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.

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

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.

Exotic Cluster Structures on $SL_n$: The Cremmer-Gervais Case
Exotic Cluster Structures on $SL_n$: The Cremmer-Gervais Case

Author: M. Gekhtman

Publisher: American Mathematical Soc.

Published: 2017-02-20

Total Pages: 106

ISBN-13: 1470422581

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This is the second paper in the series of papers dedicated to the study of natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson–Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on corresponds to a cluster structure in . The authors have shown before that this conjecture holds for any in the case of the standard Poisson–Lie structure and for all Belavin–Drinfeld classes in , . In this paper the authors establish it for the Cremmer–Gervais Poisson–Lie structure on , which is the least similar to the standard one.

Applications of Polyfold Theory I: The Polyfolds of Gromov-Witten Theory
Applications of Polyfold Theory I: The Polyfolds of Gromov-Witten Theory

Author: H. Hofer

Publisher: American Mathematical Soc.

Published: 2017-07-13

Total Pages: 230

ISBN-13: 1470422034

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In this paper the authors start with the construction of the symplectic field theory (SFT). As a general theory of symplectic invariants, SFT has been outlined in Introduction to symplectic field theory (2000), by Y. Eliashberg, A. Givental and H. Hofer who have predicted its formal properties. The actual construction of SFT is a hard analytical problem which will be overcome be means of the polyfold theory due to the present authors. The current paper addresses a significant amount of the arising issues and the general theory will be completed in part II of this paper. To illustrate the polyfold theory the authors use the results of the present paper to describe an alternative construction of the Gromov-Witten invariants for general compact symplectic manifolds.

Rationality Problem for Algebraic Tori
Rationality Problem for Algebraic Tori

Author: Akinari Hoshi

Publisher: American Mathematical Soc.

Published: 2017-07-13

Total Pages: 228

ISBN-13: 1470424096

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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 ...

Special Values of the Hypergeometric Series
Special Values of the Hypergeometric Series

Author: Akihito Ebisu

Publisher: American Mathematical Soc.

Published: 2017-07-13

Total Pages: 108

ISBN-13: 1470425335

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In this paper, the author presents a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, the author gets identities for the hypergeometric series and shows that values of at some points can be expressed in terms of gamma functions, together with certain elementary functions. The author tabulates the values of that can be obtained with this method and finds that this set includes almost all previously known values and many previously unknown values.

On Dwork's $p$-Adic Formal Congruences Theorem and Hypergeometric Mirror Maps
On Dwork's $p$-Adic Formal Congruences Theorem and Hypergeometric Mirror Maps

Author: E. Delaygue

Publisher: American Mathematical Soc.

Published: 2017-02-20

Total Pages: 106

ISBN-13: 1470423006

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Using Dwork's theory, the authors prove a broad generalization of his famous -adic formal congruences theorem. This enables them to prove certain -adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number and not only for almost all primes. Furthermore, using Christol's functions, the authors provide an explicit formula for the “Eisenstein constant” of any hypergeometric series with rational parameters. As an application of these results, the authors obtain an arithmetic statement “on average” of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It contains all the similar univariate integrality results in the literature, with the exception of certain refinements that hold only in very particular cases.

Proof of the 1-Factorization and Hamilton Decomposition Conjectures
Proof of the 1-Factorization and Hamilton Decomposition Conjectures

Author: Béla Csaba

Publisher: American Mathematical Soc.

Published: 2016-10-05

Total Pages: 176

ISBN-13: 1470420252

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In this paper the authors prove the following results (via a unified approach) for all sufficiently large n: (i) [1-factorization conjecture] Suppose that n is even and D≥2⌈n/4⌉−1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ′(G)=D. (ii) [Hamilton decomposition conjecture] Suppose that D≥⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree δ≥n/2. Then G contains at least regeven(n,δ)/2≥(n−2)/8 edge-disjoint Hamilton cycles. Here regeven(n,δ) denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. (i) was first explicitly stated by Chetwynd and Hilton. (ii) and the special case δ=⌈n/2⌉ of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.