Faithfully Quadratic Rings
Faithfully Quadratic Rings

Author: M. Dickmann

Publisher: American Mathematical Soc.

Published: 2015-10-27

Total Pages: 148

ISBN-13: 1470414686

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In this monograph the authors extend the classical algebraic theory of quadratic forms over fields to diagonal quadratic forms with invertible entries over broad classes of commutative, unitary rings where is not a sum of squares and is invertible. They accomplish this by: (1) Extending the classical notion of matrix isometry of forms to a suitable notion of -isometry, where is a preorder of the given ring, , or . (2) Introducing in this context three axioms expressing simple properties of (value) representation of elements of the ring by quadratic forms, well-known to hold in the field case.

The Local Structure Theorem for Finite Groups With a Large $p$-Subgroup
The Local Structure Theorem for Finite Groups With a Large $p$-Subgroup

Author: U. Meierfrankenfeld

Publisher: American Mathematical Soc.

Published: 2016-06-21

Total Pages: 356

ISBN-13: 1470418770

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Let p be a prime, G a finite Kp-group S a Sylow p-subgroup of G and Q a large subgroup of G in S (i.e., CG(Q)≤Q and NG(U)≤NG(Q) for 1≠U≤CG(Q)). Let L be any subgroup of G with S≤L, Op(L)≠1 and Q⋬L. In this paper the authors determine the action of L on the largest elementary abelian normal p-reduced p-subgroup YL of L.

Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces
Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces

Author: Ariel Barton:

Publisher: American Mathematical Soc.

Published: 2016-09-06

Total Pages: 122

ISBN-13: 1470419890

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This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted Lp classes. The authors establish: (1) Mapping properties for the double and single layer potentials, as well as the Newton potential; (2) Extrapolation-type solvability results: the fact that solvability of the Dirichlet or Neumann boundary value problem at any given Lp space automatically assures their solvability in an extended range of Besov spaces; (3) Well-posedness for the non-homogeneous boundary value problems. In particular, the authors prove well-posedness of the non-homogeneous Dirichlet problem with data in Besov spaces for operators with real, not necessarily symmetric, coefficients.

An Inverse Spectral Problem Related to the Geng-Xue Two-Component Peakon Equation
An Inverse Spectral Problem Related to the Geng-Xue Two-Component Peakon Equation

Author: Hans Lundmark

Publisher: American Mathematical Soc.

Published: 2016-10-05

Total Pages: 102

ISBN-13: 1470420260

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The authors solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral problems for those equations, this one is of a "discrete cubic string" type, but presents some interesting novel features.

Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting
Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting

Author: J. P. Pridham

Publisher: American Mathematical Soc.

Published: 2016-09-06

Total Pages: 190

ISBN-13: 1470419815

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The author defines and constructs mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. The author also shows that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x−i), giving new results even for simply connected varieties. The mixed Hodge structures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these structures all become real variations of mixed Hodge structure.

Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities
Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities

Author: Bart Bories

Publisher: American Mathematical Soc.

Published: 2016-06-21

Total Pages: 146

ISBN-13: 147041841X

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In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerate surface singularity. The authors start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f∈Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerate with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of f−1(0)⊂C3 close to the origin. Essentially the entire paper is dedicated to proving that, for f as above, certain candidate poles of Igusa's p-adic zeta function of f, arising from so-called B1-facets of the Newton polyhedron of f, are actually not poles. This turns out to be much harder than in the topological setting. The combinatorial proof is preceded by a study of the integral points in three-dimensional fundamental parallelepipeds. Together with the work of Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture for the p-adic and motivic zeta function of a non-degenerate surface singularity.

Quadratic and Hermitian Forms over Rings
Quadratic and Hermitian Forms over Rings

Author: Max-Albert Knus

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 536

ISBN-13: 3642754015

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From its birth (in Babylon?) till 1936 the theory of quadratic forms dealt almost exclusively with forms over the real field, the complex field or the ring of integers. Only as late as 1937 were the foundations of a theory over an arbitrary field laid. This was in a famous paper by Ernst Witt. Still too early, apparently, because it took another 25 years for the ideas of Witt to be pursued, notably by Albrecht Pfister, and expanded into a full branch of algebra. Around 1960 the development of algebraic topology and algebraic K-theory led to the study of quadratic forms over commutative rings and hermitian forms over rings with involutions. Not surprisingly, in this more general setting, algebraic K-theory plays the role that linear algebra plays in the case of fields. This book exposes the theory of quadratic and hermitian forms over rings in a very general setting. It avoids, as far as possible, any restriction on the characteristic and takes full advantage of the functorial aspects of the theory. The advantage of doing so is not only aesthetical: on the one hand, some classical proofs gain in simplicity and transparency, the most notable examples being the results on low-dimensional spinor groups; on the other hand new results are obtained, which went unnoticed even for fields, as in the case of involutions on 16-dimensional central simple algebras. The first chapter gives an introduction to the basic definitions and properties of hermitian forms which are used throughout the book.

Relative Nonhomogeneous Koszul Duality
Relative Nonhomogeneous Koszul Duality

Author: Leonid Positselski

Publisher: Springer Nature

Published: 2022-02-10

Total Pages: 303

ISBN-13: 3030895408

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This research monograph develops the theory of relative nonhomogeneous Koszul duality. Koszul duality is a fundamental phenomenon in homological algebra and related areas of mathematics, such as algebraic topology, algebraic geometry, and representation theory. Koszul duality is a popular subject of contemporary research. This book, written by one of the world's leading experts in the area, includes the homogeneous and nonhomogeneous quadratic duality theory over a nonsemisimple, noncommutative base ring, the Poincare–Birkhoff–Witt theorem generalized to this context, and triangulated equivalences between suitable exotic derived categories of modules, curved DG comodules, and curved DG contramodules. The thematic example, meaning the classical duality between the ring of differential operators and the de Rham DG algebra of differential forms, involves some of the most important objects of study in the contemporary algebraic and differential geometry. For the first time in the history of Koszul duality the derived D-\Omega duality is included into a general framework. Examples highly relevant for algebraic and differential geometry are discussed in detail.

Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equations
Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equations

Author: Volker Bach

Publisher: American Mathematical Soc.

Published: 2016-03-10

Total Pages: 134

ISBN-13: 1470417057

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The authors study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. They specify assumptions that ensure the global existence of its solutions and allow them to derive its asymptotics at temporal infinity. They demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamiltonian operators in boson quantum field theory which are quadratic in the field.

Moduli of Double EPW-Sextics
Moduli of Double EPW-Sextics

Author: Kieran G. O'Grady

Publisher: American Mathematical Soc.

Published: 2016-03-10

Total Pages: 188

ISBN-13: 1470416964

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The author studies the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of ⋀3C6 modulo the natural action of SL6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of HK 4-folds of Type K3[2] polarized by a divisor of square 2 for the Beauville-Bogomolov quadratic form. The author will determine the stable points. His work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic 4-folds.