Geometric Complexity Theory IV
Author: Jonah Blasiak
Publisher:
Published: 2014
Total Pages: 160
ISBN-13: 9781470422271
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Geometric Complexity Theory IV: Nonstandard Quantum Group for the Kronecker Problem
Author: Jonah Blasiak
Publisher: American Mathematical Soc.
Published: 2015-04-09
Total Pages: 176
ISBN-13: 1470410117
DOWNLOAD EBOOK →The Kronecker coefficient is the multiplicity of the -irreducible in the restriction of the -irreducible via the natural map , where are -vector spaces and . A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. The authors construct two quantum objects for this problem, which they call the nonstandard quantum group and nonstandard Hecke algebra. They show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality.
Recent Developments in Algebraic and Combinatorial Aspects of Representation Theory
Author: Vyjayanthi Chari
Publisher: American Mathematical Soc.
Published: 2013-11-25
Total Pages: 222
ISBN-13: 0821890379
DOWNLOAD EBOOK →This volume contains the proceedings of the International Congress of Mathematicians Satellite Conference on Algebraic and Combinatorial Approaches to Representation Theory, held August 12-16, 2010, at the National Institute of Advanced Studies, Bangalore, India, and the follow-up conference held May 18-20, 2012, at the University of California, USA. It contains original research and survey articles on various topics in the theory of representations of Lie algebras, quantum groups and algebraic groups, including crystal bases, categorification, toroidal algebras and their generalisations, vertex algebras, Hecke algebras, Kazhdan-Lusztig bases, $q$-Schur algebras, and Weyl algebras.
Numerical Approximations of Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients
Author: Martin Hutzenthaler
Publisher: American Mathematical Soc.
Published: 2015-06-26
Total Pages: 112
ISBN-13: 1470409844
DOWNLOAD EBOOK →Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, the authors establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, the authors illustrate their results for several SDEs from finance, physics, biology and chemistry.
On Non-Topological Solutions of the $A_{2}$ and $B_{2}$ Chern-Simons System
Author: Weiwei Ao
Publisher: American Mathematical Soc.
Published: 2016-01-25
Total Pages: 100
ISBN-13: 1470415437
DOWNLOAD EBOOK →Click here to view the abstract. IntroductionProof of Theorem 1.1 in the caseProof of Theorem 1.1 in the caseAppendixBibliography
Open Problems in Mathematics
Author: John Forbes Nash, Jr.
Publisher: Springer
Published: 2016-07-05
Total Pages: 547
ISBN-13: 3319321625
DOWNLOAD EBOOK →The goal in putting together this unique compilation was to present the current status of the solutions to some of the most essential open problems in pure and applied mathematics. Emphasis is also given to problems in interdisciplinary research for which mathematics plays a key role. This volume comprises highly selected contributions by some of the most eminent mathematicians in the international mathematical community on longstanding problems in very active domains of mathematical research. A joint preface by the two volume editors is followed by a personal farewell to John F. Nash, Jr. written by Michael Th. Rassias. An introduction by Mikhail Gromov highlights some of Nash’s legendary mathematical achievements. The treatment in this book includes open problems in the following fields: algebraic geometry, number theory, analysis, discrete mathematics, PDEs, differential geometry, topology, K-theory, game theory, fluid mechanics, dynamical systems and ergodic theory, cryptography, theoretical computer science, and more. Extensive discussions surrounding the progress made for each problem are designed to reach a wide community of readers, from graduate students and established research mathematicians to physicists, computer scientists, economists, and research scientists who are looking to develop essential and modern new methods and theories to solve a variety of open problems.
Irreducible Geometric Subgroups of Classical Algebraic Groups
Author: Timothy C. Burness,
Publisher: American Mathematical Soc.
Published: 2016-01-25
Total Pages: 100
ISBN-13: 1470414945
DOWNLOAD EBOOK →Let be a simple classical algebraic group over an algebraically closed field of characteristic with natural module . Let be a closed subgroup of and let be a non-trivial irreducible tensor-indecomposable -restricted rational -module such that the restriction of to is irreducible. In this paper the authors classify the triples of this form, where is a disconnected maximal positive-dimensional closed subgroup of preserving a natural geometric structure on .
Reduced Fusion Systems over 2-Groups of Sectional Rank at Most 4
Author: Bob Oliver
Publisher: American Mathematical Soc.
Published: 2016-01-25
Total Pages: 112
ISBN-13: 1470415488
DOWNLOAD EBOOK →The author classifies all reduced, indecomposable fusion systems over finite -groups of sectional rank at most . The resulting list is very similar to that by Gorenstein and Harada of all simple groups of sectional -rank at most . But this method of proof is very different from theirs, and is based on an analysis of the essential subgroups which can occur in the fusion systems.
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
DOWNLOAD EBOOK →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.
Deformation Quantization for Actions of Kahlerian Lie Groups
Author: Pierre Bieliavsky
Publisher: American Mathematical Soc.
Published: 2015-06-26
Total Pages: 166
ISBN-13: 1470414910
DOWNLOAD EBOOK →Let B be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action of B on a Fréchet algebra . Denote by the associated Fréchet algebra of smooth vectors for this action. In the Abelian case BR and isometric, Marc Rieffel proved that Weyl's operator symbol composition formula (the so called Moyal product) yields a deformation through Fréchet algebra structures R on . When is a -algebra, every deformed Fréchet algebra admits a compatible pre- -structure, hence yielding a deformation theory at the level of -algebras too. In this memoir, the authors prove both analogous statements for general negatively curved Kählerian groups. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geom,etrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calderón-Vaillancourt Theorem. In particular, the authors give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.
