1,008 results
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2. The Dilworth theorems: Selected papers of Robert P. Dilworth
- Author
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M.K Bennett
- Subjects
Combinatorics ,Mathematics(all) ,General Mathematics ,Mathematics - Published
- 1992
3. On the paper 'A ‘lost’ notebook of Ramanujan'
- Author
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R.P Agarwal
- Subjects
Algebra ,symbols.namesake ,Mathematics(all) ,General Mathematics ,symbols ,Ramanujan's sum ,Mathematics - Published
- 1984
- Full Text
- View/download PDF
4. Collected papers, vol. I: Combinatorics
- Author
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Paul R Stein
- Subjects
Mathematics(all) ,GEORGE (programming language) ,General Mathematics ,Classics ,Mathematics - Published
- 1979
5. Correction to my paper on Nakayama R-varieties
- Author
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Reiten, Idun, primary
- Published
- 1977
- Full Text
- View/download PDF
6. Collected papers
- Author
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Rota, Gian-Carlo, primary
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- 1983
- Full Text
- View/download PDF
7. Markov processes and related problems of analysis (selected papers)
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Rota, Gian-Carlo, primary
- Published
- 1985
- Full Text
- View/download PDF
8. Collected papers
- Author
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Rota, Gian-Carlo, primary
- Published
- 1977
- Full Text
- View/download PDF
9. Selected papers
- Published
- 1983
- Full Text
- View/download PDF
10. Collected papers
- Author
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Rota, Gian-Carlo, primary
- Published
- 1985
- Full Text
- View/download PDF
11. Zur algebraischen geometrie (selected papers)
- Author
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Rota, Gian-Carlo, primary
- Published
- 1985
- Full Text
- View/download PDF
12. A note on our paper “theory of decomposition in semigroups”
- Author
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Ruzsa, Imre Z., primary and Székely, Gábor J., additional
- Published
- 1986
- Full Text
- View/download PDF
13. Collected papers, vol. I: Combinatorics
- Author
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Stein, Paul R, primary
- Published
- 1979
- Full Text
- View/download PDF
14. Selected papers
- Published
- 1977
- Full Text
- View/download PDF
15. Key papers in the development of information theory
- Author
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Rota, G.-C, primary
- Published
- 1975
- Full Text
- View/download PDF
16. The Bellman continuum, a collection of the works of Richard E. Bellman. Selected papers
- Published
- 1989
- Full Text
- View/download PDF
17. On the paper “A ‘lost’ notebook of Ramanujan”
- Author
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Agarwal, R.P, primary
- Published
- 1984
- Full Text
- View/download PDF
18. Markov processes and related problems of analysis (selected papers)
- Author
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Gian-Carlo Rota
- Subjects
Discrete mathematics ,symbols.namesake ,Mathematics(all) ,General Mathematics ,symbols ,Markov process ,Mathematics - Published
- 1985
- Full Text
- View/download PDF
19. Key papers in the development of information theory
- Author
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G.-C Rota
- Subjects
Mathematics(all) ,Development (topology) ,General Mathematics ,Information theory ,Mathematical economics ,Mathematics - Published
- 1975
- Full Text
- View/download PDF
20. Selected papers
- Author
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Gian-Carlo Rota
- Subjects
Discrete mathematics ,Mathematics(all) ,General Mathematics ,Humanities ,Mathematics - Published
- 1977
- Full Text
- View/download PDF
21. Correction to my paper on Nakayama R-varieties
- Author
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Idun Reiten
- Subjects
Pure mathematics ,Mathematics(all) ,General Mathematics ,Mathematics - Published
- 1977
- Full Text
- View/download PDF
22. A note on our paper 'theory of decomposition in semigroups'
- Author
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Gábor J. Székely and Imre Z. Ruzsa
- Subjects
Krohn–Rhodes theory ,Algebra ,Mathematics(all) ,General Mathematics ,Decomposition (computer science) ,Mathematics - Published
- 1986
- Full Text
- View/download PDF
23. Zur algebraischen geometrie (selected papers)
- Author
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Gian-Carlo Rota
- Subjects
Discrete mathematics ,Mathematics(all) ,General Mathematics ,Mathematics - Published
- 1985
- Full Text
- View/download PDF
24. The Bellman continuum, a collection of the works of Richard E. Bellman. Selected papers
- Author
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Gian-Carlo Rota
- Subjects
Mathematics(all) ,Continuum (measurement) ,General Mathematics ,Mathematical economics ,Mathematics - Published
- 1989
25. Collected papers
- Author
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Gian-Carlo Rota
- Subjects
Algebra ,Mathematics(all) ,Number theory ,General Mathematics ,Library science ,Humanities ,Classics ,Mathematical physics ,Mathematics - Published
- 1985
26. The Frobenius morphism in invariant theory II
- Author
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Spenko, Spela, Raedschelders, Theo, Van Den Bergh, Michel, RAEDSCHELDERS, Theo, SPENKO, Spela, VAN DEN BERGH, Michel, Algebra and Analysis, Mathematics, and Algebra
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Frobenius summand ,General Mathematics ,Invariant theory ,Mathematics - Rings and Algebras ,Frobenius kernel ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Mathematics - Algebraic Geometry ,Rings and Algebras (math.RA) ,Géométrie algébrique ,FOS: Mathematics ,FFRT ,Representation Theory (math.RT) ,Groupes algébriques ,Algèbre commutative et algèbre homologique ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory ,Géométrie non commutative - Abstract
Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=Gr(2,n)$ defined over an algebraically closed field $k$ of characteristic $p \geq \max\{n-2,3\}$. In this paper we give a description of the decomposition of $R$, considered as graded $R^{p^r}$-module, for $r \geq 2$. This is a companion paper to our earlier paper, where the case $r=1$ was treated, and taken together, our results imply that $R$ has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators $D_k(R)$ is simple, that $\mathbb{G}$ has global finite F-representation type (GFFRT) and that $R$ provides a noncommutative resolution for $R^{p^r}$., Comment: 52 pages
- Published
- 2022
27. Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on the Siegel domains and complex hyperbolic spaces
- Author
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Lu, Guozhen and Yang, Qiaohua
- Subjects
Mathematics - Differential Geometry ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Mathematics - Classical Analysis and ODEs ,Mathematics - Complex Variables ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,42B35, 42B15, 42B37, 35J08 ,Complex Variables (math.CV) ,Analysis of PDEs (math.AP) - Abstract
This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincar\'e-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on real hyperbolic spaces using the method of Helgason-Fourier analysis on the hyperbolic spaces. The aim of this paper is to establish such inequalities on the Siegel domains and complex hyperbolic spaces. Firstly, we prove a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller' operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group $SU(1,n)$ and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincar\'e-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain $\mathcal{U}^{n}$ and the unit ball $\mathbb{B}_{\mathbb{C}}^{n}$. Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. The factorization theorem we proved is of its independent interest in the Heisenberg group and CR sphere and CR invariant differential operators therein., Comment: 51 pages
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- 2022
28. On the Goulden–Jackson–Vakil conjecture for double Hurwitz numbers
- Author
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Norman Do and Danilo Lewański
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Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,14H10, 14N10, 05A15 ,General Mathematics ,FOS: Mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Algebraic Geometry (math.AG) ,Mathematical Physics - Abstract
Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first involves Hodge classes on moduli spaces of stable maps to classifying spaces; the second involves Chiodo classes on moduli spaces of spin curves; and the third involves tautological classes on moduli spaces of stable curves. We proceed to discuss the merits of these formulas against a list of desired properties enunciated by Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between tautological intersection numbers on moduli spaces of stable curves and hints at further structure underlying Chiodo classes. The paper concludes with generalisations of our results to the context of spin Hurwitz numbers., Comment: 20 pages. Some corollaries are added in the second version, and software numerical checks are performed
- Published
- 2022
29. Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities
- Author
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Lu Zhang, Guozhen Lu, and Nguyen Lam
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Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Function (mathematics) ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Infimum and supremum ,Symmetry (physics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics ,media_common - Abstract
Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space R n , including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in [25] , combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.
- Published
- 2019
30. Normal crossings singularities for symplectic topology
- Author
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Mark McLean, Aleksey Zinger, and Mohammad Farajzadeh Tehrani
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Pure mathematics ,Logarithm ,Divisor ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics - Symplectic Geometry ,0103 physical sciences ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,53D05, 53D45, 14N35 ,Gravitational singularity ,010307 mathematical physics ,0101 mathematics ,Equivalence (formal languages) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Symplectic sum ,Symplectic geometry ,Mathematics - Abstract
We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors., Comment: 65 pages, 4 figures; a number of typos fixed; the exposition has been significantly revised, fixing a technical error in the non-compact case in the process; this paper is now restricted to the simple normal crossings case; the arbitrary normal crossings case will be detailed in a followup paper
- Published
- 2018
31. On the mean field type bubbling solutions for Chern–Simons–Higgs equation
- Author
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Shusen Yan and Chang-Shou Lin
- Subjects
General Mathematics ,010102 general mathematics ,Chern–Simons theory ,Structure (category theory) ,Type (model theory) ,01 natural sciences ,Mean field theory ,0103 physical sciences ,Higgs boson ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Parallelogram ,Mathematical physics ,Mathematics - Abstract
This paper is the second part of our comprehensive study on the structure of the solutions for the following Chern–Simons–Higgs equation: (0.1) { Δ u + 1 e 2 e u ( 1 − e u ) = 4 π ∑ j = 1 N δ p j , in Ω , u is doubly periodic on ∂ Ω , where Ω is a parallelogram in R 2 and e > 0 is a small parameter. In part 1 [29] , we proved the non-coexistence of different bubbles in the bubbling solutions and obtained an existence result for the Chern–Simons type bubbling solutions under some nearly necessary conditions. Mean field type bubbling solutions for (0.1) have been constructed in [27] . In this paper, we shall study two other important issues for the mean field type bubbling solutions: the necessary conditions for the existence and the local uniqueness. The results in this paper lay the foundation to find the exact number of solutions for (0.1) .
- Published
- 2018
32. Quasi-elliptic cohomology I
- Author
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Zhen Huan
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Elliptic cohomology ,16. Peace & justice ,Space (mathematics) ,Mathematics::Algebraic Topology ,01 natural sciences ,Cohomology ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Equivariant map ,Mathematics - Algebraic Topology ,010307 mathematical physics ,55N34, 55P35 ,0101 mathematics ,Tate curve ,Constant (mathematics) ,Computer Science::Databases ,Quotient ,Orbifold ,Mathematics - Abstract
Quasi-elliptic cohomology is a variant of elliptic cohomology theories. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. Thus, the constructions on it can be made in a neat way. This theory reflects the geometric nature of the Tate curve. In this paper we provide a systematic introduction of its construction and definition., Comment: Final Version. 26 pages. To appear in Advances in Mathematics. In this paper we generalize the construction in arXiv:1612.00930. The subtle point of this generalization is explained in Section 2
- Published
- 2018
33. The restricted content and the d-dimensional Analyst's Travelling Salesman Theorem for general sets
- Author
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Hyde, Matthew
- Subjects
General Mathematics - Abstract
In his 1990 paper, Jones characterized subsets of rectifiable curves in via a multiscale sum of β-numbers, which measure how far a given set deviates from a straight line at each scale and location. This characterization was extended by Okikiolu to subsets of and by Schul to subsets of a Hilbert space.\ud \ud Recently, there has been some interest in characterizing subsets of higher dimensional surfaces in . Using a variant of Jones' β-number introduced by Azzam and Schul, Villa gave a characterization of lower regular subsets of a certain class of topologically stable surfaces – introduced in a 2004 paper of David – via a multiscale sum of these new β-numbers.\ud \ud In this paper we remove the lower regularity condition and prove an analogous result for general d-dimensional subsets of . To do this, we introduce the restricted content, which assigns ‘mass’ to any subset of (even to sets with zero Hausdorff measure), and use it to define new d-dimensional variant of Jones' β-number that is defined for any set in .
- Published
- 2022
34. Balanced derivatives, identities, and bounds for trigonometric and Bessel series
- Author
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Bruce C. Berndt, Sun Kim, Martino Fassina, and Alexandru Zaharescu
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symbols.namesake ,Pure mathematics ,Series (mathematics) ,General Mathematics ,symbols ,Trigonometric functions ,Divisor (algebraic geometry) ,Trigonometry ,Upper and lower bounds ,Bessel function ,Gauss circle problem ,Ramanujan's sum ,Mathematics - Abstract
Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8] . These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x → ∞ are established.
- Published
- 2022
35. Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory
- Author
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Yiannis Sakellaridis
- Subjects
Transfer (group theory) ,Pure mathematics ,Hecke algebra ,symbols.namesake ,Conjecture ,Trace (linear algebra) ,General Mathematics ,Poisson summation formula ,symbols ,Functional equation (L-function) ,Abelian group ,Fundamental lemma ,Mathematics - Abstract
The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G 1 , G 2 for every morphism G 1 L → L G 2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula. The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”. In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for SL 2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to GL 2 ). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for GL 2 , and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.
- Published
- 2022
36. Decomposition spaces, incidence algebras and Möbius inversion III: The decomposition space of Möbius intervals
- Author
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Joachim Kock, Imma Gálvez-Carrillo, Andrew Tonks, Universitat Politècnica de Catalunya. Departament de Matemàtiques, and Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions
- Subjects
Pure mathematics ,Mathematics::General Mathematics ,Mathematics::Number Theory ,General Mathematics ,Coalgebra ,18 Category theory [Classificació AMS] ,Structure (category theory) ,18G Homological algebra [homological algebra] ,Combinatorial topology ,55 Algebraic topology::55P Homotopy theory [Classificació AMS] ,Algebraic topology ,Space (mathematics) ,2-Segal space ,01 natural sciences ,Combinatorics ,decomposition space ,18G30, 16T10, 06A11, 18-XX, 55Pxx ,Mathematics::Category Theory ,0103 physical sciences ,Mathematics - Combinatorics ,Mathematics::Metric Geometry ,Matemàtiques i estadística::Topologia::Topologia algebraica [Àrees temàtiques de la UPC] ,Mathematics - Algebraic Topology ,0101 mathematics ,06 Order, lattices, ordered algebraic structures::06A Ordered sets [Classificació AMS] ,Mathematics ,Topologia combinatòria ,CULF functor ,Mathematics::Combinatorics ,Functor ,Mathematics::Complex Variables ,Homotopy ,010102 general mathematics ,Mathematics - Category Theory ,Möbius interval ,Topologia algebraica ,Hopf algebra ,18 Category theory ,homological algebra::18G Homological algebra [Classificació AMS] ,010307 mathematical physics ,Möbius inversion - Abstract
Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of M\"obius decomposition space, a far-reaching generalisation of the notion of M\"obius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of M\"obius intervals, which contains the universal M\"obius function (but is not induced by a M\"obius category), can be realised as the homotopy cardinality of a M\"obius decomposition space $U$ of all M\"obius intervals, and that in a certain sense $U$ is universal for M\"obius decomposition spaces and CULF functors., Comment: 35 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v3: minor expository improvements. Final version to appear in Adv. Math
- Published
- 2018
37. Nevanlinna theory of the Askey–Wilson divided difference operator
- Author
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Yik-Man Chiang and Shao-Ji Feng
- Subjects
Pure mathematics ,Basic hypergeometric series ,High Energy Physics::Lattice ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Zero (complex analysis) ,Infinite product ,01 natural sciences ,Nevanlinna theory ,010101 applied mathematics ,Operator (computer programming) ,0101 mathematics ,Complex plane ,Picard theorem ,Meromorphic function ,Mathematics - Abstract
This paper establishes a version of Nevanlinna theory based on Askey–Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane C . A second main theorem that we have derived allows us to define an Askey–Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole-scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey–Wilson type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey–Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey–Wilson. This paper concludes with an application to difference equations generalising the Askey–Wilson second-order divided difference equation.
- Published
- 2018
38. The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem
- Author
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Yusuke Kuno, Anton Alekseev, Florian Naef, and Nariya Kawazumi
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Pure mathematics ,Lie bialgebra ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Order (ring theory) ,Field (mathematics) ,Mathematics::Geometric Topology ,01 natural sciences ,Bracket (mathematics) ,Mathematics::Quantum Algebra ,Genus (mathematics) ,0103 physical sciences ,010307 mathematical physics ,Lie theory ,0101 mathematics ,Mathematics - Abstract
In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket { − , − } and Turaev cobracket δ on the K -span of homotopy classes of free loops on Σ. Applying an expansion θ : K π → K 〈 x 1 , … , x n 〉 yields an algebraic description of the operations { − , − } and δ in terms of non-commutative variables x 1 , … , x n . If Σ is a surface of genus g = 0 the lowest degree parts { − , − } − 1 and δ − 1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31] . It was conjectured by the second and the third authors that one can define an expansion θ such that { − , − } = { − , − } − 1 and δ = δ − 1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24] , Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2] ).
- Published
- 2018
39. Exceptional collections on Dolgachev surfaces associated with degenerations
- Author
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Yongnam Lee and Yonghwa Cho
- Subjects
Derived category ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Picard group ,Vector bundle ,Type (model theory) ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,Simply connected space ,Algebraic surface ,FOS: Mathematics ,Kodaira dimension ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics - Abstract
Dolgachev surfaces are simply connected minimal elliptic surfaces with $p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via $\mathbb Q$-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper by Lee-Park. Also, some exceptional bundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothing are constructed based on the idea of Hacking. In the case if Dolgachev surfaces were of type $(2,3)$, we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exist a nontrivial phantom category in the derived category., Comment: 35 pages; 3 figures; exposition improved; Adv. Math. (to appear)
- Published
- 2018
40. Irreducible modules over finite simple Lie pseudoalgebras III. Primitive pseudoalgebras of type H
- Author
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Bakalov, B., D'Andrea, A., and Kac, V. G.
- Subjects
General Mathematics ,Mathematics - Quantum Algebra ,Conformally symplectic geometry ,Hopf algebra ,Lie pseudoalgebra ,Lie–Cartan algebra of vector fields ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,17B35 (Primary) 16W30 (Secondary) 17B81 ,Mathematics - Representation Theory - Abstract
A Lie conformal algebra is an algebraic structure that encodes the singular part of the operator product expansion of chiral fields in conformal field theory. A Lie pseudoalgebra is a generalization of this structure, for which the algebra of polynomials k[\partial] in the indeterminate is replaced by the universal enveloping algebra U(d) of a finite-dimensional Lie algebra d over the base field k. The finite (i.e., finitely generated over U(d)) simple Lie pseudoalgebras were classified in our 2001 paper [BDK]. The complete list consists of primitive Lie pseudoalgebras of type W, S, H, and K, and of current Lie pseudoalgebras over them or over simple finite-dimensional Lie algebras. The present paper is the third in our series on representation theory of simple Lie pseudoalgebras. In the first paper, we showed that any finite irreducible module over a primitive Lie pseudoalgebra of type W or S is either an irreducible tensor module or the image of the differential in a member of the pseudo de Rham complex. In the second paper, we established a similar result for primitive Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction, called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by M. Rumin [Rum]. In the present paper, we show that for primitive Lie pseudoalgebras of type H, a similar to type K result holds with the contact pseudo de Rham complex replaced by a suitable complex. However, the type H case in more involved, since the annihilation algebra is not the corresponding Lie-Cartan algebra, as in other cases, but an irreducible central extension. When the action of the center of the annihilation algebra is trivial, this complex is related to work by M. Eastwood [E] on conformally symplectic geometry, and we call it conformally symplectic pseudo de Rham complex., Comment: 62 pages
- Published
- 2021
41. On emergence and complexity of ergodic decompositions
- Author
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Pierre Berger and Jairo Bochi
- Subjects
Pure mathematics ,Lebesgue measure ,Dynamical systems theory ,General Mathematics ,010102 general mathematics ,Dynamical Systems (math.DS) ,Lebesgue integration ,37A35, 37C05, 37C45, 37C40, 37J40 ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,Metric space ,symbols.namesake ,FOS: Mathematics ,symbols ,Ergodic theory ,Mathematics - Dynamical Systems ,0101 mathematics ,Dynamical system (definition) ,Probability measure ,Mathematics - Abstract
A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms., Comment: v3: Final version; to appear in Advances in Mathematics
- Published
- 2021
42. L-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequality
- Author
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Philip T. Gressman
- Subjects
Pure mathematics ,Brascamp–Lieb inequality ,Continuum (topology) ,General Mathematics ,010102 general mathematics ,chemistry.chemical_element ,Radon ,Type (model theory) ,01 natural sciences ,Ambient space ,Range (mathematics) ,Quadratic equation ,chemistry ,Dimension (vector space) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper considers the problem of establishing L p -improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension n of the ambient space and the dimension k of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang [28] and extended by Zorin-Kranich [29] , and on recent results for geometric nonconcentration inequalities [11] . As an initial application of this new approach, this paper establishes sharp restricted strong type L p -improving inequalities for certain model quadratic submanifolds in the range k n ≤ 2 k .
- Published
- 2021
43. GIT versus Baily-Borel compactification for K3's which are double covers of P1×P1
- Author
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Radu Laza and Kieran G. O'Grady
- Subjects
Baily–Borel compactification ,Pure mathematics ,Mathematics::Algebraic Geometry ,Simple (abstract algebra) ,General Mathematics ,Quartic function ,Complete intersection ,Birational geometry ,Type (model theory) ,Mathematics ,Moduli ,Moduli space - Abstract
In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper [36] , based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period maps from the GIT moduli spaces mentioned above to the corresponding Baily-Borel compactifications. In our second paper [35] we studied the case of quartic K3's; we have given geometric meaning to this decomposition and we have partially verified our conjectures. Here, we give a full proof of the conjectures in [36] for the moduli space of K3's which are double covers of a smooth quadric surface. The main new tool here is VGIT for ( 2 , 4 ) complete intersection curves.
- Published
- 2021
44. Covering with Chang models over derived models
- Author
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Grigor Sargsyan
- Subjects
Discrete mathematics ,Conjecture ,Current (mathematics) ,General Mathematics ,010102 general mathematics ,Mathematics - Logic ,01 natural sciences ,Mathematics::Logic ,Continuation ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Logic (math.LO) ,Mathematics - Abstract
We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in Covering with Universally Baire Functions Advances in Mathematics, and the main conjecture of the current paper is a revision of the UB Covering Conjecture of the aforementioned paper.
- Published
- 2021
45. Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes
- Author
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Jeffrey Adams, Xuhua He, and Sian Nie
- Subjects
Weyl group ,Pure mathematics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Unipotent ,Reductive group ,01 natural sciences ,Injective function ,Primary: 20G07, Secondary: 06A07, 20F55, 20E45 ,symbols.namesake ,Conjugacy class ,0103 physical sciences ,FOS: Mathematics ,symbols ,Order (group theory) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Algebraically closed field ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry., Comment: 25 pages
- Published
- 2021
46. An infinite self-dual Ramsey theorem
- Author
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Dimitris Vlitas
- Subjects
Mathematics::Logic ,Pure mathematics ,Mathematics::Combinatorics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics::General Topology ,010307 mathematical physics ,Ramsey's theorem ,0101 mathematics ,01 natural sciences ,Dual (category theory) ,Mathematics - Abstract
In a recent paper [5] S. Solecki proved a finite self-dual Ramsey theorem that extends simultaneously the classical finite Ramsey theorem [4] and the Graham–Rothschild theorem [2] . In this paper we prove the corresponding infinite dimensional version of the self-dual theorem. As a consequence, we extend the classical infinite Ramsey theorem [4] and the Carlson–Simpson theorem [1] .
- Published
- 2017
47. Answer to a 1962 question by Zappa on cosets of Sylow subgroups
- Author
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Marston Conder
- Subjects
0301 basic medicine ,Pure mathematics ,Complement (group theory) ,Finite group ,Janko group ,General Mathematics ,010102 general mathematics ,Sylow theorems ,01 natural sciences ,Combinatorics ,03 medical and health sciences ,Normal p-complement ,030104 developmental biology ,Locally finite group ,Order (group theory) ,0101 mathematics ,Zappa–Szép product ,Mathematics - Abstract
In a paper in 1962, Guido Zappa asked whether a non-trivial coset of a Sylow p-subgroup of a finite group could contain only elements whose orders are powers of p, and also in that case, at least one element of order p. The first question was raised again recently in a 2014 paper by Daniel Goldstein and Robert Guralnick, when generalising an answer by John Thompson in 1967 to a similar question by L.J. Paige. In this paper we give a positive answer to both questions of Zappa, showing somewhat surprisingly that in a number of non-abelian finite simple groups (including PSL ( 3 , 4 ) , PSU ( 5 , 2 ) and the Janko group J 3 ), some non-trivial coset of a Sylow 5-subgroup (of order 5) contains only elements of order 5. Also Zappa's first question is studied in more detail. Various possibilities for the group and its Sylow p-subgroup P are eliminated, and it then follows that | P | ≥ 5 and | P | ≠ 8 . It is an open question as to whether the order of the Sylow p-subgroup can be 7 or 9 or more.
- Published
- 2017
48. Exotic elliptic algebras of dimension 4
- Author
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Alexandru Chirvasitu and S. Paul Smith
- Subjects
Discrete mathematics ,Ring (mathematics) ,General Mathematics ,010102 general mathematics ,Homogeneous coordinate ring ,Algebraic geometry ,Automorphism ,01 natural sciences ,Combinatorics ,Elliptic curve ,Grassmannian ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Incidence (geometry) ,Mathematics - Abstract
Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A = A ( E , τ ) that depends on this data: A ( E , τ ) = ( S ( E , τ ) ⊗ M 2 ( k ) ) Γ where S ( E , τ ) is the 4-dimensional Sklyanin algebra associated to ( E , τ ) , M 2 ( k ) is the ring of 2 × 2 matrices over k, and Γ is ( Z / 2 ) × ( Z / 2 ) acting in a particular way as automorphisms of S and M 2 ( k ) . The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E [ 2 ] on E. Following the ideas and results in papers of Artin–Tate–Van den Bergh, Smith–Stafford, and Levasseur–Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P 3 that we denote by Proj n c ( A ) . Point modules and fat point modules determine “points” in Proj n c ( A ) . Line modules determine “lines” in Proj n c ( A ) . Line modules for A are in bijection with certain lines in P ( A 1 ⁎ ) ≅ P 3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G ( 1 , 3 ) . Shelton–Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plucker P 5 , and that deg ( L ) = 20 . The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P ( A 1 ⁎ ) . Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.
- Published
- 2017
49. Flattening of CR singular points and analyticity of the local hull of holomorphy II
- Author
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Wanke Yin and Xiaojun Huang
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,Codimension ,Singular point of a curve ,Submanifold ,01 natural sciences ,Plateau's problem ,Hypersurface ,Complex space ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C n with n ≥ 3 . The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [23] are taken from our earlier arxiv post [22] . We split [22] into two independent articles to avoid it being too long.
- Published
- 2017
50. Bernstein inequality and holonomic modules
- Author
-
Ivan Losev
- Subjects
Pure mathematics ,Holonomic ,General Mathematics ,010102 general mathematics ,Bernstein inequalities ,01 natural sciences ,Representation theory ,0103 physical sciences ,Bimodule ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Commutative property ,Simple module ,Mathematics ,Symplectic geometry - Abstract
In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, and spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that, provided the algebraic fundamental groups of all leaves are finite, the generalized Bernstein inequality holds for the simple modules and turns into equality for holonomic simples. Under the same finiteness assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length, then any holonomic module has finite length. This allows one to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions, for quantum Hamiltonian reductions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.
- Published
- 2017
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