Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

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Publisher :
ISBN 13 : 9781470405106
Total Pages : 69 pages
Book Rating : 4.4/5 (51 download)

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Book Synopsis Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces by : William Mark Goldman

Download or read book Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces written by William Mark Goldman and published by . This book was released on 2008 with total page 69 pages. Available in PDF, EPUB and Kindle. Book excerpt: Details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. This work constructs an equivalence between the deformation theories of flat connections and Higgs pairs, providing an identification of moduli spaces arising in different contexts.

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

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Publisher : American Mathematical Soc.
ISBN 13 : 082184136X
Total Pages : 86 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces by : William Mark Goldman

Download or read book Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces written by William Mark Goldman and published by American Mathematical Soc.. This book was released on 2008 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt: This expository article details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. The authors construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyperkähler structure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of $X$. The authors describe the moduli spaces and their geometry in terms of the Riemann period matrix of $X$.

Problems on Mapping Class Groups and Related Topics

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Publisher : American Mathematical Soc.
ISBN 13 : 0821838385
Total Pages : 384 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Problems on Mapping Class Groups and Related Topics by : Benson Farb

Download or read book Problems on Mapping Class Groups and Related Topics written by Benson Farb and published by American Mathematical Soc.. This book was released on 2006-09-12 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: The appearance of mapping class groups in mathematics is ubiquitous. The book presents 23 papers containing problems about mapping class groups, the moduli space of Riemann surfaces, Teichmuller geometry, and related areas. Each paper focusses completely on open problems and directions. The problems range in scope from specific computations, to broad programs. The goal is to have a rich source of problems which have been formulated explicitly and accessibly. The book is divided into four parts. Part I contains problems on the combinatorial and (co)homological group-theoretic aspects of mapping class groups, and the way in which these relate to problems in geometry and topology. Part II concentrates on connections with classification problems in 3-manifold theory, the theory of symplectic 4-manifolds, and algebraic geometry. A wide variety of problems, from understanding billiard trajectories to the classification of Kleinian groups, can be reduced to differential and synthetic geometry problems about moduli space. Such problems and connections are discussed in Part III. Mapping class groups are related, both concretely and philosophically, to a number of other groups, such as braid groups, lattices in semisimple Lie groups, and automorphism groups of free groups. Part IV concentrates on problems surrounding these relationships. This book should be of interest to anyone studying geometry, topology, algebraic geometry or infinite groups. It is meant to provide inspiration for everyone from graduate students to senior researchers.

Geometry and Physics: Volume 2

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Publisher : Oxford University Press
ISBN 13 : 019252237X
Total Pages : 400 pages
Book Rating : 4.1/5 (925 download)

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Book Synopsis Geometry and Physics: Volume 2 by : Jørgen Ellegaard Andersen

Download or read book Geometry and Physics: Volume 2 written by Jørgen Ellegaard Andersen and published by Oxford University Press. This book was released on 2018-10-18 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nigel Hitchin is one of the world's foremost figures in the fields of differential and algebraic geometry and their relations with mathematical physics, and he has been Savilian Professor of Geometry at Oxford since 1997. Geometry and Physics: A Festschrift in honour of Nigel Hitchin contain the proceedings of the conferences held in September 2016 in Aarhus, Oxford, and Madrid to mark Nigel Hitchin's 70th birthday, and to honour his far-reaching contributions to geometry and mathematical physics. These texts contain 29 articles by contributors to the conference and other distinguished mathematicians working in related areas, including three Fields Medallists. The articles cover a broad range of topics in differential, algebraic and symplectic geometry, and also in mathematical physics. These volumes will be of interest to researchers and graduate students in geometry and mathematical physics.

The Many Facets of Geometry

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Publisher : OUP Oxford
ISBN 13 : 0191567574
Total Pages : 456 pages
Book Rating : 4.1/5 (915 download)

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Book Synopsis The Many Facets of Geometry by : Oscar Garcia-Prada

Download or read book The Many Facets of Geometry written by Oscar Garcia-Prada and published by OUP Oxford. This book was released on 2010-07-01 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: Few people have proved more influential in the field of differential and algebraic geometry, and in showing how this links with mathematical physics, than Nigel Hitchin. Oxford University's Savilian Professor of Geometry has made fundamental contributions in areas as diverse as: spin geometry, instanton and monopole equations, twistor theory, symplectic geometry of moduli spaces, integrables systems, Higgs bundles, Einstein metrics, hyperkähler geometry, Frobenius manifolds, Painlevé equations, special Lagrangian geometry and mirror symmetry, theory of grebes, and many more. He was previously Rouse Ball Professor of Mathematics at Cambridge University, as well as Professor of Mathematics at the University of Warwick, is a Fellow of the Royal Society and has been the President of the London Mathematical Society. The chapters in this fascinating volume, written by some of the greats in their fields (including four Fields Medalists), show how Hitchin's ideas have impacted on a wide variety of subjects. The book grew out of the Geometry Conference in Honour of Nigel Hitchin, held in Madrid, with some additional contributions, and should be required reading for anyone seeking insights into the overlap between geometry and physics.

Vertex Operator Algebras, Number Theory and Related Topics

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Publisher : American Mathematical Soc.
ISBN 13 : 1470449382
Total Pages : 250 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Vertex Operator Algebras, Number Theory and Related Topics by : Matthew Krauel

Download or read book Vertex Operator Algebras, Number Theory and Related Topics written by Matthew Krauel and published by American Mathematical Soc.. This book was released on 2020-07-13 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the International Conference on Vertex Operator Algebras, Number Theory, and Related Topics, held from June 11–15, 2018, at California State University, Sacramento, California. The mathematics of vertex operator algebras, vector-valued modular forms and finite group theory continues to provide a rich and vibrant landscape in mathematics and physics. The resurgence of moonshine related to the Mathieu group and other groups, the increasing role of algebraic geometry and the development of irrational vertex operator algebras are just a few of the exciting and active areas at present. The proceedings center around active research on vertex operator algebras and vector-valued modular forms and offer original contributions to the areas of vertex algebras and number theory, surveys on some of the most important topics relevant to these fields, introductions to new fields related to these and open problems from some of the leaders in these areas.

Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories

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Publisher : American Mathematical Soc.
ISBN 13 : 0821841424
Total Pages : 208 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories by : Dominic Verity

Download or read book Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories written by Dominic Verity and published by American Mathematical Soc.. This book was released on 2008 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: The primary purpose of this work is to characterise strict $\omega$-categories as simplicial sets with structure. The author proves the Street-Roberts conjecture in the form formulated by Ross Street in his work on Orientals, which states that they are exactly the ``complicial sets'' defined and named by John Roberts in his handwritten notes of that title (circa 1978). On the way the author substantially develops Roberts' theory of complicial sets itself and makes contributions to Street's theory of parity complexes. In particular, he studies a new monoidal closed structure on the category of complicial sets which he shows to be the appropriate generalisation of the (lax) Gray tensor product of 2-categories to this context. Under Street's $\omega$-categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others.

Torus Fibrations, Gerbes, and Duality

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Publisher : American Mathematical Soc.
ISBN 13 : 0821840924
Total Pages : 104 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Torus Fibrations, Gerbes, and Duality by : Ron Donagi

Download or read book Torus Fibrations, Gerbes, and Duality written by Ron Donagi and published by American Mathematical Soc.. This book was released on 2008 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let $X$ be a smooth elliptic fibration over a smooth base $B$. Under mild assumptions, the authors establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an $\mathcal{O} DEGREES{\times}$ gerbe over a genus one fibration which is a twisted form

Invariant Differential Operators for Quantum Symmetric Spaces

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Publisher : American Mathematical Soc.
ISBN 13 : 0821841319
Total Pages : 104 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Invariant Differential Operators for Quantum Symmetric Spaces by : Gail Letzter

Download or read book Invariant Differential Operators for Quantum Symmetric Spaces written by Gail Letzter and published by American Mathematical Soc.. This book was released on 2008 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of Harish-Chandra and Helgason: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials.

Representations of Shifted Yangians and Finite $W$-algebras

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Publisher : American Mathematical Soc.
ISBN 13 : 0821842161
Total Pages : 122 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Representations of Shifted Yangians and Finite $W$-algebras by : Jonathan Brundan

Download or read book Representations of Shifted Yangians and Finite $W$-algebras written by Jonathan Brundan and published by American Mathematical Soc.. This book was released on 2008 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors study highest weight representations of shifted Yangians over an algebraically closed field of characteristic $0$. In particular, they classify the finite dimensional irreducible representations and explain how to compute their Gelfand-Tsetlin characters in terms of known characters of standard modules and certain Kazhdan-Lusztig polynomials. The authors' approach exploits the relationship between shifted Yangians and the finite W-algebras associated to nilpotent orbits in general linear Lie algebras.

The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic

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Publisher : American Mathematical Soc.
ISBN 13 : 0821843699
Total Pages : 168 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic by : Irina D. Suprunenko

Download or read book The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic written by Irina D. Suprunenko and published by American Mathematical Soc.. This book was released on 2009-06-05 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: The minimal polynomials of the images of unipotent elements in irreducible rational representations of the classical algebraic groups over fields of odd characteristic are found. These polynomials have the form $(t-1)^d$ and hence are completely determined by their degrees. In positive characteristic the degree of such polynomial cannot exceed the order of a relevant element. It occurs that for each unipotent element the degree of its minimal polynomial in an irreducible representation is equal to the order of this element provided the highest weight of the representation is large enough with respect to the ground field characteristic. On the other hand, classes of unipotent elements for which in every nontrivial representation the degree of the minimal polynomial is equal to the order of the element are indicated. In the general case the problem of computing the minimal polynomial of the image of a given element of order $p^s$ in a fixed irreducible representation of a classical group over a field of characteristic $p>2$ can be reduced to a similar problem for certain $s$ unipotent elements and a certain irreducible representation of some semisimple group over the field of complex numbers. For the latter problem an explicit algorithm is given. Results of explicit computations for groups of small ranks are contained in Tables I-XII. The article may be regarded as a contribution to the programme of extending the fundamental results of Hall and Higman (1956) on the minimal polynomials from $p$-solvable linear groups to semisimple groups.

Compactification of the Drinfeld Modular Surfaces

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Publisher : American Mathematical Soc.
ISBN 13 : 0821842447
Total Pages : 113 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Compactification of the Drinfeld Modular Surfaces by : Thomas Lehmkuhl

Download or read book Compactification of the Drinfeld Modular Surfaces written by Thomas Lehmkuhl and published by American Mathematical Soc.. This book was released on 2009-01-21 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this article the author describes in detail a compactification of the moduli schemes representing Drinfeld modules of rank 2 endowed with some level structure. The boundary is a union of copies of moduli schemes for Drinfeld modules of rank 1, and its points are interpreted as Tate data. The author also studies infinitesimal deformations of Drinfeld modules with level structure.

Yang-Mills Connections on Orientable and Nonorientable Surfaces

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Publisher : American Mathematical Soc.
ISBN 13 : 0821844911
Total Pages : 113 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Yang-Mills Connections on Orientable and Nonorientable Surfaces by : Nan-Kuo Ho

Download or read book Yang-Mills Connections on Orientable and Nonorientable Surfaces written by Nan-Kuo Ho and published by American Mathematical Soc.. This book was released on 2009-10-08 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: In ``The Yang-Mills equations over Riemann surfaces'', Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory. In ``Yang-Mills Connections on Nonorientable Surfaces'', the authors study Yang-Mills functional on the space of connections on a principal $G_{\mathbb{R}}$-bundle over a closed, connected, nonorientable surface, where $G_{\mathbb{R}}$ is any compact connected Lie group. In this monograph, the authors generalize the discussion in ``The Yang-Mills equations over Riemann surfaces'' and ``Yang-Mills Connections on Nonorientable Surfaces''. They obtain explicit descriptions of equivariant Morse stratification of Yang-Mills functional on orientable and nonorientable surfaces for non-unitary classical groups $SO(n)$ and $Sp(n)$.

Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets

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Publisher : American Mathematical Soc.
ISBN 13 : 0821842927
Total Pages : 110 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets by : Jun Kigami

Download or read book Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets written by Jun Kigami and published by American Mathematical Soc.. This book was released on 2009-04-10 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper studies the following three problems. 1. When does a measure on a self-similar set have the volume doubling property with respect to a given distance? 2. Is there any distance on a self-similar set under which the contraction mappings have the prescribed values of contractions ratios? 3. When does a heat kernel on a self-similar set associated with a self-similar Dirichlet form satisfy the Li-Yau type sub-Gaussian diagonal estimate? These three problems turn out to be closely related. The author introduces a new class of self-similar set, called rationally ramified self-similar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and gives complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper Li-Yau type sub-Gaussian diagonal estimate of a heat kernel.

Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

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Publisher : American Mathematical Soc.
ISBN 13 : 0821843184
Total Pages : 124 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces by : Volkmar Liebscher

Download or read book Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces written by Volkmar Liebscher and published by American Mathematical Soc.. This book was released on 2009-04-10 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: In a series of papers Tsirelson constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So the author connects each continuous tensor product system of Hilbert spaces with measure types of distributions of random (closed) sets in $[0,1]$ or $\mathbb R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the Tsirelson examples, that the classification scheme for product systems into types $\mathrm{I}_n$, $\mathrm{II}_n$ and $\mathrm{III}$ is not complete. Moreover, based on a detailed study of this kind of measure types, the author constructs for each stationary factorising measure type a continuous tensor product system of Hilbert spaces such that this measure type arises as the before mentioned invariant.

Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups

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Publisher : American Mathematical Soc.
ISBN 13 : 0821844903
Total Pages : 176 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups by : Drew Armstrong

Download or read book Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups written by Drew Armstrong and published by American Mathematical Soc.. This book was released on 2009-10-08 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: This memoir is a refinement of the author's PhD thesis -- written at Cornell University (2006). It is primarily a desription of new research but also includes a substantial amount of background material. At the heart of the memoir the author introduces and studies a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and each positive integer $k$. When $k=1$, his definition coincides with the generalized noncrossing partitions introduced by Brady and Watt in $K(\pi, 1)$'s for Artin groups of finite type and Bessis in The dual braid monoid. When $W$ is the symmetric group, the author obtains the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman in Chain enumeration and non-crossing partitions.

Regular Subgroups of Primitive Permutation Groups

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Publisher : American Mathematical Soc.
ISBN 13 : 082184654X
Total Pages : 87 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Regular Subgroups of Primitive Permutation Groups by : Martin W. Liebeck

Download or read book Regular Subgroups of Primitive Permutation Groups written by Martin W. Liebeck and published by American Mathematical Soc.. This book was released on 2010 with total page 87 pages. Available in PDF, EPUB and Kindle. Book excerpt: Addresses the classical problem of determining finite primitive permutation groups G with a regular subgroup B.