications of invariant theory, covering fields as disparate as graph theory, coding theory, dynamical systems, and computer vision. The book is intended for postgraduate students as well as researchers in geometry, computer algebra, and, of course, invariant theory. The text is enriched with numerous explicit examples which illustrate the theory and should be of more than passing interest. More than ten years after the first publication of the book, the second edition now provides a major update and covers many recent developments in the field. Among the roughly 100 added pages there are two appendices, authored by Vladimi r Popov, and an addendum by Norbert A'Campo and Vladimir Popov.

EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions

Author : Slava Rychkov
Publisher : Springer
Release : 2016-09-30
Category : Science
ISBN : 9783319436265

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Book EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions Description/Summary:

This primer develops Conformal Field Theory (CFT) from scratch, whereby CFT is viewed as any conformally-invariant theory that describes a fixed point of a renormalization group flow in quantum field theory. The book is divided into four lectures: Lecture 1 addresses the physical foundations of conformal invariance, while Lecture 2 examines the constraints imposed by conformal symmetry on the correlation functions of local operators, presented using the so-called projective null cone – a procedure also known as the embedding formalism. In turn, Lecture 3 focuses on the radial quantization and the operator product expansion, while Lecture 4 offers a very brief introduction to the conformal bootstrap. Derived from course-based notes, these lectures are intended as a first point of entry to this topic for Master and PhD students alike.

Invariant Theory in All Characteristics

Author : Harold Edward Alexander Eddy Campbell,David L. Wehlau
Publisher : American Mathematical Soc.
Release : 2022-06-29
Category : Science
ISBN : 0821870300

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Book Invariant Theory in All Characteristics Description/Summary:

This volume includes the proceedings of a workshop on Invariant Theory held at Queen's University (Ontario). The workshop was part of the theme year held under the auspices of the Centre de recherches mathematiques (CRM) in Montreal. The gathering brought together two communities of researchers: those working in characteristic 0 and those working in positive characteristic. The book contains three types of papers: survey articles providing introductions to computational invarianttheory, modular invariant theory of finite groups, and the invariant theory of Lie groups; expository works recounting recent research in these three areas and beyond; and open problems of current interest. The book is suitable for graduate students and researchers working in invarianttheory.

Geometric Invariant Theory for Polarized Curves

Author : Gilberto Bini,Fabio Felici,Margarida Melo,Filippo Viviani
Publisher : Springer
Release : 2014-11-07
Category : Mathematics
ISBN : 9783319113371

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Book Geometric Invariant Theory for Polarized Curves Description/Summary:

We investigate GIT quotients of polarized curves. More specifically, we study the GIT problem for the Hilbert and Chow schemes of curves of degree d and genus g in a projective space of dimension d-g, as d decreases with respect to g. We prove that the first three values of d at which the GIT quotients change are given by d=a(2g-2) where a=2, 3.5, 4. We show that, for a>4, L. Caporaso's results hold true for both Hilbert and Chow semistability. If 3.5a

The Theory of Algebraic Number Fields

Author : David Hilbert
Publisher : Springer Science & Business Media
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662035450

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Book The Theory of Algebraic Number Fields Description/Summary:

A translation of Hilberts "Theorie der algebraischen Zahlkörper" best known as the "Zahlbericht", first published in 1897, in which he provides an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht also provided a firm foundation for further research in the theory, and can be seen as the starting point for all twentieth century investigations into the subject, as well as reciprocity laws and class field theory. This English edition further contains an introduction by F. Lemmermeyer and N. Schappacher.

Differential Operators on Manifolds

Author : E. Vesentini
Publisher : Springer Science & Business Media
Release : 2011-06-07
Category : Mathematics
ISBN : 9783642111143

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Book Differential Operators on Manifolds Description/Summary:

Lectures: M.F. Atiyah: Classical groups and classical differential operators on manifolds.- R. Bott: Some aspects of invariant theory in differential geometry.- E.M. Stein: Singular integral operators and nilpotent groups.- Seminars: P. Malliavin: Diffusion et géométrie différentielle globale.- S. Helgason: Solvability of invariant differential operators on homonogeneous manifolds.

Geometric Invariant Theory

Author : David Mumford,John Fogarty,Frances Kirwan
Publisher : Springer Science & Business Media
Release : 1994
Category : Mathematics
ISBN : 3540569634

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Book Geometric Invariant Theory Description/Summary:

"Geometric Invariant Theory" by Mumford/Fogarty (the firstedition was published in 1965, a second, enlarged editonappeared in 1982) is the standard reference on applicationsof invariant theory to the construction of moduli spaces.This third, revised edition has been long awaited for by themathematical community. It is now appearing in a completelyupdated and enlarged version with an additional chapter onthe moment map by Prof. Frances Kirwan (Oxford) and a fullyupdated bibliography of work in this area.The book deals firstly with actions of algebraic groups onalgebraic varieties, separating orbits by invariants andconstructionquotient spaces; and secondly with applicationsof this theory to the construction of moduli spaces.It is a systematic exposition of the geometric aspects ofthe classical theory of polynomial invariants.

Lectures on Field Theory and Topology

Author : Daniel S. Freed
Publisher : American Mathematical Soc.
Release : 2019-08-23
Category : Algebraic topology
ISBN : 9781470452063

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Book Lectures on Field Theory and Topology Description/Summary:

These lectures recount an application of stable homotopy theory to a concrete problem in low energy physics: the classification of special phases of matter. While the joint work of the author and Michael Hopkins is a focal point, a general geometric frame of reference on quantum field theory is emphasized. Early lectures describe the geometric axiom systems introduced by Graeme Segal and Michael Atiyah in the late 1980s, as well as subsequent extensions. This material provides an entry point for mathematicians to delve into quantum field theory. Classification theorems in low dimensions are proved to illustrate the framework. The later lectures turn to more specialized topics in field theory, including the relationship between invertible field theories and stable homotopy theory, extended unitarity, anomalies, and relativistic free fermion systems. The accompanying mathematical explanations touch upon (higher) category theory, duals to the sphere spectrum, equivariant spectra, differential cohomology, and Dirac operators. The outcome of computations made using the Adams spectral sequence is presented and compared to results in the condensed matter literature obtained by very different means. The general perspectives and specific applications fuse into a compelling story at the interface of contemporary mathematics and theoretical physics.