Rationality Problem For Algebraic Tori

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Rationality Problem for Algebraic Tori

Author : Akinari Hoshi
Publisher :
ISBN 13 : 9781470440541
Total Pages : 215 pages
Book Rating : 4.4/5 (45 download)

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Book Synopsis Rationality Problem for Algebraic Tori by : Akinari Hoshi

Download or read book Rationality Problem for Algebraic Tori written by Akinari Hoshi and published by . This book was released on 2017 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt: "We give the complete stably rational classification of algebraic tori of dimensions 4 and 5 over a field k. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank 4 and 5 is given. We show that there exist exactly 487 (resp. 7, resp. 216) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 4, and there exist exactly 3051 (resp. 25, resp. 3003) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 5. We make a procedure to compute a flabby resolution of a G-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a G-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby G-lattices of rank up to 6 and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for G-lattices holds when the rank ≤ 4, and fails when the rank is 5. Indeed, there exist exactly 11 (resp. 131) G-lattices of rank 5 (resp. 6) which are decomposable into two different ranks. Moreover, when the rank is 6, there exist exactly 18 G-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that H1(G, F) = 0 for any Bravais group G of dimension n ≤ 6 where F is the flabby class of the corresponding G-lattice of rank n. In particular, H1(G, F) = 0 for any maximal finite subgroup G ≤ GL(n, Z) where n ≤ 6. As an application of the methods developed, some examples of not retract (stably) rational fields over k are given."--Page v.

Rationality Problem for Algebraic Tori

Author : Akinari Hoshi
Publisher : American Mathematical Soc.
ISBN 13 : 1470424096
Total Pages : 215 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Rationality Problem for Algebraic Tori by : Akinari Hoshi

Download or read book Rationality Problem for Algebraic Tori written by Akinari Hoshi and published by American Mathematical Soc.. This book was released on 2017-07-13 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors give the complete stably rational classification of algebraic tori of dimensions and over a field . In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank and is given. The authors show that there exist exactly (resp. , resp. ) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension , and there exist exactly (resp. , resp. ) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension . The authors make a procedure to compute a flabby resolution of a -lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a -lattice is invertible (resp. zero) or not. Using the algorithms, the suthors determine all the flabby and coflabby -lattices of rank up to and verify that they are stably permutation. The authors also show that the Krull-Schmidt theorem for -lattices holds when the rank , and fails when the rank is ...

Rationality Problems in Algebraic Geometry

Author : Arnaud Beauville
Publisher : Springer
ISBN 13 : 3319462091
Total Pages : 170 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis Rationality Problems in Algebraic Geometry by : Arnaud Beauville

Download or read book Rationality Problems in Algebraic Geometry written by Arnaud Beauville and published by Springer. This book was released on 2016-12-06 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: Providing an overview of the state of the art on rationality questions in algebraic geometry, this volume gives an update on the most recent developments. It offers a comprehensive introduction to this fascinating topic, and will certainly become an essential reference for anybody working in the field. Rationality problems are of fundamental importance both in algebra and algebraic geometry. Historically, rationality problems motivated significant developments in the theory of abelian integrals, Riemann surfaces and the Abel–Jacobi map, among other areas, and they have strong links with modern notions such as moduli spaces, Hodge theory, algebraic cycles and derived categories. This text is aimed at researchers and graduate students in algebraic geometry.

Multiplicative Invariant Fields of Dimension $leq 6$

Author : Akinari Hoshi
Publisher : American Mathematical Society
ISBN 13 : 147046022X
Total Pages : 150 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Multiplicative Invariant Fields of Dimension $leq 6$ by : Akinari Hoshi

Download or read book Multiplicative Invariant Fields of Dimension $leq 6$ written by Akinari Hoshi and published by American Mathematical Society. This book was released on 2023-03-09 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Encyclopaedia of Mathematics

Author : M. Hazewinkel
Publisher : Springer
ISBN 13 : 1489937919
Total Pages : 932 pages
Book Rating : 4.4/5 (899 download)

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Book Synopsis Encyclopaedia of Mathematics by : M. Hazewinkel

Download or read book Encyclopaedia of Mathematics written by M. Hazewinkel and published by Springer. This book was released on 2013-12-01 with total page 932 pages. Available in PDF, EPUB and Kindle. Book excerpt:

20 Lectures Delivered at the International Congress of Mathematicians in Vancouver, 1974

Author :
Publisher : American Mathematical Soc.
ISBN 13 : 9780821895467
Total Pages : 138 pages
Book Rating : 4.8/5 (954 download)

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Book Synopsis 20 Lectures Delivered at the International Congress of Mathematicians in Vancouver, 1974 by :

Download or read book 20 Lectures Delivered at the International Congress of Mathematicians in Vancouver, 1974 written by and published by American Mathematical Soc.. This book was released on 1977-12-31 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Affine Algebraic Geometry

Author : Kayo Masuda
Publisher : World Scientific
ISBN 13 : 9814436712
Total Pages : 352 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Affine Algebraic Geometry by : Kayo Masuda

Download or read book Affine Algebraic Geometry written by Kayo Masuda and published by World Scientific. This book was released on 2013-05-20 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present volume grew out of an international conference on affine algebraic geometry held in Osaka, Japan during 3–6 March 2011 and is dedicated to Professor Masayoshi Miyanishi on the occasion of his 70th birthday. It contains 16 refereed articles in the areas of affine algebraic geometry, commutative algebra and related fields, which have been the working fields of Professor Miyanishi for almost 50 years. Readers will be able to find recent trends in these areas too. The topics contain both algebraic and analytic, as well as both affine and projective, problems. All the results treated in this volume are new and original which subsequently will provide fresh research problems to explore. This volume is suitable for graduate students and researchers in these areas. Contents:Acyclic Curves and Group Actions on Affine Toric Surfaces (Ivan Arzhantsev and Mikhail Zaidenberg)Hirzeburch Surfaces and Compactifications of ℂ2 (M Furushima and A Ishida)Cyclic Multiple Planes, Branched Covers of Sn and a Result of D L Goldsmith (R V Gurjar)𝔸1*-Fibrations on Affine Threefolds (R V Gurjar, M Koras, K Masuda, M Miyanishi and P Russell)Miyanishi's Characterization of Singularities Appearing on 𝔸1-Fibrations Does Not Hold in Higher Dimensions (Takashi Kishimoto)A Galois Counterexample to Hilbert's Fourteenth Problem in Dimension Three with Rational Coefficients (Ei Kobayashi and Shigeru Kuroda)Open Algebraic Surfaces of Logarithmic Kodaira Dimension One (Hideo Kojima)Some Properties of ℂ* in ℂ2 (M Koras and P Russell)Abhyankar-Sathaye Embedding Conjecture for a Geometric Case (Tomoaki Ohta)Some Subgroups of the Cremona Groups (Vladimir L Popov)The Gonality of Singular Plane Curves II (Fumio Sakai)Examples of Non-Uniruled Surfaces with Pre-Tango Structures Involving Non-Closed Global Differential 1-Forms (Yoshifumi Takeda)Representations of 𝔾a of Codimension Two (Ryuji Tanimoto)The Projective Characterization of Genus Two Plane Curves Which Have One Place at Infinity (Keita Tono)Sextic Variety as Galois Closure Variety of Smooth Cubic (Hisao Yoshihara)Invariant Hypersurfaces of Endomorphisms of the Projective 3-Space (De-Qi Zhang) Readership: Graduate students and researchers in affine algebraic geometry. Keywords:Affine algebraic geometry;Commutative algebra;Polynomial algebra;Group actions;FibrationsKey Features:Many active researchers such as M Zaidenberg, R V Gurjar, M Koras, P Russell, F Sakai, V Popov, H Yoshihara, and D-Q Zhang contributed articles to this proceedingsMany viewpoints are taken into consideration to study surfaces and threefolds. These will give hints for further research in neighboring fields

Encyclopaedia of Mathematics

Author : Michiel Hazewinkel
Publisher : Springer Science & Business Media
ISBN 13 : 940151237X
Total Pages : 506 pages
Book Rating : 4.4/5 (15 download)

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Book Synopsis Encyclopaedia of Mathematics by : Michiel Hazewinkel

Download or read book Encyclopaedia of Mathematics written by Michiel Hazewinkel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: This ENCYCLOPAEDIA OF MA THEMA TICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

Author : Xiao Xiong
Publisher : American Mathematical Soc.
ISBN 13 : 1470428067
Total Pages : 118 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori by : Xiao Xiong

Download or read book Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori written by Xiao Xiong and published by American Mathematical Soc.. This book was released on 2018-03-19 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Rational Points on Varieties

Author : Bjorn Poonen
Publisher : American Mathematical Society
ISBN 13 : 1470474581
Total Pages : 357 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Rational Points on Varieties by : Bjorn Poonen

Download or read book Rational Points on Varieties written by Bjorn Poonen and published by American Mathematical Society. This book was released on 2023-08-10 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere. The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points. —Yuri Manin, Max-Planck-Institute, Bonn It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student. —Hendrik Lenstra, University Leiden Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics. —Jean-Louis Colliot-Thélène, Université Paris-Sud A beautiful subject, handled by a master. —Joseph Silverman, Brown University

European Congress of Mathematics

Author : Antal Balog
Publisher : Birkhäuser
ISBN 13 : 3034888988
Total Pages : 412 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis European Congress of Mathematics by : Antal Balog

Download or read book European Congress of Mathematics written by Antal Balog and published by Birkhäuser. This book was released on 2012-12-06 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second volume of the procedings of the second European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners. Together with volume II it contains a collection of contributions by the invited lecturers. Finally, volume II also presents reports on some of the Round Table discussions. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician. Contributors: Vol. I: N. Alon, L. Ambrosio, K. Astala, R. Benedetti, Ch. Bessenrodt, F. Bethuel, P. Bjørstad, E. Bolthausen, J. Bricmont, A. Kupiainen, D. Burago, L. Caporaso, U. Dierkes, I. Dynnikov, L.H. Eliasson, W.T. Gowers, H. Hedenmalm, A. Huber, J. Kaczorowski, J. Kollár, D.O. Kramkov, A.N. Shiryaev, C. Lescop, R. März. Vol. II: J. Matousek, D. McDuff, A.S. Merkurjev, V. Milman, St. Müller, T. Nowicki, E. Olivieri, E. Scoppola, V.P. Platonov, J. Pöschel, L. Polterovich , L. Pyber, N. Simányi, J.P. Solovej, A. Stipsicz, G. Tardos, J.-P. Tignol, A.P. Veselov, E. Zuazua.

Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem

Author : Donatella Daniell
Publisher : American Mathematical Soc.
ISBN 13 : 1470425475
Total Pages : 103 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem by : Donatella Daniell

Download or read book Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem written by Donatella Daniell and published by American Mathematical Soc.. This book was released on 2017-09-25 with total page 103 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Algebraic Groups and Their Birational Invariants

Author : V. E. Voskresenskii
Publisher : American Mathematical Soc.
ISBN 13 : 0821872885
Total Pages : 218 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Algebraic Groups and Their Birational Invariants by : V. E. Voskresenskii

Download or read book Algebraic Groups and Their Birational Invariants written by V. E. Voskresenskii and published by American Mathematical Soc.. This book was released on 2011-10-06 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the late 1960s, methods of birational geometry have been used successfully in the theory of linear algebraic groups, especially in arithmetic problems. This book--which can be viewed as a significant revision of the author's book, Algebraic Tori (Nauka, Moscow, 1977)--studies birational properties of linear algebraic groups focusing on arithmetic applications. The main topics are forms and Galois cohomology, the Picard group and the Brauer group, birational geometry of algebraic tori, arithmetic of algebraic groups, Tamagawa numbers, $R$-equivalence, projective toric varieties, invariants of finite transformation groups, and index-formulas. Results and applications are recent. There is an extensive bibliography with additional comments that can serve as a guide for further reading.

Multiplicative Invariant Theory

Author : Martin Lorenz
Publisher : Springer Science & Business Media
ISBN 13 : 3540273581
Total Pages : 179 pages
Book Rating : 4.5/5 (42 download)

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Book Synopsis Multiplicative Invariant Theory by : Martin Lorenz

Download or read book Multiplicative Invariant Theory written by Martin Lorenz and published by Springer Science & Business Media. This book was released on 2005-12-08 with total page 179 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multiplicative invariant theory, as a research area in its own right within the wider spectrum of invariant theory, is of relatively recent vintage. The present text offers a coherent account of the basic results achieved thus far.. Multiplicative invariant theory is intimately tied to integral representations of finite groups. Therefore, the field has a predominantly discrete, algebraic flavor. Geometry, specifically the theory of algebraic groups, enters through Weyl groups and their root lattices as well as via character lattices of algebraic tori. Throughout the text, numerous explicit examples of multiplicative invariant algebras and fields are presented, including the complete list of all multiplicative invariant algebras for lattices of rank 2. The book is intended for graduate and postgraduate students as well as researchers in integral representation theory, commutative algebra and, mostly, invariant theory.

An Invitation to Computational Homotopy

Author : Graham Ellis
Publisher : Oxford University Press
ISBN 13 : 0192569414
Total Pages : 640 pages
Book Rating : 4.1/5 (925 download)

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Book Synopsis An Invitation to Computational Homotopy by : Graham Ellis

Download or read book An Invitation to Computational Homotopy written by Graham Ellis and published by Oxford University Press. This book was released on 2019-08-14 with total page 640 pages. Available in PDF, EPUB and Kindle. Book excerpt: An Invitation to Computational Homotopy is an introduction to elementary algebraic topology for those with an interest in computers and computer programming. It expertly illustrates how the basics of the subject can be implemented on a computer through its focus on fully-worked examples designed to develop problem solving techniques. The transition from basic theory to practical computation raises a range of non-trivial algorithmic issues which will appeal to readers already familiar with basic theory and who are interested in developing computational aspects. The book covers a subset of standard introductory material on fundamental groups, covering spaces, homology, cohomology and classifying spaces as well as some less standard material on crossed modules. These topics are covered in a way that hints at potential applications of topology in areas of computer science and engineering outside the usual territory of pure mathematics, and also in a way that demonstrates how computers can be used to perform explicit calculations within the domain of pure algebraic topology itself. The initial chapters include in-depth examples from data mining, biology and digital image analysis, while the later chapters cover a range of computational examples on the cohomology of classifying spaces that are likely beyond the reach of a purely paper-and-pen approach to the subject. An Invitation to Computational Homotopy serves as a self-contained and informal introduction to these topics and their implementation in the sphere of computer science. Written in a dynamic and engaging style, it skilfully showcases a range of useful machine computations, and will serve as an invaluable aid to graduate students working with algebraic topology.

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

Author : Stefano Bianchini
Publisher : American Mathematical Soc.
ISBN 13 : 1470427664
Total Pages : 112 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis On Sudakov’s Type Decomposition of Transference Plans with Norm Costs by : Stefano Bianchini

Download or read book On Sudakov’s Type Decomposition of Transference Plans with Norm Costs written by Stefano Bianchini and published by American Mathematical Soc.. This book was released on 2018-02-23 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Orthogonal and Symplectic -level Densities

Author : A. M. Mason
Publisher : American Mathematical Soc.
ISBN 13 : 1470426854
Total Pages : 93 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Orthogonal and Symplectic -level Densities by : A. M. Mason

Download or read book Orthogonal and Symplectic -level Densities written by A. M. Mason and published by American Mathematical Soc.. This book was released on 2018-02-23 with total page 93 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper the authors apply to the zeros of families of -functions with orthogonal or symplectic symmetry the method that Conrey and Snaith (Correlations of eigenvalues and Riemann zeros, 2008) used to calculate the -correlation of the zeros of the Riemann zeta function. This method uses the Ratios Conjectures (Conrey, Farmer, and Zimbauer, 2008) for averages of ratios of zeta or -functions. Katz and Sarnak (Zeroes of zeta functions and symmetry, 1999) conjecture that the zero statistics of families of -functions have an underlying symmetry relating to one of the classical compact groups , and . Here the authors complete the work already done with (Conrey and Snaith, Correlations of eigenvalues and Riemann zeros, 2008) to show how new methods for calculating the -level densities of eigenangles of random orthogonal or symplectic matrices can be used to create explicit conjectures for the -level densities of zeros of -functions with orthogonal or symplectic symmetry, including all the lower order terms. They show how the method used here results in formulae that are easily modified when the test function used has a restricted range of support, and this will facilitate comparison with rigorous number theoretic -level density results.