128 results
Search Results
2. Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials
- Author
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Daniel Erman, Steven V Sam, and Andrew Snowden
- Subjects
Pure mathematics ,General Mathematics ,media_common.quotation_subject ,MathematicsofComputing_GENERAL ,Hilbert's basis theorem ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematics - Algebraic Geometry ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Ideal (ring theory) ,0101 mathematics ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics ,media_common ,Conjecture ,Hilbert's syzygy theorem ,Mathematics::Commutative Algebra ,Degree (graph theory) ,Applied Mathematics ,010102 general mathematics ,13A02, 13D02 ,Mathematics - Commutative Algebra ,Infinity ,Bounded function ,symbols ,010307 mathematical physics - Abstract
Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin
- Published
- 2018
3. On period relations for automorphic 𝐿-functions I
- Author
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Fabian Januszewski
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Statistics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Period (music) ,Mathematics - Abstract
This paper is the first in a series of two dedicated to the study of period relations of the type L ( 1 2 + k , Π ) ∈ ( 2 π i ) d ⋅ k Ω ( − 1 ) k \bf Q ( Π ) , 1 2 + k critical , \begin{equation*} L\Big (\frac {1}{2}+k,\Pi \Big )\;\in \;(2\pi i)^{d\cdot k}\Omega _{(-1)^k}\textrm {\bf Q}(\Pi ),\quad \frac {1}{2}+k\;\text {critical}, \end{equation*} for certain automorphic representations Π \Pi of a reductive group G . G. In this paper we discuss the case G = G L ( n + 1 ) × G L ( n ) . G=\mathrm {GL}(n+1)\times \mathrm {GL}(n). The case G = G L ( 2 n ) G=\mathrm {GL}(2n) is discussed in part two. Our method is representation theoretic and relies on the author’s recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation Π \Pi under consideration. The new period relations we prove are in accordance with Deligne’s Conjecture on special values of L L -functions, and the author expects this method to apply to other cases as well.
- Published
- 2018
4. Functions of triples of noncommuting self-adjoint operators under perturbations of class $\boldsymbol {S}_p$
- Author
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V. V. Peller
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,16. Peace & justice ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Mathematics - Spectral Theory ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,010307 mathematical physics ,Complex Variables (math.CV) ,0101 mathematics ,Spectral Theory (math.SP) ,Self-adjoint operator ,Mathematics - Abstract
In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten--von Neumann norm $\boldsymbol S_p$, $1\le p\le\infty$, for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$. In other words, we prove that for $p\in[1,\infty]$, there is no constant $K>0$ such that the inequality \begin{align*} \|f(A_1,B_1,C_1)&-f(A_2,B_2,C_2)\|_{\boldsymbol S_p}\\[.1cm] &\le K\|f\|_{B_{\infty,1}^1} \max\big\{\|A_1-A_2\|_{\boldsymbol S_p},\|B_1-B_2\|_{\boldsymbol S_p},\|C_1-C_2\|_{\boldsymbol S_p}\big\} \end{align*} holds for an arbitrary function $f$ in $B_{\infty,1}^1({\Bbb R}^3)$ and for arbitrary finite rank self-adjoint operators $A_1,\,B_1,\,C_1,\,A_2,\,B_2$ and $C_2$., 14 pages. arXiv admin note: substantial text overlap with arXiv:1606.08961
- Published
- 2018
5. Wave front sets of reductive Lie group representations II
- Author
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Benjamin Harris
- Subjects
Wavefront ,Induced representation ,Applied Mathematics ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Wave front set ,Lie group ,(g,K)-module ,01 natural sciences ,Algebra ,Representation of a Lie group ,0103 physical sciences ,FOS: Mathematics ,Tempered representation ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups., Accepted to Transactions of the American Mathematical Society
- Published
- 2017
6. Gromov hyperbolicity, the Kobayashi metric, and $\mathbb {C}$-convex sets
- Author
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Andrew Zimmer
- Subjects
Pure mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Regular polygon ,Boundary (topology) ,Codimension ,01 natural sciences ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,Affine transformation ,Ball (mathematics) ,0101 mathematics ,Mathematics - Abstract
In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.
- Published
- 2017
7. Tame circle actions
- Author
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Jordan Watts and Susan Tolman
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Kähler manifold ,Fixed point ,01 natural sciences ,Mathematics - Symplectic Geometry ,0103 physical sciences ,Symplectic category ,Slice theorem ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,53D20 (Primary) 53D05, 53B35 (Secondary) ,0101 mathematics ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic manifold ,Symplectic geometry ,Mathematics - Abstract
In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases., Comment: 25 pages
- Published
- 2017
8. Isoperimetric properties of the mean curvature flow
- Author
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Or Hershkovits
- Subjects
Pure mathematics ,Mean curvature flow ,Mean curvature ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,Upper and lower bounds ,Geometric measure theory ,0103 physical sciences ,Hausdorff measure ,010307 mathematical physics ,0101 mathematics ,Isoperimetric inequality ,Constant (mathematics) ,Mathematics - Abstract
In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self-evident. The first is a genuine, 5-line proof, for the isoperimetric inequality for k k -cycles in R n \mathbb {R}^n , with a constant differing from the optimal constant by a factor of only k \sqrt {k} , as opposed to a factor of k k k^k produced by all of the other soft methods. The second is a 3-line proof of a lower bound for extinction for arbitrary co-dimensional, singular, mean curvature flows starting from cycles, generalizing the main result of Giga and Yama-uchi (1993). We then turn to use the above-mentioned relation to prove a bound on the parabolic Hausdorff measure of the space-time track of high co-dimensional, singular, mean curvature flow starting from a cycle, in terms of the mass of that cycle. This bound is also reminiscent of a Michael-Simon isoperimetric inequality. To prove it, we are led to study the geometric measure theory of Euclidean rectifiable sets in parabolic space and prove a co-area formula in that setting. This formula, the proof of which occupies most of this paper, may be of independent interest.
- Published
- 2017
9. Rectifiable measures, square functions involving densities, and the Cauchy transform
- Author
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Xavier Tolsa
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Cauchy distribution ,01 natural sciences ,Square (algebra) ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Radon measure ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,010307 mathematical physics ,28A75, 42B20 ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{ and }\quad \int_0^1\left|\frac{\mu(B(x,r))}{r} - \frac{\mu(B(x,2r))}{2r}\right|^2\,\frac{dr}r< \infty$$ for $\mu$-a.e. $x\in\mathbb R^d$, then $\mu$ is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set $E\subset\mathbb R^d$ with finite $1$-dimensional Hausdorff measure $H^1$ is rectifiable if and only $$\int_0^1\left|\frac{H^1(E\cap B(x,r))}{r} - \frac{H^1(E\cap B(x,2r))}{2r}\right|^2\,\frac{dr}r< \infty \quad\mbox{ for $H^1$-a.e. $x\in E$.}$$ The second result of the paper deals with the relationship between a similar square function in the complex plane and the Cauchy transform $C_\mu f(z) = \int \frac1{z-\xi}\,f(\xi)\,d\mu(\xi)$. Suppose that $\mu$ has linear growth, that is, $\mu(B(z,r))\leq c\,r$ for all $z\in\mathbb C$ and all $r>0$. It is proved that $C_\mu$ is bounded in $L^2(\mu)$ if and only if $$ \int_{z\in Q}\int_0^\infty\left|\frac{\mu(Q\cap B(z,r))}{r} - \frac{\mu(Q\cap B(z,2r))}{2r}\right|^2\,\frac{dr}r\,d\mu(z)\leq c\,\mu(Q) \quad\mbox{ for every square $Q\subset\mathbb C$.} $$, Comment: Minor corrections and adjustments
- Published
- 2017
10. Modular curvature for noncommutative two-tori
- Author
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Alain Connes and Henri Moscovici
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Curvature ,01 natural sciences ,Noncommutative geometry ,Constant curvature ,46L87, 58B34 ,Differential Geometry (math.DG) ,Differential geometry ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,010307 mathematical physics ,0101 mathematics ,Operator Algebras (math.OA) ,Spectral triple ,Conformal geometry ,Ricci curvature ,Mathematics ,Scalar curvature - Abstract
In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two functions in the modular operator corresponding to the non-tracial weight defined by the conformal factor, applied to expressions involving derivatives of the same factor. The first is a generating function for the Bernoulli numbers and is applied to the noncommutative Laplacian of the conformal factor, while the second is a two-variable function and is applied to a quadratic form in the first derivatives of the factor. Further outcomes of the paper include a variational proof of the Gauss-Bonnet theorem for noncommutative 2-tori, the modular analogue of Polyakov's conformal anomaly formula for regularized determinants of Laplacians, a conceptual understanding of the modular curvature as gradient of the Ray-Singer analytic torsion, and the proof using operator positivity that the scale invariant version of the latter assumes its extreme value only at the flat metric., Comment: 44 pages, 6 figures; minor changes. Two Mathematica notebooks detailing the computations added as ancillary files
- Published
- 2014
11. Transverse LS category for Riemannian foliations
- Author
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Steven Hurder and Dirk Töben
- Subjects
Pure mathematics ,Closed manifold ,Riemannian submersion ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Lie group ,01 natural sciences ,Upper and lower bounds ,symbols.namesake ,Compact group ,Mathematics::Category Theory ,0103 physical sciences ,symbols ,Foliation (geology) ,Lusternik–Schnirelmann category ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Category theory ,Mathematics - Abstract
We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation F on a closed manifold M. The essential transverse category cat e (M, F) is introduced in this paper, and we prove that cat e (M, F) is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category cat (M, F) is finite, and thus cat e (M, F) = cat(M, F) holds. A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of cat e (M, F) into a standard problem about O(q)-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated O(q)-manifold W, we have that cat e (M, F) = cat O(q) (Ŵ). Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. A generalization of the Lusternik-Schnirelmann theorem is derived: given a C 1 -function f: M → R which is constant along the leaves of a Riemannian foliation F, the essential transverse category cat e (M, F) is a lower bound for the number of critical leaf closures of f.
- Published
- 2009
12. A curvature-free 𝐿𝑜𝑔(2𝑘-1) theorem
- Author
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Florent Balacheff and Louis Merlin
- Subjects
Discrete mathematics ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,16. Peace & justice ,Curvature ,Mathematics::Geometric Topology ,01 natural sciences ,Volume entropy ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper presents a curvature-free version of the Log ( 2 k − 1 ) \text {Log}(2k-1) Theorem of Anderson, Canary, Culler, and Shalen [J. Differential Geometry 44 (1996), pp. 738–782]. It generalizes a result by Hou [J. Differential Geometry 57 (2001), no. 1, pp. 173–193] and its proof is rather straightforward once we know the work by Lim [Trans. Amer. Math. Soc. 360 (2008), no. 10, pp. 5089–5100] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the Collar Lemma in all dimensions.
- Published
- 2023
13. Blow-up phenomena for the Yamabe equation
- Author
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Simon Brendle
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Conjecture ,Applied Mathematics ,General Mathematics ,Prescribed scalar curvature problem ,Yamabe flow ,010102 general mathematics ,Mathematical analysis ,Yamabe problem ,01 natural sciences ,Mathematics - Analysis of PDEs ,Compact space ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Differential Geometry ,010307 mathematical physics ,Sectional curvature ,0101 mathematics ,Ricci curvature ,Analysis of PDEs (math.AP) ,Scalar curvature ,Mathematics - Abstract
Let (M,g) be a compact Riemannian manifold of dimension n \geq 3. The Compactness Conjecture asserts that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M,g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dimensions n \geq 52., Published paper
- Published
- 2007
14. Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman
- Author
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Tobias H. Colding and William P. Minicozzi
- Subjects
Pure mathematics ,Minimal surface ,Curve-shortening flow ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Prime decomposition ,Ricci flow ,Surface (topology) ,01 natural sciences ,Upper and lower bounds ,Zero (linguistics) ,Algebra ,Mathematics - Analysis of PDEs ,Flow (mathematics) ,AP, DG, GT ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this note we prove some bounds for the extinction time for the Ricci flow on certain 3-manifolds. Our interest in this comes from a question of Grisha Perelman asked to the first author at a dinner in New York City on April 25th of 2003. His question was ``what happens to the Ricci flow on the 3-sphere when one starts with an arbitrary metric? In particular does the flow become extinct in finite time?'' He then went on to say that one of the difficulties in answering this is that he knew of no good way of constructing minimal surfaces for such a metric in general. However, there is a natural way of constructing such surfaces and that comes from the min--max argument where the minimal of all maximal slices of sweep-outs is a minimal surface; see, for instance, [CD]. The idea is then to look at how the area of this min-max surface changes under the flow. Geometrically the area measures a kind of width of the 3-manifold and as we will see for certain 3-manifolds (those, like the 3-sphere, whose prime decomposition contains no aspherical factors) the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time. Moreover, we will discuss a possible lower bound for how fast the area becomes zero. Very recently Perelman posted a paper (see [Pe1]) answering his original question about finite extinction time. However, even after the appearance of his paper, then we still think that our slightly different approach may be of interest. In part because it is in some ways geometrically more natural, in part because it also indicates that lower bounds should hold, and in part because it avoids using the curve shortening flow that he simultaneously with the Ricci flow needed to invoke and thus our approach is in some respects technically easier., published version
- Published
- 2005
15. Approximating spectral invariants of Harper operators on graphs II
- Author
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Varghese Mathai, Thomas Schick, and Stuart Yates
- Subjects
Dirichlet problem ,Pure mathematics ,Discrete group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Amenable group ,58G25(Primary) 39A12 (Secondary) ,Mathematics::Spectral Theory ,Differential operator ,01 natural sciences ,Mathematics - Spectral Theory ,symbols.namesake ,Von Neumann algebra ,0103 physical sciences ,FOS: Mathematics ,symbols ,Neumann boundary condition ,010307 mathematical physics ,Boundary value problem ,0101 mathematics ,Spectral Theory (math.SP) ,Self-adjoint operator ,Mathematics - Abstract
We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation., LaTeX2e, 7 pages
- Published
- 2002
16. Problèmes de petites valeurs propres sur les surfaces de courbure moyenne constante
- Author
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Philippe Castillon, Institut de Mathématiques et de Modélisation de Montpellier (I3M), and Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)
- Subjects
Surface (mathematics) ,Laplace transform ,Applied Mathematics ,General Mathematics ,Hyperbolic space ,010102 general mathematics ,Mathematical analysis ,Geometry ,01 natural sciences ,Stability (probability) ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,0103 physical sciences ,Constant-mean-curvature surface ,Total curvature ,010307 mathematical physics ,0101 mathematics ,Finite set ,ComputingMilieux_MISCELLANEOUS ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper deals with the spectra of the Laplace and stability operators of a constant mean curvature surface in the hyperbolic space. In a preceding work, the author described the essential spectra of these operators, assuming that the surface is of finite total curvature. In this paper, we prove that these two operators have a finite number of eigenvalues below their essential spectra.
- Published
- 2001
17. On pro-2 identities of 2×2 linear groups
- Author
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Efim Zelmanov and David BenEzra
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics education ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let F ^ \hat {F} be a free pro- p p non-abelian group, and let Δ \Delta be a commutative Noetherian complete local ring with a maximal ideal I I such that c h a r ( Δ / I ) = p > 0 \mathrm {char}(\Delta /I)=p>0 . Zubkov [Sibirsk. Mat. Zh. 28 (1987), pp. 64–69] showed that when p ≠ 2 p\neq 2 , the pro- p p congruence subgroup \[ G L 2 1 ( Δ ) = ker ( G L 2 ( Δ ) ⟶ Δ → Δ / I G L 2 ( Δ / I ) ) GL_{2}^{1}(\Delta )=\ker (GL_{2}(\Delta )\overset {\Delta \to \Delta /I}{\longrightarrow }GL_{2}(\Delta /I)) \] admits a pro- p p identity, i.e., there exists an element 1 ≠ w ∈ F ^ 1\neq w\in \hat {F} that vanishes under any continuous homomorphism F ^ → G L 2 1 ( Δ ) \hat {F}\to GL_{2}^{1}(\Delta ) . In this paper we investigate the case p = 2 p=2 . The main result is that when c h a r ( Δ ) = 2 \mathrm {char}(\Delta )=2 , the pro- 2 2 group G L 2 1 ( Δ ) GL_{2}^{1}(\Delta ) admits a pro- 2 2 identity. This result was obtained by the use of trace identities that originate in PI-theory.
- Published
- 2021
18. Scattering for the 𝐿² supercritical point NLS
- Author
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Riccardo Adami, Reika Fukuizumi, and Justin Holmer
- Subjects
Scattering ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Schrödinger equation ,Nonlinear point interaction ,01 natural sciences ,symbols.namesake ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,symbols ,NLS ,Point (geometry) ,010307 mathematical physics ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematical physics ,Mathematics - Abstract
We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. “Point” means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ± ∞ t\to \pm \infty in the L 2 L^2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.
- Published
- 2020
19. Tame topology of arithmetic quotients and algebraicity of Hodge loci
- Author
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Benjamin Bakker, Bruno Klingler, and Jacob Tsimerman
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics - Logic ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Logic (math.LO) ,Algebraic Geometry (math.AG) ,Quotient ,Topology (chemistry) ,Mathematics - Abstract
In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. $2)$ We prove that the period map associated to any pure polarized variation of integral Hodge structures $\mathbb{V}$ on a smooth complex quasi-projective variety $S$ is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. $3)$ As a corollary of $2)$ and of Peterzil-Starchenko's o-minimal Chow theorem we recover that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$, a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable $SL_2$-orbit theorem of Cattani-Kaplan-Schmid., 23 pages, final version. arXiv admin note: substantial text overlap with arXiv:1803.09384
- Published
- 2020
20. The family of perfect ideals of codimension 3, of type 2 with 5 generators
- Author
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Witold Kraśkiewicz, Jerzy Weyman, Ela Celikbas, and Jai Laxmi
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Complete intersection ,Codimension ,Type (model theory) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,16. Peace & justice ,01 natural sciences ,13C40, 13D02, 13H10 ,0103 physical sciences ,FOS: Mathematics ,Key (cryptography) ,Multiplication ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
In this paper we define an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the Tor algebra. This family is likely to play a key role in classifying perfect ideals with five generators of type two., 11 pages
- Published
- 2020
21. A negative answer to a question of Bass
- Author
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Christian Haesemeyer, Guillermo Cortiñas, Mark E. Walker, and Charles A. Weibel
- Subjects
Rational number ,Matemáticas ,General Mathematics ,Field (mathematics) ,Transcendence degree ,Type (model theory) ,01 natural sciences ,Matemática Pura ,purl.org/becyt/ford/1 [https] ,Combinatorics ,Singularity ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics ,19D35 (Primary) 14F10, 14J17, 19D55 (Secondary) ,Applied Mathematics ,010102 general mathematics ,purl.org/becyt/ford/1.1 [https] ,K-Theory and Homology (math.KT) ,Algebraic number field ,Bass (sound) ,K-regularity ,Finite field ,Du Bois invariants ,Mathematics - K-Theory and Homology ,010307 mathematical physics ,Algorithm ,CIENCIAS NATURALES Y EXACTAS - Abstract
In this companion paper to arXiv:0802.1928 we provide an example of an isolated surface singularity $R$ over a number field such that $K_0(R) = K_0(R[t])$ but $K_0(R) \neq K_0(R[t_1,t_2])$. This answers, negatively, a question of Bass., Comment: The paper was previously part of arXiv:0802.1928
- Published
- 2011
22. A geometric formula for multiplicities of 𝐾-types of tempered representations
- Author
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Yanli Song, Shilin Yu, and Peter Hochs
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Multiplicity (mathematics) ,Tempered representation ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let G G be a connected, linear, real reductive Lie group with compact centre. Let K > G K>G be compact. Under a condition on K K , which holds in particular if K K is maximal compact, we give a geometric expression for the multiplicities of the K K -types of any tempered representation (in fact, any standard representation) π \pi of G G . This expression is in the spirit of Kirillov’s orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of π | K \pi |_K obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of SU ( p , 1 ) \textrm {SU}(p,1) , SO 0 ( p , 1 ) \textrm {SO}_0(p,1) , and SO 0 ( 2 , 2 ) \textrm {SO}_0(2,2) restrict multiplicity freely to maximal compact subgroups.
- Published
- 2019
23. Regular supercuspidal representations
- Author
-
Tasho Kaletha
- Subjects
Pure mathematics ,Property (philosophy) ,Root of unity ,Mathematics::Number Theory ,General Mathematics ,MathematicsofComputing_GENERAL ,01 natural sciences ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics ,Mathematics - Number Theory ,Group (mathematics) ,Applied Mathematics ,010102 general mathematics ,Expression (computer science) ,16. Peace & justice ,Transfer (group theory) ,Character (mathematics) ,Maximal torus ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups., Comment: v2: Removed assumption that ground field has characteristic zero from most of paper. Added results towards Hypothesis C(G) of Hakim-Murnaghan. Simplified definition of regular Yu-datum. Simplified character computations in depth-zero case. Other minor improvements. 96pp
- Published
- 2019
24. Asymptotic gcd and divisible sequences for entire functions
- Author
-
Julie Wang and Ji Guo
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Entire function ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let f f and g g be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of f n − 1 f^n-1 and g n − 1 g^n-1 for sufficiently large n n . We then apply this estimate to study divisible sequences in the sense that f n − 1 f^n-1 is divisible by g n − 1 g^n-1 for infinitely many n n . For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying a theorem from a paper by Hussein and Ru and explicitly computing the constants involved for a blowup of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 .
- Published
- 2019
25. Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I
- Author
-
William Elbæk Petersen and Jørgen Ellegaard Andersen
- Subjects
Geometric quantization ,Pure mathematics ,Topological quantum field theory ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Torus ,Fixed point ,Mathematics::Geometric Topology ,01 natural sciences ,Mapping class group ,Moduli space ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Asymptotic expansion ,Mathematics - Abstract
In this paper we engage in a general study of the asymptoticexpansion of the Witten–Reshetikhin–Turaev invariants of mapping tori ofsurface mapping class group elements. We use the geometric constructionof the Witten–Reshetikhin–Turaev topological quantum field theory via thegeometric quantization of moduli spaces of flat connections on surfaces. Weidentify assumptions on the mapping class group elements that allow us toprovide a full asymptotic expansion. In particular, we show that our resultsapply to all pseudo-Anosov mapping classes on a punctured torus and show byexample that our assumptions on the mapping class group elements are strictlyweaker than hitherto successfully considered assumptions in this context. Theproof of our main theorem relies on our new results regarding asymptoticexpansions of oscillatory integrals, which allows us to go significantly beyondthe standard transversely cut out assumption on the fixed point set. Thismakes use of the Picard–Lefschetz theory for Laplace integrals.
- Published
- 2018
26. Potentially 𝐺𝐿₂-type Galois representations associated to noncongruence modular forms
- Author
-
Tong Liu, Ling Long, and Wen-Ching Winnie Li
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Modular form ,010307 mathematical physics ,0101 mathematics ,Type (model theory) ,Galois module ,01 natural sciences ,Mathematics - Abstract
In this paper, we consider representations of the absolute Galois group Gal ( Q ¯ / Q ) \text {Gal}(\overline {\mathbb Q}/\mathbb Q) attached to modular forms for noncongruence subgroups of SL 2 ( Z ) \text {SL}_2(\mathbb Z) . When the underlying modular curves have a model over Q \mathbb {Q} , these representations are constructed by Scholl in [Invent. Math. 99 (1985), pp. 49–77] and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over Q \mathbb Q . As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments on automorphy lifting theorem, we obtain various automorphy and potential automorphy results for potentially G L 2 \mathrm {GL}_2 -type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. In particular, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions.
- Published
- 2018
27. On Calabi’s extremal metric and properness
- Author
-
Weiyong He
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Modulo ,010102 general mathematics ,Scalar (mathematics) ,01 natural sciences ,Manifold ,Flow (mathematics) ,0103 physical sciences ,Metric (mathematics) ,010307 mathematical physics ,Identity component ,0101 mathematics ,Invariant (mathematics) ,Constant (mathematics) ,Mathematics - Abstract
In this paper we extend a recent breakthrough of Chen and Cheng on the existence of a constant scalar Kähler metric on a compact Kähler manifold to Calabi’s extremal metric. There are no new a priori estimates needed, but rather there are necessary modifications adapted to the extremal case. We prove that there exists an extremal metric with extremal vector V V if and only if the modified Mabuchi energy is proper, modulo the action of the subgroup in the identity component of the automorphism group which commutes with the flow of V V . We introduce two essentially equivalent notions, called reductive properness and reduced properness. We observe that one can test reductive properness/reduced properness only for invariant metrics. We prove that existence of an extremal metric is equivalent to reductive properness/reduced properness of the modified Mabuchi energy.
- Published
- 2018
28. Discontinuous homomorphisms, selectors, and automorphisms of the complex field
- Author
-
Jindrich Zapletal and Paul Larson
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Ultrafilter ,16. Peace & justice ,Automorphism ,01 natural sciences ,Mathematics::Logic ,0103 physical sciences ,Equivalence relation ,Homomorphism ,Axiom of choice ,010307 mathematical physics ,Set theory ,0101 mathematics ,Mathematics ,Additive group ,Real number - Abstract
We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$. This shows that a nonprincipal ultrafilter on the integers is not sufficient to construct a discontinuous automorphism of the complex field, confirming a conjecture of Simon Thomas. This is an improved version of our paper in the Proceedings of the American Mathematical Society, which used a weak version of the Axiom of Choice for the same result.
- Published
- 2018
29. Vector-valued modular forms on a three-dimensional ball
- Author
-
Riccardo Salvati Manni and Eberhard Freitag
- Subjects
Discrete mathematics ,Conjecture ,business.industry ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Modular form ,Modular design ,01 natural sciences ,Picard modular group ,0103 physical sciences ,vector valued modular forms ,010307 mathematical physics ,Ball (mathematics) ,0101 mathematics ,business ,Mathematics ,Siegel modular form - Abstract
In this paper we give a structure theorem for the module of vector valued modular forms in the case of a three dimensional ball with the action of the Picard modular group Γ 3 [ − 3 ] \Gamma _3 [\sqrt {-3}] . The corresponding modular variety of dimension 3 3 is a copy of the Segre cubic.
- Published
- 2018
30. Admissible sequences of positive operators
- Author
-
David R. Larson and Victor Kaftal
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Schur–Horn theorem ,Mathematics - Abstract
A scalar sequence ξ \xi is said to be admissible for a positive operator A A if A = ∑ ξ j P j A= \sum \xi _jP_j for some rank-one projections P j P_j , or, equivalently, if diag ξ \xi is the diagonal of V A V ∗ VAV^* for some partial isometry V V having as domain the closure of the range of A A . The main result of this paper is that if A A is the sum of infinitely many projections (converging in the strong operator topology) and ξ \xi is a nonsummable sequence in [ 0 , 1 ] [0,1] that satisfies the Kadison condition that requires that either ∑ { ξ i ∣ ξ i ≤ 1 2 } + ∑ { ( 1 − ξ i ) ∣ ξ i > 1 2 } = ∞ \sum \{\xi _i \mid \xi _i\le \frac {1}{2}\}+ \sum \{(1-\xi _i) \mid \xi _i> \frac {1}{2}\} = \infty or the difference ∑ { ξ i ∣ ξ i ≤ 1 2 } − ∑ { ( 1 − ξ i ) ∣ ξ i > 1 2 } \sum \{\xi _i \mid \xi _i\le \frac {1}{2}\}- \sum \{(1-\xi _i) \mid \xi _i> \frac {1}{2}\} is an integer, then ξ \xi is admissible for A A . This result extends Kadison’s carpenter’s theorem and provides an independent proof of it.
- Published
- 2018
31. Relative Manin–Mumford in additive extensions
- Author
-
Harry Schmidt
- Subjects
Pure mathematics ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In recent papers Masser and Zannier have proved various results of “relative Manin–Mumford” type for various families of abelian varieties, some with field of definition restricted to the algebraic numbers. Typically these imply the finiteness of the set of torsion points on a curve in the family. After Bertrand, Masser, and Zannier discovered some surprising counterexamples for multiplicative extensions of elliptic families, the three authors together with Pillay settled completely the situation for this case over the algebraic numbers. Here we treat the last remaining case of surfaces, that of additive extensions of elliptic families, and even over the field of all complex numbers. In particular analogous counterexamples do not exist. There are finiteness consequences for Pell’s equation over polynomial rings and integration in elementary terms. Our work can be made effective (as opposed to most of that preceding), mainly because we use counting results only for analytic curves.
- Published
- 2018
32. Rectifiability of the singular set of multiple-valued energy minimizing harmonic maps
- Author
-
Jonas Hirsch, Daniele Valtorta, Salvatore Stuvard, Hirsch, J, Stuvard, S, and Valtorta, D
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Reifenberg theorem ,General Mathematics ,Boundary (topology) ,Quantitative stratification ,01 natural sciences ,Set (abstract data type) ,Mathematics - Analysis of PDEs ,0103 physical sciences ,Minkowski space ,Simply connected space ,FOS: Mathematics ,Rectifiability ,Harmonic map ,0101 mathematics ,Q-valued function ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Manifold ,49Q20, 58E20 ,Differential Geometry (math.DG) ,Singular set ,010307 mathematical physics ,Energy (signal processing) ,Analysis of PDEs (math.AP) - Abstract
In this paper we study the singular set of Dirichlet-minimizing $Q$-valued maps from $\mathbb{R}^m$ into a smooth compact manifold $\mathcal{N}$ without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform Minkowski bounds. Moreover, as opposed to the single valued case, we prove that the target $\mathcal{N}$ being non-positively curved but not simply connected does not imply continuity of the map., Comment: 43 pages
- Published
- 2018
33. Generalizing the MVW involution, and the contragredient
- Author
-
Dipendra Prasad
- Subjects
Classical group ,Involution (mathematics) ,Conjecture ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Automorphism ,01 natural sciences ,Admissible representation ,Combinatorics ,Algebraic group ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,Representation Theory (math.RT) ,11F70, 22E55 ,0101 mathematics ,Local field ,Mathematics - Representation Theory ,Mathematics - Abstract
For certain quasi-split reductive groups $G$ over a general field $F$, we construct an automorphism $\iota_G$ of $G$ over $F$, well-defined as an element of ${\rm Aut}(G)(F)/jG(F)$ where $j:G(F) \rightarrow {\rm Aut}(G)(F)$ is the inner-conjugation action of $G(F)$ on $G$. The automorphism $\iota_G$ generalizes (although only for quasi-split groups) an involution due to Moeglin-Vigneras-Waldspurger in [MVW] for classical groups which takes any irreducible admissible representation $\pi$ of $G(F)$ for $G$ a classical group and $F$ a local field, to its contragredient $\pi^\vee$. The paper also formulates a conjecture on the contragredient of an irreducible admissible representation of $G(F)$ for $G$ a reductive algebraic group over a local field $F$ in terms of the (enhanced) Langlands parameter of the representation., Comment: arXiv admin note: substantial text overlap with arXiv:1512.04347
- Published
- 2018
34. Quantitative stratification for some free-boundary problems
- Author
-
Max Engelstein and Nick Edelen
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Stratification (mathematics) ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Metric Geometry ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we prove the rectifiability of and measure bounds on the singular set of the free boundary for minimizers of a functional first considered by Alt-Caffarelli. Our main tools are the Quantitative Stratification and Rectifiable-Reifenberg framework of Naber-Valtorta, which allow us to do a type of "effective dimension-reduction." The arguments are sufficiently robust that they apply to a broad class of related free boundary problems as well., Comment: This version has minor corrections and additional references
- Published
- 2018
35. Bounding Harish-Chandra series
- Author
-
Olivier Dudas and Gunter Malle
- Subjects
Discrete mathematics ,Pure mathematics ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Group Theory (math.GR) ,Unipotent ,Reductive group ,01 natural sciences ,Matrix (mathematics) ,Character (mathematics) ,Bounding overwatch ,0103 physical sciences ,FOS: Mathematics ,Irreducibility ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,20C33, 20C08 ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Simple module ,Mathematics - Representation Theory ,Mathematics - Abstract
We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees., Comment: arXiv admin note: text overlap with arXiv:1611.07373
- Published
- 2018
36. Undecidability of equations in free Lie algebras
- Author
-
Olga Kharlampovich and Alexei Myasnikov
- Subjects
Pure mathematics ,Rank (linear algebra) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Finite system ,Zero (complex analysis) ,Field (mathematics) ,Mathematics - Logic ,Mathematics - Rings and Algebras ,01 natural sciences ,Ring of integers ,Rings and Algebras (math.RA) ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,010307 mathematical physics ,03C60 ,0101 mathematics ,Logic (math.LO) ,Mathematics - Abstract
In this paper we prove undecidability of finite systems of equations in free Lie algebras of rank at least three over an arbitrary field. We show that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero., Comment: arXiv admin note: text overlap with arXiv:1606.03617
- Published
- 2018
37. Difference of modular functions and their CM value factorization
- Author
-
Tonghai Yang and Hongbo Yin
- Subjects
Conjecture ,Mathematics - Number Theory ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,14G35, 14G40, 11G18, 11F27 ,01 natural sciences ,Modular curve ,Omega ,Moduli ,Combinatorics ,Factorization ,Genus (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In this paper, we use Borcherds lifting and the big CM value formula of Bruinier, Kudla, and Yang to give an explicit factorization formula for the norm of $\Psi(\frac{d_1+\sqrt{d_1}}2) -\Psi(\frac{d_2+\sqrt{d_2}}2)$, where $\Psi$ is the $j$-invariant or the Weber invariant $\omega_2$. The $j$-invariant case gives another proof of the well-known Gross-Zagier factorization formula of singular moduli, while the Weber invariant case gives a proof of the Yui-Zagier conjecture for $\omega_2$. The method used here could be extended to deal with other modular functions on a genus zero modular curve., Comment: accepted to appear in Trans. AMS
- Published
- 2018
38. Solving \overline{∂} with prescribed support on Hartogs triangles in ℂ² and ℂℙ²
- Author
-
Christine Laurent-Thiébaut and Mei-Chi Shaw
- Subjects
Algebra ,Overline ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we consider the problem of solving the Cauchy–Riemann equation with prescribed support in a domain of a complex manifold for forms or currents. We are especially interested in the case when the domain is a Hartogs triangle in C 2 \mathbb {C}^2 or C P 2 \mathbb {C}\mathbb {P}^2 . In particular, we show that the strong L 2 L^2 Dolbeault cohomology group on the Hartogs triangle in C P 2 \mathbb {C}\mathbb {P}^2 is infinitely dimensional.
- Published
- 2018
39. A restricted Magnus property for profinite surface groups
- Author
-
Pavel Zalesskii and Marco Boggi
- Subjects
Mathematics::Functional Analysis ,Class (set theory) ,Property (philosophy) ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,High Energy Physics::Phenomenology ,010102 general mathematics ,Mathematics::General Topology ,Group Theory (math.GR) ,Surface (topology) ,01 natural sciences ,Combinatorics ,Mathematics::Group Theory ,Simple (abstract algebra) ,0103 physical sciences ,Free group ,FOS: Mathematics ,010307 mathematical physics ,Finitely-generated abelian group ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
Magnus proved that, given two elements $x$ and $y$ of a finitely generated free group $F$ with equal normal closures $\langle x\rangle^F=\langle y\rangle^F$, then $x$ is conjugated either to $y$ or $y^{-1}$. More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for ${\mathscr S}$ a class of finite groups, we prove that, if $x$ and $y$ are \emph{algebraically simple} elements of the pro-${\mathscr S}$ completion $\hat{\Pi}^{\mathscr S}$ of an orientable surface group $\Pi$, such that, for all $n\in{\mathbb N}$, there holds $\langle x^n\rangle^{\hat{\Pi}^{\mathscr S}}=\langle y^n\rangle^{\hat{\Pi}^{\mathscr S}}$, then $x$ is conjugated to $y^s$ for some $s\in(\hat{\mathbb Z}^{\mathscr S})^\ast$. As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions. The most important application of the theory above is a generalization of the description of centralizers of profinite Dehn twists to profinite Dehn multitwists., Comment: 27 pages. Final version, to appear on Transactions of the American Mathematical Society
- Published
- 2018
40. Contractive inequalities for Bergman spaces and multiplicative Hankel forms
- Author
-
Ole Fredrik Brevig, Joaquim Ortega-Cerdà, Karl-Mikael Perfekt, Antti Haimi, Frédéric Bayart, and Universitat de Barcelona
- Subjects
Pure mathematics ,Inequality ,Function algebras ,Funcions de variables complexes ,General Mathematics ,media_common.quotation_subject ,Àlgebres de funcions ,Polydisc ,Type (model theory) ,Teoria d'operadors ,01 natural sciences ,Functions of complex variables ,symbols.namesake ,0103 physical sciences ,Analytic functions ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Dirichlet series ,media_common ,Mathematics ,Mathematics::Functional Analysis ,Mathematics - Number Theory ,Mathematics::Complex Variables ,Applied Mathematics ,010102 general mathematics ,Multiplicative function ,Operator theory ,Hardy space ,Linear operators ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Funcions analítiques ,symbols ,010307 mathematical physics ,Isoperimetric inequality ,Operadors lineals ,Unit (ring theory) - Abstract
We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two type of forms which does not exist in lower dimensions. Finally, we produce some counter-examples concerning Carleson measures on the infinite polydisc., Comment: This paper has been accepted for publication in Transactions of the AMS
- Published
- 2018
41. Tensor product of cyclic 𝐴_{∞}-algebras and their Kontsevich classes
- Author
-
Lino Amorim and Junwu Tu
- Subjects
Pure mathematics ,Tensor product ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,MathematicsofComputing_GENERAL ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Given two cyclic A ∞ A_\infty -algebras A A and B B , in this paper we prove that there exists a cyclic A ∞ A_\infty -algebra structure on their tensor product A ⊗ B A\otimes B which is unique up to a cyclic A ∞ A_\infty -quasi-isomorphism. Furthermore, the Kontsevich class of A ⊗ B A\otimes B is equal to the cup product of the Kontsevich classes of A A and B B on the moduli space of curves.
- Published
- 2018
42. Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type
- Author
-
The Bui, Xuan Thinh Duong, and Fu Ken Ly
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hardy space ,Type (model theory) ,01 natural sciences ,Atomic decomposition ,symbols.namesake ,Homogeneous ,0103 physical sciences ,symbols ,Maximal function ,Critical function ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let X X be a space of homogeneous type and let L \mathfrak {L} be a nonnegative self-adjoint operator on L 2 ( X ) L^2(X) enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove (local) nontangential and radial maximal function characterizations for the local Hardy spaces associated to L \mathfrak {L} . This gives the maximal function characterization for local Hardy spaces in the sense of Coifman and Weiss provided that L \mathfrak {L} satisfies certain extra conditions. Secondly we introduce local Hardy spaces associated with a critical function ρ \rho which are motivated by the theory of Hardy spaces related to Schrödinger operators and of which include the local Hardy spaces of Coifman and Weiss as a special case. We then prove that these local Hardy spaces can be characterized by (local) nontangential and radial maximal functions related to L \mathfrak {L} and ρ \rho , and by global maximal functions associated to ‘perturbations’ of L \mathfrak {L} . We apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Schrödinger operators on manifolds to Schrödinger operators on connected and simply connected nilpotent Lie groups.
- Published
- 2018
43. A gap theorem for the complex geometry of convex domains
- Author
-
Andrew Zimmer
- Subjects
Unit sphere ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Boundary (topology) ,01 natural sciences ,Domain (mathematical analysis) ,Compact space ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,Sectional curvature ,0101 mathematics ,Bergman metric ,Mathematics - Abstract
In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the following: for any dimension there exists some ϵ > 0 \epsilon > 0 so that if the squeezing function on a smoothly bounded convex domain is greater than 1 − ϵ 1-\epsilon outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another consequence is the following: for any dimension d d there exists some ϵ > 0 \epsilon > 0 so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within ϵ \epsilon of − 4 / ( d + 1 ) -4/(d+1) outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to − 4 / ( d + 1 ) -4/(d+1) on the boundary).
- Published
- 2018
44. Transformation properties for Dyson’s rank function
- Author
-
Frank G. Garvan
- Subjects
Conjecture ,Rank (linear algebra) ,Root of unity ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Modular form ,Generating function ,01 natural sciences ,Ramanujan's sum ,Combinatorics ,Ramanujan theta function ,symbols.namesake ,Identity (mathematics) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(\zeta,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $\zeta$ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon and McIntosh, and motivated by Dyson's question, Bringmann, Ono and Rhoades studied transformation properties of $R(\zeta,q)$. In this paper we strengthen and extend the results of Bringmann, Rhoades and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo $5$ from the Lost Notebook has an analogue for all primes greater than $3$. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod $7$.
- Published
- 2018
45. Langlands correspondence for isocrystals and the existence of crystalline companions for curves
- Author
-
Tomoyuki Abe
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Diagonal ,MathematicsofComputing_GENERAL ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Morphism ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper, we show the Langlands correspondence for isocrystals on curves, which asserts the existence of crystalline companions in the case of curves. For the proof we generalize the theory of arithmetic D \mathscr {D} -modules to algebraic stacks whose diagonal morphisms are finite. Finally, combining with methods of Deligne and Drinfeld, we show the existence of an “ ℓ \ell -adic companion” for any isocrystal on a smooth scheme of any dimension under the assumption of a Bertini-type conjecture.
- Published
- 2018
46. Exotic elliptic algebras
- Author
-
Alexandru Chirvasitu and S. Paul Smith
- Subjects
Pure mathematics ,General method ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Order (ring theory) ,Hopf algebra ,01 natural sciences ,Comodule ,Mathematics::Quantum Algebra ,0103 physical sciences ,Torsor ,010307 mathematical physics ,0101 mathematics ,Special case ,Descent (mathematics) ,Mathematics - Abstract
This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "exotic elliptic algebra". We show that the twisted algebra has many of the good properties that the Sklyanin algebra has, and that it has some new properties that make it quite unusual by comparison.
- Published
- 2018
47. Relative Morita equivalence of Cuntz–Krieger algebras and flow equivalence of topological Markov shifts
- Author
-
Kengo Matsumoto
- Subjects
Markov chain ,Mathematics::Operator Algebras ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Picard group ,Dynamical Systems (math.DS) ,Topology ,01 natural sciences ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Morita equivalence ,Operator Algebras (math.OA) ,Mathematics - Abstract
In this paper, we will introduce notions of relative version of imprimitivity bimodules and relative version of strong Morita equivalence for pairs of $C^*$-algebras $(\mathcal{A}, \mathcal{D})$ such that $\mathcal{D}$ is a $C^*$-subalgebra of $\mathcal{A}$ with certain conditions. We will then prove that two pairs $(\mathcal{A}_1, \mathcal{D}_1)$ and $(\mathcal{A}_2, \mathcal{D}_2)$ are relatively Morita equivalent if and only if their relative stabilizations are isomorphic. In particularly, for two pairs $(\mathcal{O}_A, \mathcal{D}_A)$ and $(\mathcal{O}_B, \mathcal{D}_B)$ of Cuntz--Krieger algebras with their canonical masas, they are relatively Morita equivalent if and only if their underlying two-sided topological Markov shifts $(\bar{X}_A,\bar{\sigma}_A)$ and $(\bar{X}_B,\bar{\sigma}_B)$ are flow equivalent. We also introduce a relative version of the Picard group ${\operatorname{Pic}}(\mathcal{A}, \mathcal{D})$ for the pair $(\mathcal{A}, \mathcal{D})$ of $C^*$-algebras and study them for the Cuntz--Krieger pair $(\mathcal{O}_A, \mathcal{D}_A)$., Comment: 38 pages
- Published
- 2018
48. The degenerate Eisenstein series attached to the Heisenberg parabolic subgroups of quasi-split forms of 𝑆𝑝𝑖𝑛₈
- Author
-
Avner Segal
- Subjects
Pure mathematics ,symbols.namesake ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Degenerate energy levels ,Eisenstein series ,symbols ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In [J. Inst. Math. Jussieu 14 (2015), 149–184] and [Int. Math. Res. Not. IMRN 7 (2017), 2014–2099] a family of Rankin-Selberg integrals was shown to represent the twisted standard L \mathcal {L} -function L ( s , π , χ , s t ) \mathcal {L}(s,\pi ,\chi ,\mathfrak {st}) of a cuspidal representation π \pi of the exceptional group of type G 2 G_2 . These integral representations bind the analytic behavior of this L \mathcal {L} -function with that of a family of degenerate Eisenstein series for quasi-split forms of S p i n 8 Spin_8 associated to an induction from a character on the Heisenberg parabolic subgroup. This paper is divided into two parts. In Part 1 we study the poles of these degenerate Eisenstein series in the right half-plane R e ( s ) > 0 \mathfrak {Re}(s)>0 . In Part 2 we use the results of Part 1 to prove the conjecture, made by J. Hundley and D. Ginzburg in [Israel J. Math. 207 (2015), 835–879], for stable poles and also to give a criterion for π \pi to be a CAP representation with respect to the Borel subgroup of G 2 G_2 in terms of the analytic behavior of L ( s , π , χ , s t ) \mathcal {L}(s,\pi ,\chi ,\mathfrak {st}) at s = 3 2 s=\frac {3}{2} .
- Published
- 2018
49. 𝔸¹-equivalence of zero cycles on surfaces
- Author
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Yi Zhu
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Zero (complex analysis) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Equivalence (measure theory) ,Mathematics - Abstract
In this paper, we study A 1 \mathbb {A}^1 -equivalence classes of zero cycles on open algebraic surfaces. We prove the logarithmic version of Mumford’s theorem on zero cycles. We also prove that the log Bloch conjecture holds for surfaces with log Kodaira dimension − ∞ -\infty .
- Published
- 2018
50. An application of non-positively curved cubings of alternating links
- Author
-
Makoto Sakuma and Yoshiyuki Yokota
- Subjects
Ideal (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Volume conjecture ,Geometric Topology (math.GT) ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Arc (geometry) ,Mathematics - Geometric Topology ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Link (knot theory) ,Mathematics - Abstract
By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges is homotopic relative its endpoints to a peripheral arc. This guarantees that the hyperbolicity equations for those triangulations for hyperbolic alternating links have solutions corresponding to the complete hyperbolic structures. Since the ideal triangulations considered in this paper are often used in the study of the volume conjecture, this result has a potential application to the volume conjecture.
- Published
- 2018
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