Maximal Cohen-Macaulay Modules Over Non-Isolated Surface Singularities and Matrix Problems

Maximal Cohen-Macaulay Modules Over Non-Isolated Surface Singularities and Matrix Problems PDF

Author: Igor Burban

Publisher: American Mathematical Soc.

Published: 2017-07-13

Total Pages: 134

ISBN-13: 1470425378

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In this article the authors develop a new method to deal with maximal Cohen–Macaulay modules over non–isolated surface singularities. In particular, they give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen–Macaulay modules. Next, the authors prove that the degenerate cusp singularities have tame Cohen–Macaulay representation type. The authors' approach is illustrated on the case of k as well as several other rings. This study of maximal Cohen–Macaulay modules over non–isolated singularities leads to a new class of problems of linear algebra, which the authors call representations of decorated bunches of chains. They prove that these matrix problems have tame representation type and describe the underlying canonical forms.

Maximal Abelian Sets of Roots

Maximal Abelian Sets of Roots PDF

Author: R. Lawther

Publisher: American Mathematical Soc.

Published: 2018-01-16

Total Pages: 234

ISBN-13: 147042679X

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In this work the author lets be an irreducible root system, with Coxeter group . He considers subsets of which are abelian, meaning that no two roots in the set have sum in . He classifies all maximal abelian sets (i.e., abelian sets properly contained in no other) up to the action of : for each -orbit of maximal abelian sets we provide an explicit representative , identify the (setwise) stabilizer of in , and decompose into -orbits. Abelian sets of roots are closely related to abelian unipotent subgroups of simple algebraic groups, and thus to abelian -subgroups of finite groups of Lie type over fields of characteristic . Parts of the work presented here have been used to confirm the -rank of , and (somewhat unexpectedly) to obtain for the first time the -ranks of the Monster and Baby Monster sporadic groups, together with the double cover of the latter. Root systems of classical type are dealt with quickly here; the vast majority of the present work concerns those of exceptional type. In these root systems the author introduces the notion of a radical set; such a set corresponds to a subgroup of a simple algebraic group lying in the unipotent radical of a certain maximal parabolic subgroup. The classification of radical maximal abelian sets for the larger root systems of exceptional type presents an interesting challenge; it is accomplished by converting the problem to that of classifying certain graphs modulo a particular equivalence relation.

Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem

Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem PDF

Author: Donatella Daniell

Publisher: American Mathematical Soc.

Published: 2017-09-25

Total Pages: 116

ISBN-13: 1470425475

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The authors give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori PDF

Author: Xiao Xiong

Publisher: American Mathematical Soc.

Published: 2018-03-19

Total Pages: 118

ISBN-13: 1470428067

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This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative -torus (with a skew symmetric real -matrix). These spaces share many properties with their classical counterparts. The authors prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces.

Boundary Conditions and Subelliptic Estimates for Geometric Kramers-Fokker-Planck Operators on Manifolds with Boundaries

Boundary Conditions and Subelliptic Estimates for Geometric Kramers-Fokker-Planck Operators on Manifolds with Boundaries PDF

Author: Francis Nier

Publisher: American Mathematical Soc.

Published: 2018-03-19

Total Pages: 142

ISBN-13: 1470428024

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This article is concerned with the maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic laplacian.

On Operads, Bimodules and Analytic Functors

On Operads, Bimodules and Analytic Functors PDF

Author: Nicola Gambino

Publisher: American Mathematical Soc.

Published: 2017-09-25

Total Pages: 122

ISBN-13: 1470425769

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The authors develop further the theory of operads and analytic functors. In particular, they introduce the bicategory of operad bimodules, that has operads as -cells, operad bimodules as -cells and operad bimodule maps as 2-cells, and prove that it is cartesian closed. In order to obtain this result, the authors extend the theory of distributors and the formal theory of monads.

On Sudakov's Type Decomposition of Transference Plans with Norm Costs

On Sudakov's Type Decomposition of Transference Plans with Norm Costs PDF

Author: Stefano Bianchini

Publisher: American Mathematical Soc.

Published: 2018-02-23

Total Pages: 124

ISBN-13: 1470427664

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The authors consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost with , probability measures in and absolutely continuous w.r.t. . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in , where are disjoint regions such that the construction of an optimal map is simpler than in the original problem, and then to obtain by piecing together the maps . When the norm is strictly convex, the sets are a family of -dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map is straightforward provided one can show that the disintegration of (and thus of ) on such segments is absolutely continuous w.r.t. the -dimensional Hausdorff measure. When the norm is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper the authors show how these difficulties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set and then in . The strategy is sufficiently powerful to be applied to other optimal transportation problems.

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