Download or read book The Beilinson Complex and Canonical Rings of Irregular Surfaces written by Alberto Canonaco and published by American Mathematical Soc.. This book was released on 2006 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: An important theorem by Beilinson describes the bounded derived category of coherent sheaves on $\mathbb{P n$, yielding in particular a resolution of every coherent sheaf on $\mathbb{P n$ in terms of the vector bundles $\Omega {\mathbb{P n j(j)$ for $0\le j\le n$. This theorem is here extended to weighted projective spaces. To this purpose we consider, instead of the usual category of coherent sheaves on $\mathbb{P ({\rm w )$ (the weighted projective space of weights $\rm w=({\rm w 0,\dots,{\rm w n)$), a suitable category of graded coherent sheaves (the two categories are equivalent if and only if ${\rm w 0=\cdots={\rm w n=1$, i.e. $\mathbb{P ({\rm w )= \mathbb{P n$), obtained by endowing $\mathbb{P ({\rm w )$ with a natural graded structure sheaf. The resulting graded ringed space $\overline{\mathbb{P ({\rm w )$ is an example of graded scheme (in chapter 1 graded schemes are defined and studied in some greater generality than is needed in the rest of the work). Then in chapter 2 we prove This weighted version of Beilinson's theorem is then applied in chapter 3 to prove a structure theorem for good birational weighted canonical projections of surfaces of general type (i.e., for morphisms, which are birational onto the image, from a minimal surface of general type $S$ into a $3$-dimensional $\mathbb{P ({\rm w )$, induced by $4$ sections $\sigma i\in H0(S,\mathcal{O S({\rm w iK S))$). This is a generalization of a theorem by Catanese and Schreyer (who treated the case of projections into $\mathbb{P 3$), and is mainly interesting for irregular surfaces, since in the regular case a similar but simpler result (due to Catanese) was already known. The theorem essentially states that giving a good birational weighted canonical projection is equivalent to giving a symmetric morphism of (graded) vector bundles on $\overline{\mathbb{P ({\rm w )$, satisfying some suitable conditions. Such a morphism is then explicitly determined in chapter 4 for a family of surfaces with numerical invariant
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 $X$.
Download or read book Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I Generic Covers and Covers with Many Branch Points written by Robert M. Guralnick and published by American Mathematical Soc.. This book was released on 2007 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: Considers indecomposable degree $n$ covers of Riemann surfaces with monodromy group an alternating or symmetric group of degree $d$. The authors show that if the cover has five or more branch points then the genus grows rapidly with $n$ unless either $d = n$ or the curves have genus zero, there are precisely five branch points and $n =d(d-1)/2$.
Download or read book The Legacy of Niels Henrik Abel written by Olav Arnfinn Laudal and published by Springer Science & Business Media. This book was released on 2011-06-28 with total page 785 pages. Available in PDF, EPUB and Kindle. Book excerpt: A unique series of fascinating research papers on subjects related to the work of Niels Henrik Abel, written by some of the foremost specialists in their fields. Some of the authors have been specifically invited to present papers, discussing the influence of Abel in a mathematical-historical context. Others have submitted papers presented at the Abel Bicentennial Conference, Oslo June 3-8, 2002. The idea behind the book has been to produce a text covering a substantial part of the legacy of Abel, as perceived at the beginning of the 21st century.
Download or read book Galois Extensions of Structured Ring Spectra Stably Dualizable Groups written by John Rognes and published by American Mathematical Soc.. This book was released on 2008 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.
Download or read book Eigenvalues and Completeness for Regular and Simply Irregular Two Point Differential Operators written by John Locker and published by American Mathematical Soc.. This book was released on 2008 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph the author develops the spectral theory for an $n$th order two-point differential operator $L$ in the Hilbert space $L2[0,1]$, where $L$ is determined by an $n$th order formal differential operator $\ell$ having variable coefficients and by $n$ linearly independent boundary values $B 1, \ldots, B n$. Using the Birkhoff approximate solutions of the differential equation $(\rhon I - \ell)u = 0$, the differential operator $L$ is classified as belonging to one of threepossible classes: regular, simply irregular, or degenerate irregular. For the regular and simply irregular classes, the author develops asymptotic expansions of solutions of the differential equation $(\rhon I - \ell)u = 0$, constructs the characteristic determinant and Green's function,characterizes the eigenvalues and the corresponding algebraic multiplicities and ascents, and shows that the generalized eigenfunctions of $L$ are complete in $L2[0,1]$. He also gives examples of degenerate irregular differential operators illustrating some of the unusual features of this class.
Download or read book The Structure of the Rational Concordance Group of Knots written by Jae Choon Cha and published by American Mathematical Soc.. This book was released on 2007 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author studies the group of rational concordance classes of codimension two knots in rational homology spheres. He gives a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, he relates these invariants with limiting behaviour of the Artin reciprocity over an infinite tower of number fields and analyzes it using tools from algebraic number theory. In higher dimensions it classifies the rational concordance group of knots whose ambient space satisfies a certain cobordism theoretic condition. In particular, he constructs infinitely many torsion elements. He shows that the structure of the rational concordance group is much more complicated than the integral concordance group from a topological viewpoint. He also investigates the structure peculiar to knots in rational homology 3-spheres. To obtain further nontrivial obstructions in this dimension, he develops a technique of controlling a certain limit of the von Neumann $L 2$-signature invariants.
Download or read book Projective Group Structures as Absolute Galois Structures with Block Approximation written by Dan Haran and published by American Mathematical Soc.. This book was released on 2007 with total page 70 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors prove: A proper profinite group structure G is projective if and only if G is the absolute Galois group structure of a proper field-valuation structure with block approximation.
Download or read book The Universal Kobayashi Hitchin Correspondence on Hermitian Manifolds written by Martin Lübke and published by American Mathematical Soc.. This book was released on 2006 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds, and we discuss differential geometric properties of the corresponding moduli spaces. This correspondence refers to moduli spaces of ``universal holomorphic oriented pairs''. Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pairs, holomorphic Higgs pairs, Witten triples, arbitrary quiver moduli problems) are special cases of this universal classification problem. Our Kobayashi-Hitchin correspondence relates the complex geometric concept ``polystable oriented holomorphic pair'' to the existence of a reduction solving a generalized Hermitian-Einstein equation. The proof is based on the Uhlenbeck-Yau continuity method. Using ideas from Donaldson theory, we further introduce and investigate canonical Hermitian metrics on such moduli spaces. We discuss in detail remarkable classes of moduli spaces in the non-Kahlerian framework: Oriented holomorphic structures, Quot-spaces, oriented holomorphic pairs and oriented vortices, non-abelian Seiberg-Witten monopoles.
Download or read book Weil Petersson Metric on the Universal Teichmuller Space written by Leon Armenovich Takhtadzhi︠a︡n and published by American Mathematical Soc.. This book was released on 2006 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this memoir, we prove that the universal Teichmuller space $T(1)$ carries a new structure of a complex Hilbert manifold and show that the connected component of the identity of $T(1)$ -- the Hilbert submanifold $T {0 (1)$ -- is a topological group. We define a Weil-Petersson metric on $T(1)$ by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that $T(1)$ is a Kahler-Einstein manifold with negative Ricci and sectional curvatures. We introduce and compute Mumford-Miller-Morita characteristic forms for the vertical tangent bundle of the universal Teichmuller curve fibration over the universal Teichmuller space. As an application, we derive Wolpert curvature formulas for the finite-dimensional Teichmuller spaces from the formulas for the universal Teichmuller space. We study in detail the Hilbert manifold structure on $T {0 (1)$ and characterize points on $T {0 (1)$ in terms of Bers and pre-Bers embeddings by proving that the Grunsky operators $B {1 $ and The results of this memoir were presented in our e-prints: Weil-Petersson metric on the universal Teichmuller space I. Curvature properties and Chern forms, arXiv:math.CV/0312172 (2003), and Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping, arXiv:math.CV/0406408 (2004).
Download or read book Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds written by John M. Lee and published by American Mathematical Soc.. This book was released on 2006 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Volume 183, number 864 (end of volume)."
Download or read book Noncommutative Maslov Index and Eta Forms written by Charlotte Wahl and published by American Mathematical Soc.. This book was released on 2007 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author defines and proves a noncommutative generalization of a formula relating the Maslov index of a triple of Lagrangian subspaces of a symplectic vector space to eta-invariants associated to a pair of Lagrangian subspaces. The noncommutative Maslov index, defined for modules over a $C *$-algebra $\mathcal{A}$, is an element in $K_0(\mathcal{A})$. The generalized formula calculates its Chern character in the de Rham homology of certain dense subalgebras of $\mathcal{A}$. The proof is a noncommutative Atiyah-Patodi-Singer index theorem for a particular Dirac operator twisted by an $\mathcal{A}$-vector bundle. The author develops an analytic framework for this type of index problem.
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.
Download or read book Semisolvability of Semisimple Hopf Algebras of Low Dimension written by Sonia Natale and published by American Mathematical Soc.. This book was released on 2007 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author proves that every semisimple Hopf algebra of dimension less than $60$ over an algebraically closed field $k$ of characteristic zero is either upper or lower semisolvable up to a cocycle twist.
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.
Download or read book Entropy and Multivariable Interpolation written by Gelu Popescu and published by American Mathematical Soc.. This book was released on 2006 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: We define a new notion of entropy for operators on Fock spaces and positive multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (e.g., multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive multi-Toeplitz kernels on free semigroups (resp. multi-analytic operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra $F ninfty$. We obtain several geometric characterizations of the central intertwining lifting, a maximal principle, and a permanence principle for the noncommutative commutant lifting theorem. Under certain natural conditions, we find explicit forms for the maximal entropy solution of this multivariable commutant lifting theorem. All these results are used to solve maximal entropy interpolation problems in several variables. We obtain explicit forms for the maximal entropy solution (as well as its entropy) of the Sarason, Caratheodory-Schur, and Nevanlinna-Pick type interpolation problems for the noncommutative (resp. commutative) analytic Toeplitz algebra $F ninfty$ (resp. $W ninfty$) and their tensor products with $B({\mathcal H , {\mathcal K )$. In particular, we provide explicit forms for the maximal entropy solutions of several interpolation problems on the unit ball of $\mathbb{C n$.
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
