66 results
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2. $p$ -adic Eisenstein–Kronecker series for CM elliptic curves and the Kronecker limit formulas
- Author
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Kenichi Bannai, Shinichi Kobayashi, and Hidekazu Furusho
- Subjects
Pure mathematics ,Distribution (number theory) ,14G10 ,General Mathematics ,Mathematics::Number Theory ,Theta function ,Eisenstein–Kronecker series ,Ring of integers ,14F30 ,symbols.namesake ,Kronecker limit formula ,Kronecker delta ,FOS: Mathematics ,Number Theory (math.NT) ,distribution relation ,11G15 ,Mathematics ,Coleman’s $p$-adic integration ,Mathematics - Number Theory ,Series (mathematics) ,11G55 ,Elliptic curve ,11G55, 11G07, 11G15, 14F30, 14G10 ,symbols ,Quadratic field ,11G07 - Abstract
Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function., v2. The current version is the synthesis of {\S}1-{\S}3 of the first version of this article with the content of arXiv:0807.4008 "The Kronecker limit formulas via the distribution relation." {\S}4,{\S}5 of the first version of this paper will be treated in a subsequent article
- Published
- 2015
3. FI-modules and stability for representations of symmetric groups
- Author
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Thomas Church, Jordan S. Ellenberg, and Benson Farb
- Subjects
Pure mathematics ,General Mathematics ,symmetric groups ,Mathematics - Geometric Topology ,Symmetric group ,FI-modules ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Combinatorics ,Mathematics - Algebraic Topology ,Representation Theory (math.RT) ,Mathematics ,Ring (mathematics) ,Group (mathematics) ,Subalgebra ,representations ,Geometric Topology (math.GT) ,20J06 ,Cohomology ,Moduli space ,Combinatorics (math.CO) ,Configuration space ,55N25 ,05E10 ,Mathematics - Representation Theory ,Vector space - Abstract
In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module., 54 pages. v4: new title, paper completely reorganized; final version, to appear in Duke Math Journal
- Published
- 2015
4. Growth of the Weil–Petersson diameter of moduli space
- Author
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William Cavendish and Hugo Parlier
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,0102 computer and information sciences ,01 natural sciences ,Upper and lower bounds ,symbols.namesake ,Mathematics - Geometric Topology ,Mathematics::Algebraic Geometry ,Genus (mathematics) ,32F15 ,FOS: Mathematics ,32G15 ,0101 mathematics ,Mathematics ,Riemann surface ,010102 general mathematics ,Geometric Topology (math.GT) ,Auxiliary function ,Function (mathematics) ,Moduli space ,Differential Geometry (math.DG) ,30FXX ,010201 computation theory & mathematics ,symbols ,Constant (mathematics) - Abstract
In this paper we study the Weil-Petersson geometry of $\overline{\mathcal{M}_{g,n}}$, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of $\overline{\mathcal{M}_{g,n}}$ as a function of $g$ and $n$. We show that this diameter grows as $\sqrt{n}$ in $n$, and is bounded above by $C \sqrt{g}\log g$ in $g$ for some constant $C$. We also give a lower bound on the growth in $g$ of the diameter of $\overline{\mathcal{M}_{g,n}}$ in terms of an auxiliary function that measures the extent to which the thick part of moduli space admits radial coordinates., 26 pages, 7 figures
- Published
- 2012
5. Morphisms determined by objects. The case of modules over Artin algebras
- Author
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Claus Michael Ringel
- Subjects
Pure mathematics ,16D90, 16G10, 16G70 ,16G10 ,General Mathematics ,Mathematics::Rings and Algebras ,Assertion ,16G70 ,Mathematics - Rings and Algebras ,Morphism ,Artin algebra ,Rings and Algebras (math.RA) ,FOS: Mathematics ,16D90 ,Determiner ,Homomorphism ,Representation Theory (math.RT) ,Indecomposable module ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
We deal with finitely generated modules over an artin algebra. In his Philadelphia Notes, M.Auslander showed that any homomorphism is right determined by a module C, but a formula for C which he wrote down has to be modified. The paper includes now complete and direct proofs of the main results concerning right determiners of morphisms. We discuss the role of indecomposable projective direct summands of a minimal right determiner and provide a detailed analysis of those morphisms which are right determined by a module without any non-zero projective direct summand., The paper has been revised and expanded. The terminology has been changed as follows: "essential kernel" is replaced by "intrinsic kernel", "determinator" is replaced by "determiner". Sections 3, 4 and 5 are new
- Published
- 2012
6. New estimates for a time-dependent Schrödinger equation
- Author
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Marius Beceanu
- Subjects
symbols.namesake ,Mathematics - Analysis of PDEs ,General Mathematics ,35Q41 ,FOS: Mathematics ,symbols ,Analysis of PDEs (math.AP) ,Mathematical physics ,Mathematics ,Schrödinger equation - Abstract
This paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold for potentials in scaling-critical, translation-invariant spaces. The proof of the time-independent results uses a novel method based on an abstract version of Wiener's Theorem., 49 pages; this is an expanded and improved version of the older paper
- Published
- 2011
7. Non-commutative varieties with curvature having bounded signature
- Author
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J. William Helton, Scott McCullough, and Harry Dym
- Subjects
Polynomial ,47L07 ,Degree (graph theory) ,Zero set ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Curvature ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Combinatorics ,47Axx ,Bounded function ,47A63 ,FOS: Mathematics ,Irreducibility ,47L30 ,14P10 ,0101 mathematics ,Variety (universal algebra) ,Signature (topology) ,Mathematics - Abstract
A natural notion for the signature $C_{\pm}({\mathcal V}(p))$ of the curvature of the zero set ${\mathcal V}(p)$ of a non-commutative polynomial $p$ is introduced. The main result of this paper is the bound \[ \operatorname{deg} p \leq2 C_\pm \bigl({\mathcal V}(p) \bigr) + 2. \] It is obtained under some irreducibility and nonsingularity conditions, and shows that the signature of the curvature of the zero set of $p$ dominates its degree. ¶ The condition $C_+({\mathcal V}(p))=0$ means that the non-commutative variety ${\mathcal V}(p)$ has positive curvature. In this case, the preceding inequality implies that the degree of $p$ is at most two. Non-commutative varieties ${\mathcal V}(p)$ with positive curvature were introduced in Indiana Univ. Math. J. 56 (2007) 1189-1231). There a slightly weaker irreducibility hypothesis plus a number of additional hypotheses yielded a weaker result on $p$. The approach here is quite different; it is cleaner, and allows for the treatment of arbitrary signatures. ¶ In J. Anal. Math. 108 (2009) 19-59), the degree of a non-commutative polynomial $p$ was bounded by twice the signature of its Hessian plus two. In this paper, we introduce a modified version of this non-commutative Hessian of $p$ which turns out to be very appropriate for analyzing the variety ${\mathcal V}(p)$.
- Published
- 2011
8. The foundational inequalities of D. L. Burkholder and some of their ramifications
- Author
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Rodrigo Bañuelos
- Subjects
Class (set theory) ,Pure mathematics ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,01 natural sciences ,010104 statistics & probability ,Quasiconvex function ,Riesz transform ,Operator (computer programming) ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,60G46 ,0101 mathematics ,Mathematics ,Mathematics::Functional Analysis ,Conjecture ,Probability (math.PR) ,010102 general mathematics ,Singular integral ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Areas of mathematics ,42B20 ,Martingale (probability theory) ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
This paper presents an overview of some of the applications of the martingale inequalities of D. L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms, the Beurling–Ahlfors operator and other multipliers obtained by projections (conditional expectations) of transformations of stochastic integrals. While martingale inequalities can be used to prove the boundedness of a wider class of Calderón–Zygmund singular integrals, the aim of this paper is to show results which give optimal or near optimal bounds in the norms, hence our restriction to the above operators. ¶ Connections of Burkholder’s foundational work on sharp martingale inequalities to other areas of mathematics where either the results themselves or techniques to prove them have become of considerable interest in recent years, are discussed. These include the 1952 conjecture of C. B. Morrey on rank-one convex and quasiconvex functions with connections to problems in the calculus of variations and the 1982 conjecture of T. Iwaniec on the $L^p$-norm of the Beurling–Ahlfors operator with connections to problems in the theory of qasiconformal mappings. Open questions, problems and conjectures are listed throughout the paper and copious references are provided.
- Published
- 2010
9. More mixed Tsirelson spaces that are not isomorphic to their modified versions
- Author
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Denny H. Leung and Wee-Kee Tang
- Subjects
Discrete mathematics ,Large class ,Class (set theory) ,Mathematics::Functional Analysis ,General Mathematics ,Banach space ,46B20 ,46B45 ,Space (mathematics) ,Tsirelson space ,Sequence space ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Combinatorics ,Development (topology) ,FOS: Mathematics ,Isomorphism ,Mathematics - Abstract
The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present paper, we investigate the problem of comparing isomorphically the mixed Tsirelson space T((Sn, θn) ∞=1) and its modified version TM((Sn, θn) ∞=1). It is shown that these spaces are not isomorphic for a large class of parameters (θn). 1 ≤ p < ∞. Figiel and Johnson (7) provided an analytic description, based on iteration, of the norm of the dual of Tsirelson's original space. Subse- quently, other examples of spaces were constructed with norms described it- eratively, notable among them were Tzafriri's spaces (20) and Schlumprecht's space(18). Gowers' and Maurey's solution to the unconditional basic se- quence problem (8) is a variation based on the same theme. It has emerged in recent years that, far from being isolated examples, Tsirelson's space and its variants from an important class of Banach spaces. Argyros and Deliyanni (2) were the first to provide a general framework for such spaces by defining the class of mixed Tsirelson spaces. Among the earliest vari- ants of Tsirelson's space was its modified version introduced by Johnson (9). Casazza and Odell (6) showed that Tsirelson's space is isomorphic to its modified version. This isomorphism was exploited to study the struc- ture of the space. The modification can be extended directly to the class of mixed Tsirelson spaces, forming the class of modified mixed Tsirelson spaces. It is thus of natural interest to determine if a mixed Tsirelson space is isomorphic to its modified version. This question has been considered by various authors, e.g., (3, 12), who provided answers in what may be con- sidered "extremal" cases. In the present paper, we show that for a large class of parameters, a mixed Tsirelson space and its modified version are not isomorphic.
- Published
- 2008
10. Kronecker-Weber plus epsilon
- Author
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Greg W. Anderson
- Subjects
Discrete mathematics ,Rational number ,Mathematics - Number Theory ,Root of unity ,General Mathematics ,Abelian extension ,Galois group ,Field (mathematics) ,Algebraic closure ,11R32 ,11R34 ,11R37 ,FOS: Mathematics ,Galois extension ,Number Theory (math.NT) ,Abelian group ,Mathematics ,11R20 - Abstract
We say that a group is {\em almost abelian} if every commutator is central and squares to the identity. Now let $G$ be the Galois group of the algebraic closure of the field $\QQ$ of rational numbers in the field of complex numbers. Let $G^{\ab+\epsilon}$ be the quotient of $G$ universal for homomorphisms to almost abelian profinite groups and let $\QQ^{\ab+\epsilon}/\QQ$ be the corresponding Galois extension. We prove that $\QQ^{\ab+\epsilon}$ is generated by the roots of unity, the fourth roots of the (rational) prime numbers and the square roots of certain sine-monomials. The inspiration for the paper came from recent studies of algebraic $\Gamma$-monomials by P.~Das and by S.~Seo. This paper has appeared as Duke Math. J. 114 (2002) 439-475.
- Published
- 2002
11. Traces of intertwiners for quantum groups and difference equations, I
- Author
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Pavel Etingof and Alexander Varchenko
- Subjects
General Mathematics ,010102 general mathematics ,39A10 ,01 natural sciences ,17B37 ,33D52 ,Algebra ,High Energy Physics::Theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,32G34 ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Quantum ,Mathematics - Abstract
The main object considered in this paper is the trace function, defined as a suitably normalized trace of a product of intertwining operators for the Drinfeld-Jimbo quantum group, multiplied by the exponential of an element of the Cartan subalgebra. This function depends of two parameters -- the element of the Cartan subalgebra, and the highest weight of the Verma module in which the trace is taken. The main results of the paper are that the trace function satisfies two systems of difference equations with respect to the first parameter (the quantum Knizhnik-Zamolodchikov-Bernard and Macdonald-Ruijsenaars equations), and that it is symmetric with respect to the two parameters. In particular, this implies that for each of the above two systems of equations there is the dual system with respect to the second parameter, which is also satisfied by the trace function. The paper establishes a connection between the I.Frenkel-Reshetikhin theory of quantum conformal blocks, the work of Felder-Mukhin-Tarasov-Varchenko on the quantum KZB and Ruijsenaars equations, the work of Etingof-I.Frenkel- Kirillov Jr.-Styrkas on traces of intetwining operators, and the Macdonald- Cherednik theory. The methods of the paper are based on the theory of dynamical twists and R-matrices., Comment: 38 pages, amstex; some misprints and small errors were corrected in the new version
- Published
- 2000
12. Commutators of free random variables
- Author
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Alexandru Nica and Roland Speicher
- Subjects
Multivariate random variable ,General Mathematics ,01 natural sciences ,Free algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,46L50 ,0101 mathematics ,Operator Algebras (math.OA) ,010306 general physics ,Mathematics ,Probability measure ,Discrete mathematics ,Mathematics::Operator Algebras ,010102 general mathematics ,Mathematics - Operator Algebras ,Random element ,State (functional analysis) ,16. Peace & justice ,Free probability ,Algebra of random variables ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Random variable - Abstract
Let A be a unital $C^*$-algebra, given together with a specified state $\phi:A \to C$. Consider two selfadjoint elements a,b of A, which are free with respect to $\phi$ (in the sense of the free probability theory of Voiculescu). Let us denote $c:=i(ab-ba)$, where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper funct-an/9604012., Comment: LaTeX, 38 pages with 2 figures
- Published
- 1998
13. On isometric and minimal isometric embeddings
- Author
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Thomas A. Ivey and Joseph M. Landsberg
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Rigidity (psychology) ,Isometric exercise ,Construct (python library) ,53C42 ,Space (mathematics) ,01 natural sciences ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity., 21 pages, AMSTeX. Significantly changed version of paper originally Titled "On minimal isometric embeddings"
- Published
- 1997
14. Linear independence result for $p$ -adic $L$ -values
- Author
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Johannes Sprang
- Subjects
General Mathematics ,Mathematics::Number Theory ,01 natural sciences ,Omega ,11F85 ,Dirichlet distribution ,Volkenborn integration ,Combinatorics ,Hurwitz zeta function ,Mathematics - Algebraic Geometry ,symbols.namesake ,p-adic L-functions ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Mathematics - Number Theory ,11J72 ,010102 general mathematics ,Sigma ,Algebraic number field ,linear independence criterion ,Dirichlet character ,11M06 ,Mathematik ,symbols ,010307 mathematical physics ,Linear independence ,irrationality - Abstract
The aim of this paper is to provide an analogue of the Ball-Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a Dirichlet character $\chi$ and a number field $K$ the formula $\dim_{K}(K+\sum_{i=2}^{s+1} L_p(i,\chi\omega^{1-i}) K )\geq \frac{(1-\epsilon)\log (s)}{2[K:\mathbb{Q}](1+\log 2)}$. As a byproduct, we establish an asymptotic linear independence result for the values of the $p$-adic Hurwitz zeta function., Comment: 26 pages, final version; Duke Math. J. (accepted)
- Published
- 2020
15. Primality of multiply connected polyominoes
- Author
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Giancarlo Rinaldo, Carla Mascia, and Francesco Romeo
- Subjects
13A02 ,Polyomino ,General Mathematics ,Computer Science::Computational Geometry ,Commutative Algebra (math.AC) ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Simple (abstract algebra) ,Computer Science::Discrete Mathematics ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,Rank (graph theory) ,05E40 ,0101 mathematics ,Primality test ,Mathematics ,Sequence ,Ideal (set theory) ,Mathematics::Combinatorics ,010102 general mathematics ,Mathematics - Commutative Algebra ,13A02 05E40 ,010307 mathematical physics ,Combinatorics (math.CO) ,Focus (optics) ,Computer Science::Formal Languages and Automata Theory - Abstract
It is known that the polyomino ideal of simple polyominoes is prime. In this paper, we focus on multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14 the above condition is also sufficient. Lastly, we present an infinite class of prime polyomino ideals., Comment: In this version we proved that the grid polyominoes are primes without the use of Groebner basis (see previous version). In particular, we prove that the polyomino ideal is equal to the toric ideal J_P associated to the polyomino as we defined in Section 3
- Published
- 2020
16. Local rings with self-dual maximal ideal
- Author
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Toshinori Kobayashi
- Subjects
Pure mathematics ,Endomorphism ,Mathematics::Commutative Algebra ,13C14 ,13H10 ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,13E15 ,Local ring ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Dual (category theory) ,0103 physical sciences ,FOS: Mathematics ,Maximal ideal ,010307 mathematical physics ,0101 mathematics ,Connection (algebraic framework) ,Mathematics - Abstract
Let R be a Cohen-Macaulay local ring possessing a canonical module. In this paper we consider when the maximal ideal of R is self-dual, i.e. it is isomorphic to its canonical dual as an R-module. local rings satisfying this condition are called Teter rings, and studied by Teter, Huneke-Vraciu, Ananthnarayan-Avramov-Moore, and so on. On the positive dimensional case, we show such rings are exactly the endomorphism rings of the maximal ideals of some Gorenstein local rings of dimension one. We also provide some connection between the self-duality of the maximal ideal and near Gorensteinness., 14 pages
- Published
- 2020
17. Noncommutative boundaries and the ideal structure of reduced crossed products
- Author
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Matthew Kennedy and Christopher Schafhauser
- Subjects
Pure mathematics ,General Mathematics ,45L55 ,Dynamical Systems (math.DS) ,01 natural sciences ,Prime (order theory) ,C$^{*}$-dynamical system ,Crossed product ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,47L65 ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,43A65 ,Operator Algebras (math.OA) ,Commutative property ,Mathematics ,Ideal (set theory) ,Group (mathematics) ,010102 general mathematics ,ideal structure ,Mathematics - Operator Algebras ,46L35 ,Noncommutative geometry ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010307 mathematical physics ,reduced crossed product ,noncommutative boundary - Abstract
A C*-dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C*-algebra. In this paper we characterize this property for unital C*-dynamical systems over discrete groups. To every C*-dynamical system we associate a "twisted" partial C*-dynamical system that encodes much of the structure of the action. This system can often be "untwisted," for example when the algebra is commutative, or when the algebra is prime and a certain specific subgroup has vanishing Mackey obstruction. In this case, we obtain relatively simple necessary and sufficient conditions for the ideal separation property. A key idea is a notion of noncommutative boundary for a C*-dynamical system that generalizes Furstenberg's notion of topological boundary for a group., Comment: 43 pages; revised
- Published
- 2019
18. Two generalizations of Auslander–Reiten duality and applications
- Author
-
Ryo Takahashi and Arash Sadeghi
- Subjects
Pure mathematics ,Conjecture ,Mathematics::Commutative Algebra ,13C14 ,13H10 ,General Mathematics ,010102 general mathematics ,Mathematics::Rings and Algebras ,Local ring ,Duality (optimization) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,0103 physical sciences ,FOS: Mathematics ,13D07 ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
This paper extends Auslander-Reiten duality in two directions. As an application, we obtain various criteria for freeness of modules over local rings in terms of vanishing of Ext modules, which recover a lot of known results on the Auslander-Reiten conjecture.
- Published
- 2019
19. A tropical motivic Fubini theorem with applications to Donaldson–Thomas theory
- Author
-
Sam Payne and Johannes Nicaise
- Subjects
Pure mathematics ,motivic integration ,General Mathematics ,Donaldson–Thomas theory ,nearby cycles ,Context (language use) ,01 natural sciences ,Identity (music) ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,14T05 ,Fubini's theorem ,0103 physical sciences ,Tropical geometry ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Conjecture ,Fiber (mathematics) ,010102 general mathematics ,Mathematics::Logic ,tropical geometry ,14E18 ,010307 mathematical physics ,Motivic integration ,14N35 - Abstract
We present a new tool for the calculation of Denef and Loeser's motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan's theory of motivic volumes of semi-algebraic sets. As applications, we prove a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and give a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by L\^e Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson-Thomas theory., Comment: Grant information completed, no changes to the body of the paper
- Published
- 2019
20. The Markoff group of transformations in prime and composite moduli
- Author
-
Doron Puder, Dan Carmon, and Chen Meiri
- Subjects
General Mathematics ,11D25 ,Group Theory (math.GR) ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Markoff equation ,Markoff triples ,Morphism ,Integer ,Symmetric group ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,20B25 ,Mathematics ,Conjecture ,Mathematics - Number Theory ,Group (mathematics) ,010102 general mathematics ,20E05 ,Alternating group ,16. Peace & justice ,T-systems ,11D25 (Primary) 20B15, 20B25, 20E05 (Secondary) ,Simple group ,010307 mathematical physics ,20B15 ,Mathematics - Group Theory - Abstract
The Markoff group of transformations is a group $\Gamma$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$. The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$, the group $\Gamma$ acts transitively on the set $X^{*}\left(p\right)$ of non-zero solutions to the same equation over $\mathbb{Z}/p\mathbb{Z}$. Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside a small exceptional set. In the current paper, we study a group of permutations obtained by the action of $\Gamma$ on $X^{*}\left(p\right)$, and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that $\Gamma$ acts transitively also on the set of non-zero solutions in a big class of composite moduli. Our result is also related to a well-known theorem of Gilman, stating that for any finite non-abelian simple group $G$ and $r\ge3$, the group $\mathrm{Aut}\left(F_{r}\right)$ acts on at least one $T_{r}$-system of $G$ as the alternating or symmetric group. In this language, our main result translates to that for most primes $p$, the group $\mathrm{Aut}\left(F_{2}\right)$ acts on a particular $T_{2}$-system of $\mathrm{PSL}\left(2,p\right)$ as the alternating or symmetric group., Comment: 31 pages, by Chen Meiri and Doron Puder, with an appendix by Dan Carmon. Better exposition than in last version, and some non-accurate statements fixed
- Published
- 2018
21. Linear differential equations on the Riemann sphere and representations of quivers
- Author
-
Kazuki Hiroe
- Subjects
Pure mathematics ,Generalization ,Differential equation ,General Mathematics ,Riemann sphere ,01 natural sciences ,moduli spaces of meromorphic connections ,Mathematics - Algebraic Geometry ,symbols.namesake ,34M56 ,Mathematics::Algebraic Geometry ,Linear differential equation ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,representations of quivers ,0101 mathematics ,Representation Theory (math.RT) ,additive Deligne–Simpson problem ,Algebraic Geometry (math.AG) ,Mathematics ,Meromorphic function ,010102 general mathematics ,Quiver ,linear ODE with irregular singular points ,16G20 ,Moduli space ,34M25 ,Mathematics - Classical Analysis and ODEs ,symbols ,Embedding ,010307 mathematical physics ,middle convolution ,Mathematics - Representation Theory - Abstract
Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an arbitrary number of unramified irregular singular points and determine the existence of solutions of the generalized additive Deligne-Simpson problems. Moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere. Namely, open embedding of the moduli spaces into quiver varieties is given and the non-emptiness condition of the moduli spaces is determined. Furthermore the connectedness of the moduli spaces is shown., Comment: 62 pages, some errors and typos are corrected
- Published
- 2017
22. Spectral instability of characteristic boundary layer flows
- Author
-
Emmanuel Grenier, Toan T. Nguyen, and Yan Guo
- Subjects
General Mathematics ,Tollmien–Schlichting instability waves ,Boundary (topology) ,FOS: Physical sciences ,boundary layers ,35B35 ,01 natural sciences ,Instability ,spectral instability ,010305 fluids & plasmas ,Physics::Fluid Dynamics ,symbols.namesake ,Navier–Stokes equations ,Singularity ,Mathematics - Analysis of PDEs ,Airy function ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematical Physics ,Mathematics ,010102 general mathematics ,Mathematical analysis ,Reynolds number ,Laminar flow ,Mathematical Physics (math-ph) ,35B25 ,Boundary layer ,35Q30 ,symbols ,Analysis of PDEs (math.AP) - Abstract
In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number: $R \to \infty$. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well-documented in physics literature, going back to the work of Heisenberg, C.C. Lin, Tollmien, Drazin and Reid, but a rigorous construction requires delicate mathematical details, involving for instance a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of $e^{t/\sqrt {R}}$. A new, operator-based approach is introduced, avoiding to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators., 55 pages. arXiv admin note: substantial text overlap with arXiv:1402.1395
- Published
- 2016
23. Gaussian fluctuations of Young diagrams and structure constants of Jack characters
- Author
-
Maciej Dołęga, Valentin Féray, and University of Zurich
- Subjects
Pure mathematics ,Structure constants ,Distribution (number theory) ,General Mathematics ,Gaussian ,Stein’s method ,0102 computer and information sciences ,01 natural sciences ,Measure (mathematics) ,symbols.namesake ,510 Mathematics ,FOS: Mathematics ,05E05 ,Mathematics - Combinatorics ,60C05 ,0101 mathematics ,Jack polynomials ,Mathematics::Representation Theory ,Mathematics ,2600 General Mathematics ,60B20 ,Probability (math.PR) ,010102 general mathematics ,Diagram ,Stein's method ,random partitions ,60C05, 05E05 (secondary:60B20) ,Jack measure ,Connection (mathematics) ,10123 Institute of Mathematics ,polynomial functions on Young diagrams ,010201 computation theory & mathematics ,bulk fluctuations ,symbols ,Combinatorics (math.CO) ,Mathematics - Probability - Abstract
In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the first order asymptotics) and Kerov (for the second order asymptotics). This gives more evidence of the connection with Gaussian $\beta$-ensemble, already suggested by some work of Matsumoto. Our main tool is a polynomiality result for the structure constant of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interested in itself and we give several other applications of it., Comment: 71 pages. Minor modifications from version 1. An extended abstract of this work, with significantly fewer results and a different title, is available as arXiv:1201.1806
- Published
- 2016
24. The symplectic arc algebra is formal
- Author
-
Ivan Smith, Mohammed Abouzaid, and Apollo - University of Cambridge Repository
- Subjects
Khovanov homology ,General Mathematics ,nilpotent slice ,53D40 ,01 natural sciences ,Mathematics::Algebraic Topology ,Arc (geometry) ,Mathematics - Geometric Topology ,Mathematics::K-Theory and Homology ,Fukaya category ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Algebra over a field ,53D40, 57M25 ,Mathematics::Symplectic Geometry ,Mathematics ,Final version ,010102 general mathematics ,Geometric Topology (math.GT) ,Mathematics::Geometric Topology ,Algebra ,symplectic topology ,Mathematics - Symplectic Geometry ,57M25 ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,Symplectic geometry - Abstract
We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology, over fields of characteristic zero. The key ingredient is the construction of a degree one Hochschild cohomology class on a Floer A-infinity algebra associated to the (k,k)-nilpotent slice Y, obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactification of Y. The partial compactification is obtained as the Hilbert scheme of a partial compactification of a Milnor fibre. A sequel to this paper will prove formality of the symplectic cup and cap bimodules, and infer that symplectic Khovanov cohomology and Khovanov cohomology have the same total rank over characteristic zero fields., Comment: 58 pages, 15 figures. Final version: minor corrections
- Published
- 2016
25. Multiplicity estimates: A Morse-theoretic approach
- Author
-
Gal Binyamini
- Subjects
0209 industrial biotechnology ,Pure mathematics ,General Mathematics ,Polytope ,02 engineering and technology ,Dynamical Systems (math.DS) ,Morse code ,01 natural sciences ,law.invention ,Mathematics - Algebraic Geometry ,020901 industrial engineering & automation ,law ,34M99 ,FOS: Mathematics ,multiplicity estimates for vector fields ,11J99 ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - Dynamical Systems ,Algebraic Geometry (math.AG) ,Mathematics ,Bernstein–Kushnirenko–Khovanskii bound ,Mathematics - Number Theory ,010102 general mathematics ,Polar varieties ,Multiplicity (mathematics) ,Invertible matrix ,Milnor fibers ,Vector field - Abstract
The problem of estimating the multiplicity of the zero of a polynomial when restricted to the trajectory of a non-singular polynomial vector field, at one or several points, has been considered by authors in several different fields. The two best (incomparable) estimates are due to Gabrielov and Nesterenko. In this paper we present a refinement of Gabrielov's method which simultaneously improves these two estimates. Moreover, we give a geometric description of the multiplicity function in terms certain naturally associated polar varieties, giving a topological explanation for an asymptotic phenomenon that was previously obtained by elimination theoretic methods in the works of Brownawell, Masser and Nesterenko. We also give estimates in terms of Newton polytopes, strongly generalizing the classical estimates., Comment: Minor revision; To appear in Duke Math. Journal
- Published
- 2016
26. The geometry of Newton strata in the reduction modulo $p$ of Shimura varieties of PEL type
- Author
-
Paul Hamacher
- Subjects
Shimura variety ,Pure mathematics ,14G35 ,General Mathematics ,Modulo ,Mathematics::Number Theory ,010102 general mathematics ,Deformation theory ,20G25 ,01 natural sciences ,Stratification (mathematics) ,14G35, 14L05 (Primary), 20G25 (Secondary) ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Rapoport–Zink space ,0103 physical sciences ,FOS: Mathematics ,Newton stratification ,Level structure ,010307 mathematical physics ,0101 mathematics ,14L05 ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we study the Newton stratification on the reduction of Shimura varieties of PEL type with hyperspecial level structure and the Newton stratification on the deformation space of a Barsotti-Tate group with PEL structure. Our main result is a formula for the dimension of Newton strata and the description their closure in each of the two cases. Furthermore, we calculate the dimension of some Rapoport-Zink spaces as an intermediate result., Comment: 56 pages; v2: corrected some typos, gave more detailed introduction and added a sketch for the idea behind the definition of "numerical dimension" at the beginning of section 10
- Published
- 2015
27. Lagrangian Floer theory on compact toric manifolds, I
- Author
-
Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono, and Kenji Fukaya
- Subjects
Pure mathematics ,14J32 ,General Mathematics ,53D40 ,FOS: Physical sciences ,symbols.namesake ,Mathematics - Algebraic Geometry ,14J45, 14J32 (Secondary) ,FOS: Mathematics ,53D12, 53D40 (Primary) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Superpotential ,14J45 ,Mathematical Physics (math-ph) ,53D12 ,Mathematics - Symplectic Geometry ,Jacobian matrix and determinant ,symbols ,Symplectic Geometry (math.SG) ,Hamiltonian (quantum mechanics) ,Lagrangian ,Symplectic geometry ,Quantum cohomology - Abstract
The present authors introduced the notion of \emph{weakly unobstructed} Lagrangian submanifolds and constructed their \emph{potential function} $\mathfrak{PO}$ purely in terms of $A$-model data in [FOOO2]. In this paper, we carry out explicit calculations involving $\mathfrak{PO}$ on toric manifolds and study the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [Gi1] which advocates that quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the Landau-Ginzburg superpotential. Combining this study with the results from [FOOO2], we also apply the study to various examples to illustrate its implications to symplectic topology of Lagrangian fibers of toric manifolds. In particular we relate it to Hamiltonian displacement property of Lagrangian fibers and to Entov-Polterovich's symplectic quasi-states., Comment: 84 pages, submitted version ; more examples and new results added, exposition polished, minor typos corrected; v3) to appear in Duke Math.J., Example 10.19 modified, citations from the book [FOOO2,3] updated accoding to the final version of [FOOO3] to be published
- Published
- 2010
28. Prime polynomials in short intervals and in arithmetic progressions
- Author
-
Efrat Bank, Lior Bary-Soroker, and Lior Rosenzweig
- Subjects
Mathematics - Number Theory ,Coprime integers ,General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,Prime number ,Euler's totient function ,11T06 ,0102 computer and information sciences ,01 natural sciences ,Prime (order theory) ,symbols.namesake ,Factorization ,Integer ,010201 computation theory & mathematics ,Arithmetic progression ,symbols ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Arithmetic ,Monic polynomial ,Mathematics - Abstract
In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes p1 and m\geq 3 if q is even and deg f' \leq 1. We show that this estimation fails in the neglected cases. Let \pi_q(k) be the number of monic prime polynomials of degree k with coefficients in the finite field with q elements \FF_q. For relatively prime polynomials f,D\in \FF_q[t] we prove that the number N' of monic prime polynomials g that are congruent to f modulo D and of degree k satisfies |N'-\pi_q(k)/\phi(D)|\leq c(k)\pi_q(k)q^{-1/2}/\phi(D), as long as 1\leq \deg D\leq k-3 (or \leq k-4 if p=2 and (f/D)' is constant). We also generalize these results to other factorization types., Comment: Changes in the introduction, accepted to Duke
- Published
- 2015
29. Kadison-Kastler stable factors
- Author
-
Alan D. Wiggins, Erik Christensen, Roger R. Smith, Stuart White, Jan Cameron, and Allan M. Sinclair
- Subjects
von Neumann algebras ,General Mathematics ,Group cohomology ,01 natural sciences ,Combinatorics ,symbols.namesake ,Crossed product ,perturbations ,0103 physical sciences ,FOS: Mathematics ,Ergodic theory ,Kadison–Kastler stability ,0101 mathematics ,Operator Algebras (math.OA) ,Mathematics ,Probability measure ,Mathematics::Operator Algebras ,46L10 ,010102 general mathematics ,Mathematics - Operator Algebras ,Hilbert space ,Von Neumann algebra ,Operator algebra ,Free group ,symbols ,010307 mathematical physics - Abstract
A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For $n\geq 3$ and a free ergodic probability measure preserving action of $SL_n(\mathbb Z)$ on a standard nonatomic probability space $(X,\mu)$, write $M=((L^\infty(X,\mu)\rtimes SL_n(\mathbb Z))\,\overline{\otimes}\, R$, where $R$ is the hyperfinite II$_1$ factor. We show that whenever $M$ is represented as a von Neumann algebra on some Hilbert space $\mathcal H$ and $N\subseteq\mathcal B(\mathcal H)$ is sufficiently close to $M$, then there is a unitary $u$ on $\mathcal H$ close to the identity operator with $uMu^*=N$. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler's conjecture. We also obtain stability results for crossed products $L^\infty(X,\mu)\rtimes\Gamma$ whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module $L^2(X,\mu)$. In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when $\Gamma$ is a free group., Comment: 33 pages. Paper restructured. Some of the material removed will appear in a future article. Duke Math. J., to appear
- Published
- 2014
30. Integrability of the pentagram map
- Author
-
Fedor Soloviev
- Subjects
Pure mathematics ,General Mathematics ,FOS: Physical sciences ,37J35 ,14H70 ,01 natural sciences ,symbols.namesake ,Poisson bracket ,Mathematics - Algebraic Geometry ,Pentagram ,Poisson manifold ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematical Physics ,Mathematics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,010102 general mathematics ,Regular polygon ,Mathematical Physics (math-ph) ,37K20 ,Monodromy ,Jacobian matrix and determinant ,Pentagram map ,symbols ,010307 mathematical physics ,Exactly Solvable and Integrable Systems (nlin.SI) ,Symplectic geometry - Abstract
The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, study the corresponding spectral curve, and the dynamics on its Jacobian. We also prove that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever-Phong's universal formula., 33 pages, 1 figure; v3: substantially revised
- Published
- 2013
31. Loop spaces and representations
- Author
-
David Nadler and David Ben-Zvi
- Subjects
Pure mathematics ,General Mathematics ,01 natural sciences ,Representation theory ,Coherent sheaf ,Infinitesimal character ,symbols.namesake ,Mathematics - Algebraic Geometry ,0103 physical sciences ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,22E46 ,Möbius strip ,0101 mathematics ,Representation Theory (math.RT) ,55P35 ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Mathematics ,Nilpotent cone ,010102 general mathematics ,Derived algebraic geometry ,symbols ,22E57 ,010307 mathematical physics ,Variety (universal algebra) ,Mathematics - Representation Theory ,Flag (geometry) - Abstract
We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups. In particular, we apply the theory developed in our previous paper arXiv:1002.3636 to flag varieties, and obtain new insights into fundamental categories in representation theory. First, we show that one can recover finite Hecke categories (realized by D-modules on flag varieties) from affine Hecke categories (realized by coherent sheaves on Steinberg varieties) via S^1-equivariant localization. Similarly, one can recover D-modules on the nilpotent cone from coherent sheaves on the commuting variety. We also show that the categorical Langlands parameters for real groups studied by Adams-Barbasch-Vogan and Soergel arise naturally from the study of loop spaces of flag varieties and their Jordan decomposition (or in an alternative formulation, from the study of local systems on a Moebius strip). This provides a unifying framework that overcomes a discomforting aspect of the traditional approach to the Langlands parameters, namely their evidently strange behavior with respect to changes in infinitesimal character., A strengthened version of the second half of arXiv:0706.0322, with significant new material. v2: minor revisions. v3: more minor revisions
- Published
- 2013
32. Quasisplit Hecke algebras and symmetric spaces
- Author
-
David A. Vogan, George Lusztig, Massachusetts Institute of Technology. Department of Mathematics, Vogan, David A., and Lusztig, George
- Subjects
Pure mathematics ,Hecke algebra ,20C08 ,General Mathematics ,010102 general mathematics ,Automorphism ,01 natural sciences ,Cohomology ,Intersection homology ,Local system ,0103 physical sciences ,FOS: Mathematics ,Generalized flag variety ,Sheaf ,20G40 ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Algebraically closed field ,Mathematics - Representation Theory ,Mathematics - Abstract
Let (G,K) be a symmetric pair over an algebraically closed field of characteristic different from 2, and let σ be an automorphism with square 1 of G preserving K. In this paper we consider the set of pairs (O,L) where O is a σ-stable K-orbit on the flag manifold of G and L is an irreducible K-equivariant local system on O which is “fixed” by σ. Given two such pairs (O,L), (O',L'), with O' in the closure [bar over O] of O, the multiplicity space of L' in a cohomology sheaf of the intersection cohomology of [bar over O] with coefficients in L (restricted to O') carries an involution induced by σ, and we are interested in computing the dimensions of its +1 and −1 eigenspaces. We show that this computation can be done in terms of a certain module structure over a quasisplit Hecke algebra on a space spanned by the pairs (O,L) as above., National Science Foundation (U.S.) (Grant DMS-0758262), National Science Foundation (U.S.) (Grant DMS-0967272)
- Published
- 2013
33. Hypersurfaces of prescribed curvature measure
- Author
-
Yanyan Li, Pengfei Guan, and Junfang Li
- Subjects
Mathematics - Differential Geometry ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,A priori estimate ,53C42 ,Curvature ,53C23 ,01 natural sciences ,Measure (mathematics) ,35J60 ,010101 applied mathematics ,53C23, 35J60, 53C42 ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,FOS: Mathematics ,Mathematics::Differential Geometry ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We consider the corresponding Christoffel–Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$ -convex bodies with prescribed $k$ th curvature measures ( $k\textgreater 0$ ) has been a longstanding problem. This is settled in this paper through the establishment of a crucial a priori $C^{2}$ -estimate for the corresponding curvature equation on $\mathbb{S}^{n}$ .
- Published
- 2012
34. K3格子への有限シンプレクティック作用
- Author
-
Kenji Hashimoto
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,14J28, 11H56 ,0102 computer and information sciences ,01 natural sciences ,Mathematics - Algebraic Geometry ,010201 computation theory & mathematics ,Lattice (order) ,FOS: Mathematics ,11H56 ,0101 mathematics ,Algebraic Geometry (math.AG) ,14J28 ,Symplectic geometry ,Mathematics - Abstract
In this paper, we study finite symplectic actions on K3 surfaces X, i.e. actions of finite groups G on X which act on H^{2,0}(X) trivially. We show that the action on the K3 lattice H^2(X,Z) induced by a symplectic action of G on X depends only on G up to isomorphism, except for five groups., 46 pages. #13, #19, #25 and #37 in Table 10.2, and #62 in Table 10.3 were corrected
- Published
- 2012
35. The Picard group of the moduli space of curves with level structures
- Author
-
Andrew Putman
- Subjects
Pure mathematics ,14D22 ,Group (mathematics) ,General Mathematics ,Adjoint representation ,Picard group ,Geometric Topology (math.GT) ,Group Theory (math.GR) ,Divisibility rule ,Mapping class group ,Cohomology ,Moduli space ,Mathematics - Algebraic Geometry ,Mathematics - Geometric Topology ,Mathematics::Algebraic Geometry ,FOS: Mathematics ,32G15 ,11G15 ,Abelian group ,57N05 ,Algebraic Geometry (math.AG) ,Mathematics - Group Theory ,57S05 ,Mathematics - Abstract
For $4 \nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level $L$ subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level $L$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of the mod $L$ symplectic group with coefficients in the adjoint representation., 32 pages, 1 figure; substantial revision. To appear in Duke Math. J
- Published
- 2012
36. Bounds on the Hilbert-Kunz multiplicity
- Author
-
Olgur Celikbas, Yi Zhang, Craig Huneke, and Hailong Dao
- Subjects
13A35, 13B22, 13H15, 14B05 ,Discrete mathematics ,13A35 ,14B05 ,Mathematics::Commutative Algebra ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Local ring ,Multiplicity (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,13B22 ,Bounding overwatch ,0103 physical sciences ,FOS: Mathematics ,13H15 ,0101 mathematics ,Mathematics - Abstract
In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed non-regular local rings, bounding them uniformly away from one. Our results improve previous work of Aberbach and Enescu., Comment: Some minor typos fixed, especially Theorem 3.7 (min instead of max). Corollary 2.4 is added by the referee's suggestion
- Published
- 2012
37. The geometry of Markov traces
- Author
-
Geordie Williamson and Ben Webster
- Subjects
14F10 ,Pure mathematics ,Hecke algebra ,Trace (linear algebra) ,General Mathematics ,Homology (mathematics) ,Mathematics::Algebraic Topology ,Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Equivariant cohomology ,20G40 ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,20C08, 14F10, 20G40, 16T99 ,Mathematics ,20C08 ,Hochschild homology ,Mathematics::Geometric Topology ,Algebraic group ,Simple group ,Bimodule ,16T99 ,Mathematics - Representation Theory - Abstract
We give a geometric interpretation of the Jones-Ocneanu trace on the Hecke algebra, using the equivariant cohomology of sheaves on SL(n). This construction makes sense for all simple algebraic groups, so we obtain a generalization of the Jones-Ocneanu trace to Hecke algebras of other types. We give a geometric expansion of this trace in terms of the irreducible characters of the Hecke algebra, and conclude that it agrees with a trace defined independently by Gomi. Based on our proof, we also prove that certain simple perverse sheaves on a reductive algebraic group G are equivariantly formal for the conjugation action of a Borel B, or equivalently, that the Hochschild homology of any Soergel bimodule is free, as the authors had previously conjectured. This construction is closely tied to knot homology. This interpretation of the Jones-Ocneanu trace is a more elementary manifestation of the geometric construction of HOMFLYPT homology given by the authors in a previous paper., 14 pages. v2; typos and minor errors fixed, simplified argument in final section. DVI may not render correctly on all computers; PDF is prefered
- Published
- 2011
38. Stickelberger elements and Kolyvagin systems
- Author
-
Kâzım Büyükboduk, Büyükboduk, Kazım, College of Sciences, and Department of Department of Mathematics
- Subjects
Pure mathematics ,Conjecture ,Mathematics - Number Theory ,11R23, 11R29, 11R42, 11R80 (Primary) 11F80, 11R34 (Secondary) ,General Mathematics ,010102 general mathematics ,11R29 ,Euler system ,01 natural sciences ,symbols.namesake ,11R23 ,11R34 ,0103 physical sciences ,11R27 ,FOS: Mathematics ,Euler's formula ,symbols ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Mathematics ,Real field ,11R42 - Abstract
In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is r >1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author., Scientific and Technological Research Council of Turkey (TÜBİTAK); Marie Curie grant
- Published
- 2011
39. Internal DLA and the Gaussian free field
- Author
-
Lionel Levine, Scott Sheffield, David Jerison, Massachusetts Institute of Technology. Department of Mathematics, Jerison, David S., and Sheffield, Scott Roger
- Subjects
60G50 ,Statistical Mechanics (cond-mat.stat-mech) ,General Mathematics ,Probability (math.PR) ,Mathematical analysis ,FOS: Physical sciences ,Order (ring theory) ,Scale (descriptive set theory) ,Mathematical Physics (math-ph) ,Radius ,Sense (electronics) ,Mathematics - Analysis of PDEs ,60K35 ,Gaussian free field ,FOS: Mathematics ,Cluster (physics) ,Almost surely ,82C24 ,Constant (mathematics) ,Mathematics - Probability ,Condensed Matter - Statistical Mechanics ,Mathematical Physics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In previous works, we showed that the internal diffusion-limited aggregation (DLA) cluster on Z[superscript d] with t particles is almost surely spherical up to a maximal error of O(logt) if d=2 and O(logt√) if d≥3. This paper addresses average error: in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when d=2 and of order r[superscript 1−d/2] (for a radius r cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field., National Science Foundation (U.S.) (grant DMS-0645585), National Science Foundation (U.S.) (grant DMS-1105960), National Science Foundation (U.S.) (Postdoctoral Research Fellowship), National Science Foundation (U.S.) (grant DMS-1069225)
- Published
- 2011
40. Existence of regular neighborhoods for H-surfaces
- Author
-
William H. Meeks and Giuseppe Tinaglia
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mean curvature ,General Mathematics ,010102 general mathematics ,53A10 ,53C42 ,01 natural sciences ,49Q05 ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Mathematics::Differential Geometry ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
In this paper, we study the global geometry of complete, constant mean curvature hypersurfaces embedded in n-manifolds. More precisely, we give conditions that imply properness of such surfaces and prove the existence of fixed size one-sided regular neighborhoods for certain constant mean curvature hypersurfaces in certain n-manifolds., Comment: 11 pages
- Published
- 2011
41. The norm of a Ree group
- Author
-
Tom De Medts and Richard M. Weiss
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Group Theory (math.GR) ,Type (model theory) ,Ree group ,Mathematical proof ,Non-abelian group ,Combinatorics ,51E12 ,51E24 ,Group of Lie type ,FOS: Mathematics ,(B, N) pair ,20E42 ,Mathematics - Group Theory ,Mathematics - Abstract
We give an explicit construction of the Ree groups of type G2 as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject (see [7] and [8]). We also give a short proof that the norm of a Ree group is anisotropic.
- Published
- 2010
42. On the conjugacy growth functions of groups
- Author
-
Victor Guba and Mark Sapir
- Subjects
Pure mathematics ,General Mathematics ,20F69 ,Radius ,Group Theory (math.GR) ,Exponential function ,Mathematics::Group Theory ,Conjugacy class ,Exponential growth ,Growth function ,FOS: Mathematics ,Ball (mathematics) ,Finitely generated group ,20F65 ,Constant (mathematics) ,Mathematics - Group Theory ,Mathematics - Abstract
To every finitely generated group one can assign the conjugacy growth function that counts the number of conjugacy classes intersecting a ball of radius $n$. Results of Ivanov and Osin show that the conjugacy growth function may be constant even if the (ordinary) growth function is exponential. The aim of this paper is to provide conjectures, examples and statements that show that in "normal" cases, groups with exponential growth functions also have exponential conjugacy growth functions., Comment: 10 pages
- Published
- 2010
43. Exponentially generic subsets of groups
- Author
-
Alexei Miasnikov, Robert H. Gilman, and Denis Osin
- Subjects
Discrete mathematics ,Presentation of a group ,43A07 ,Hyperbolic group ,Primary 20F10. Secondary 20F67, 43A07 ,General Mathematics ,20F67 ,Amenable group ,Small cancellation theory ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Group Theory (math.GR) ,Relatively hyperbolic group ,Combinatorics ,Random group ,FOS: Mathematics ,Word problem (mathematics) ,Word problem for groups ,20F10 ,Mathematics - Group Theory ,Computer Science::Formal Languages and Automata Theory ,Mathematics - Abstract
In this paper we study the generic, i.e., typical, behavior of finitely generated subgroups of hyperbolic groups and also the generic behavior of the word problem for amenable groups. We show that a random set of elements of a nonelementary word hyperbolic group is very likely to be a set of free generators for a nicely embedded free subgroup. We also exhibit some finitely presented amenable groups for which the restriction of the word problem is unsolvable on every sufficiently large subset of words., 17 pages, 1 figure
- Published
- 2010
44. The base change fundamental lemma for central elements in parahoric Hecke algebras
- Author
-
Thomas J. Haines
- Subjects
Hecke algebra ,Pure mathematics ,General Mathematics ,Mathematics::Number Theory ,Field (mathematics) ,Center (group theory) ,01 natural sciences ,Base change ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Commutative property ,Mathematics ,Mathematics - Number Theory ,Group (mathematics) ,010102 general mathematics ,20G25 ,Fundamental lemma ,22E50 ,Homomorphism ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p., Comment: No figures; 53 pages. Statement of Lemma 4.2.1 modified; minor corrections in section 5; minor expositional changes; final version -- to appear in Duke Math J
- Published
- 2009
45. Appell polynomials and their relatives III. Conditionally free theory
- Author
-
Michael Anshelevich
- Subjects
Discrete mathematics ,Polynomial ,46L53 (Primary), 46L54, 05E35 (Secondary) ,Mathematics::Operator Algebras ,46L54 ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Free probability ,01 natural sciences ,010101 applied mathematics ,05E35 ,Difference polynomials ,Probability theory ,Orthogonal polynomials ,Probability mass function ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,46L53 ,Operator Algebras (math.OA) ,Random variable ,Mathematics ,Probability measure - Abstract
We extend to the multivariate non-commutative context the descriptions of a "once-stripped" probability measure in terms of Jacobi parameters, orthogonal polynomials, and the moment generating function. The corresponding map Phi on states was introduced previously by Belinschi and Nica. We then relate these constructions to the c-free probability theory, which is a version of free probability for algebras with two states, introduced by Bozejko, Leinert, and Speicher. This theory includes as two extreme cases the free and Boolean probability theories. The main objects in the paper are the analogs of the Appell polynomial families in the two state context. They arise as fixed points of the transformation which takes a polynomial family to the associated polynomial family (in several variables), and their orthogonality is also related to the map Phi above. In addition, we prove recursions, generating functions, and factorization and martingale properties for these polynomials, and describe the c-free version of the Kailath-Segall polynomials, their combinatorics, and Hilbert space representations., A major revision: same theorems, different emphasis
- Published
- 2009
46. Isoparametric hypersurfaces with four principal curvatures revisited
- Author
-
Quo-Shin Chi
- Subjects
Mathematics - Differential Geometry ,Polynomial ,Pure mathematics ,isoparametric hypersurface ,Partial differential equation ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,53C40 ,Characterization (mathematics) ,Type (model theory) ,Submanifold ,01 natural sciences ,Hypersurface ,Differential Geometry (math.DG) ,Principal curvature ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics - Abstract
The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and M\"{u}nzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles. In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-M\"{u}nzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear., Comment: 25 pages
- Published
- 2009
47. On tilting modules over cluster-tilted algebras
- Author
-
David Smith
- Subjects
Pure mathematics ,Endomorphism ,16G20 ,18E30 ,General Mathematics ,01 natural sciences ,Representation theory ,Cluster algebra ,Lift (mathematics) ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics ,Discrete mathematics ,Derived category ,Functor ,010102 general mathematics ,Tilting theory ,Quiver ,Mathematics - Rings and Algebras ,Rings and Algebras (math.RA) ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
In this paper, we show that the tilting modules over a cluster-tilted algebra $A$ lift to tilting objects in the associated cluster category $\mathcal{C}_H$. As a first application, we describe the induced exchange relation for tilting $A$-modules arising from the exchange relation for tilting object in $\mathcal{C}_H$. As a second application, we exhibit tilting $A$-modules having cluster-tilted endomorphism algebras., Comment: 19 pages, 1 figure
- Published
- 2008
48. The characteristic polynomial of a random unitary matrix: A probabilistic approach
- Author
-
Paul Bourgade, Ashkan Nikeghbali, Marc Yor, C. P. Hughes, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Benassù, Serena
- Subjects
Pure mathematics ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Multivariate random variable ,General Mathematics ,Mathematics::Number Theory ,FOS: Physical sciences ,01 natural sciences ,Matrix polynomial ,010104 statistics & probability ,60F05 ,FOS: Mathematics ,0101 mathematics ,Mathematical Physics ,ComputingMilieux_MISCELLANEOUS ,Central limit theorem ,Characteristic polynomial ,Mathematics ,Discrete mathematics ,Probability (math.PR) ,010102 general mathematics ,15A52 ,Law of the iterated logarithm ,Mathematical Physics (math-ph) ,Unitary matrix ,16. Peace & justice ,15A52, 60F05, 60F15 ,Polynomial matrix ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Convergence of random variables ,60F15 ,Mathematics - Probability - Abstract
In this paper, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin Fourier transform of such a random polynomial, first obtained by Keating and Snaith in (7), using a simple recursion formula, and from there we are able to obtain the joint law of its radial and an- gular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of in- dependent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in (7) is now obtained from the classical central limit theorems of Probability Theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm type results.
- Published
- 2008
49. The geometric theory of the fundamental germ
- Author
-
T. M. Gendron
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Dynamical Systems ,Covering space ,General Mathematics ,01 natural sciences ,diophantine approximation ,Group action ,57R30, 14B0 ,Mathematics::Algebraic Geometry ,0103 physical sciences ,fundamental germ ,FOS: Mathematics ,57R30 ,Algebraic Topology (math.AT) ,Germ ,Mathematics - Algebraic Topology ,0101 mathematics ,Algebraic number ,Invariant (mathematics) ,Smooth structure ,Mathematics ,foliations ,010308 nuclear & particles physics ,Mathematics::Complex Variables ,010102 general mathematics ,Mathematical analysis ,mapping class group ,Mathematics::Geometric Topology ,Mapping class group ,Geometric group theory ,Differential Geometry (math.DG) ,laminations - Abstract
The fundamental germ is a generalization of $\pi_{1}$, first defined for laminations which arise through group actions in math.DG/0506270. In this paper, the fundamental germ is extended to any lamination having a dense leaf admitting a smooth structure. In addition, an amplification of the fundamental germ called the mother germ is constructed, which is, unlike the fundamental germ, a topological invariant. The fundamental germs of the antenna lamination and the $PSL(2,\Z)$ lamination are calculated, laminations for which the definition in math.DG/0506270 was not available. The mother germ is used to give a proof of a Nielsen theorem for the algebraic universal cover of a closed surface of hyperbolic type., Comment: 23 pages, 2 figures, submitted for publication
- Published
- 2008
50. Finite dimensional point derivations for graph algebras
- Author
-
Benton L. Duncan
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,47L55 ,Voltage graph ,Mathematics - Operator Algebras ,Directed graph ,Graph algebra ,Tensor algebra ,law.invention ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,47L40, 47L55, 47L75, 46L80 ,law ,Line graph ,FOS: Mathematics ,Graph (abstract data type) ,47L40 ,47L75 ,Graph property ,Null graph ,Operator Algebras (math.OA) ,46L80 ,Mathematics - Abstract
This paper focuses on certain finite dimensional point derivations for the non-selfadjoint operator algebras corresponding to directed graphs. We begin by analyzing the derivations corresponding to full matrix representations of the tensor algebra of a directed graph. We determine when such a derivation is inner, and describe situations that give rise to non-inner derivations. We also analyze the situation when the derivation corresponds to a multiplicative linear functional., 18 pages
- Published
- 2008
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