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The mathematical work of H. Blaine Lawson, Jr. Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Robert Bryant, Jeff Cheeger, Paulo Lima-Filho, Jonathan Rosenberg, Brian White
In this article, we celebrate the 80th birthday and remarkable career of H. Blaine Lawson, Jr. For more than half a century, Lawson has been a leading figure in mathematics. His work, a masterful combination of differential geometry, topology, algebraic geometry and analysis, has been enormously influential. He has made numerous fundamental contributions to diverse areas involving these subjects. He
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A remark on calibrations and Lie groups Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Nigel Hitchin
We use the notion of the principal three-dimensional subgroup of a simple Lie group to identify certain special subspaces of the Lie algebra and address the question of whether these are calibrated for invariant forms on the group.
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A note on the primitive cohomology lattice of a projective surface Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Chris Peters
The isometry class of the intersection form of a compact complex surface can be easily determined from complex-analytic invariants. For projective surfaces the primitive lattice is another naturally occurring lattice. The goal of this note is to show that it can be determined from the intersection lattice and the self-intersection of a primitive ample class, at least when the primitive lattice is indefinite
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Hyperbolic domains in real Euclidean spaces Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Barbara Drinovec Drnovšek, Franc Forstnerič
The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb{R}^n$, $n \geq 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi’s pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $\mathbb{R}^n$, this minimal metric coincides with the classical Beltrami–Cayley–Klein
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Curvature in the balance: the Weyl functional and scalar curvature of $4$-manifolds Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Claude LeBrun
The infimum of the Weyl functional is shown to be surprisingly small on many compact $4$-manifolds that admit positive-scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kähler $4$-manifolds.
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$RO(C_2)$-graded equivariant cohomology and classical Steenrod squares Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Pedro F. dos Santos, Paulo Lima-Filho
We investigate the restriction to fixed-points and change of coefficient functors in $RO(C_2)$-graded equivariant cohomology, with applications to the equivariant cohomology of spaces with a trivial $C_2$-action for $\underline{\mathbb{Z}}$ and $\underline{\mathbb{F}_2}$ coefficients. To this end, we study the nonequivariant spectra representing these theories and the corresponding functors. In particular
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The generality of closed $\mathrm{G}_2$ solitons Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Robert L. Bryant
The local generality of the space of solitons for the Laplacian flow of closed $\mathrm{G}_2$-structures is analyzed, and it is shown that the germs of such structures depend, up to diffeomorphism, on $16$ functions of $6$ variables (in the sense of É. Cartan). The method is to construct a natural exterior differential system whose integral manifolds describe such solitons and to show that it is involutive
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The $L^\infty$ estimates for parabolic complex Monge–Ampère and Hessian equations Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Xiuxiong Chen, Jingrui Cheng
In this paper, we consider a version of parabolic complex Monge–Ampère equations, and use a PDE approach similar to Phong et al. to establish $L^\infty$ and Hölder estimates. We also generalize the $L^\infty$ estimates to parabolic Hessian equations.
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Any oriented non-closed connected $4$-manifold can be spread holomorphically over the complex projective plane minus a point Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Dennis Sullivan
$\def\spinc{\operatorname{spin}^\mathrm{c}}$We give a 1965 era proof of the title assuming $M$ is $\spinc$. The fact that any oriented four manifold is $\spinc$ is a challenging result from 1995 whose interesting argument by Teichner–Vogt is analyzed and used in the appendix to show an analogous integral lift result about the top Wu class in $\dim 4k$. This will be used in future work to study related
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Positive scalar curvature on manifolds with boundary and their doubles Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Jonathan Rosenberg, Shmuel Weinberger
This paper is about positive scalar curvature on a compact manifold $X$ with non-empty boundary $\partial X$. In some cases, we completely answer the question of when $X$ has a positive scalar curvature metric which is a product metric near $\partial X$, or when $X$ has a positive scalar curvature metric with positive mean curvature on the boundary, and more generally, we study the relationship between
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Covering complexity, scalar curvature, and quantitative $K$-theory Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Hao Guo, Guoliang Yu
We establish a relationship between a certain notion of covering complexity of a Riemannian spin manifold and positive lower bounds on its scalar curvature. This makes use of a pairing between quantitative operator $K$-theory and Lipschitz topological $K$-theory, combined with an earlier vanishing theorem for the quantitative higher index.
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Interplay between nonlinear potential theory and fully nonlinear elliptic PDEs Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 F. Reese Harvey, Kevin R. Payne
We discuss one of the many topics that illustrate the interaction of Blaine Lawson’s deep geometric and analytic insights. The first author is extremely grateful to have had the pleasure of collaborating with Blaine over many enjoyable years. The topic to be discussed concerns the fruitful interplay between nonlinear potential theory; that is, the study of subharmonics with respect to a general constraint
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Global behavior at infinity of period mappings defined on algebraic surface Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Mark Green, Phillip Griffiths
The global behavior of period mappings defined on generally non-complete algebraic varieties $B$ as well as their local behavior around points in the boundary $Z = \overline{B}\setminus B$ of smooth completions of $B$ have been extensively investigated. In this paper we shall study the global behavior of period mappings in neighborhoods of the entire boundary $Z$ when $\dim B = 2$. One method will
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Spectral measures for $Sp(2)$ Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 David Evans, Mathew Pugh
Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group $SO(5)$ and its double cover the compact connected, simply-connected rank two Lie group $Sp(2)$, including the McKay graphs for the irreducible representations of $Sp(2)$ and $SO(5)$ and their maximal tori, and fusion modules
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The descendant colored Jones polynomials Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Stavros Garoufalidis, Rinat Kashaev
We discuss two realizations of the colored Jones polynomials of a knot, one appearing in an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another formulated in terms of a sequence of elements of the Habiro ring appearing in recent work of D. Zagier and the first author on the Refined Quantum Modularity Conjecture
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Graded extensions of generalized Haagerup categories Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Pinhas Grossman, Masaki Izumi, Noah Snyder
$\def\Z{\mathbb{Z}}$We classify certain $\Z_2$-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: $\Z_2$-graded extensions of $\Z_{2n}$ generalized Haagerup categories for all $n \leq 5$; $\Z_2 \times \Z_2$-graded extensions of the Asaeda-Haagerup
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Planar diagrammatics of self-adjoint functors and recognizable tree series Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Mikhail Khovanov, Robert Laugwitz
A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the ground field, then one assigns an element of that field to a diagram of nested circles. We focus on the self-adjoint functor case of this construction and study the reverse
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A note on continuous entropy Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Roberto Longo, Edward Witten
Von Neumann entropy has a natural extension to the case of an arbitrary semifinite von Neumann algebra, as was considered by I. E. Segal. We relate this entropy to the relative entropy and show that the entropy increase for an inclusion of von Neumann factors is bounded by the logarithm of the Jones index. The bound is optimal if the factors are infinite dimensional.
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On the paving size of a subfactor Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Sora Popin
Given an inclusion of $\mathrm{II}_1$ factors $N \subset M$ with finite Jones index, $[M:N] \lt \infty$, we prove that for any $F \subset M$ finite and $\varepsilon \gt 0$, there exists a partition of $1$ with $r \leq \lceil 16 \varepsilon^{-2} \rceil \cdot {\lceil 4 [M:N] \varepsilon}^{-2} \rceil$ projections $p_1, \dotsc , p_r \in N$ such that ${\lVert \sum^r_{i=1} p_i xp_i - E_{N^\prime \cap M}
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Spin Calogero-Moser periodic chains and two dimensional Yang-Mills theory with corners Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Nicolai Reshetikhin
The Quantum Calogero–Moser spin system is a superintegrable system with the spectrum of commuting Hamiltonians that can be described entirely in terms of representation theory of the corresponding simple Lie group. Here we describe its natural generalization known as quantum Calogero–Moser spin chain or $N$-spin Calogero–Moser system. In the first part of this paper we show that quantum Calogero–Moser
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On module categories related to $Sp(N-1) \subset Sl(N)$ Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Hans Wenzl
$\def\End{\operatorname{End}}$$\def\Rep{\operatorname{Rep}}$$\def\sl{\mathfrak{sl}}$Let $V = \mathbb{C}^N$ with $N$ odd.We construct a $q$-deformation of $\End_{Sp(N-1)}(V^{\otimes n})$ which contains $\End_{U_q \sl_N} (V^{\otimes n})$. It is a quotient of an abstract two-variable algebra which is defined by adding one more generator to the generators of the Hecke algebras $H_n$. These results suggest
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On three homework problems from Vaughan Jones Pure Appl. Math. Q. (IF 0.7) Pub Date : 2024-01-30 Feng Xu
This paper contains my previously unpublished work on three problems proposed by Vaughan Jones.
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Horn conditions for quiver subrepresentations and the moment map Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Velleda Baldoni, Michèle Vergne, Michael Walter
We give inductive conditions that characterize the Schubert positions of subrepresentations of a general quiver representation. Our results generalize Belkale’s criterion for the intersection of Schubert varieties in Grassmannians and refine Schofield’s characterization of the dimension vectors of general subrepresentations. This implies Horn type inequalities for the moment cone associated to the
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A Lie–Rinehart algebra in general relativity Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Christian Blohmann, Michele Schiavina, Alan Weinstein
We construct a Lie–Rinehart algebra over an infinitesimal extension of the space of initial value fields for Einstein’s equations. The bracket relations in this algebra are precisely those of the constraints for the initial value problem. The Lie–Rinehart algebra comes from a slight generalization of a Lie algebroid in which the algebra consists of sections of a sheaf rather than a vector bundle. (An
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Dedekind sums via Atiyah–Bott–Lefschetz Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Ana Cannas da Silva
This paper, written for differential geometers, shows how to deduce the reciprocity laws of Dedekind and Rademacher, as well as $n$-dimensional generalizations of these, from the Atiyah–Bott–Lefschetz formula, by applying this formula to appropriate elliptic complexes on weighted projective spaces.
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Generalizing the Mukai Conjecture to the symplectic category and the Kostant game Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Alexander Caviedes Castro, Milena Pabiniak, Silvia Sabatini
In this paper we pose the question of whether the (generalized) Mukai inequalities hold for compact, positive monotone symplectic manifolds. We first provide a method that enables one to check whether the (generalized) Mukai inequalities hold true. This only makes use of the almost complex structure of the manifold and the analysis of the zeros of the so-called generalized Hilbert polynomial, which
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Classical and Quantum mechanics on 3D contact manifolds Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Yves Colin de Verdière
In this survey paper, I describe some aspects of the dynamics and the spectral theory of sub-Riemannian 3D contact manifolds. We use Toeplitz quantization of the characteristic cone as introduced by Louis Boutet de Monvel and Victor Guillemin. We also discuss trace formulae following our work as well as the Duistermaat–Guillemin trace formula.
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Symplectic geometric flows Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Teng Fei, Duong H. Phong
Several geometric flows on symplectic manifolds are introduced which are potentially of interest in symplectic geometry and topology. They are motivated by the Type IIA flow and T‑duality between flows in symplectic geometry and flows in complex geometry. Examples include the Hitchin gradient flow on symplectic manifolds, and a new flow which is called the dual Ricci flow.
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Lower bounds for Steklov eigenfunctions Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Jeffrey Galkowski, John A. Toth
Let $(\Omega,g)$ be a compact, real analytic Riemannian manifold with real analytic boundary $\partial \Omega = M$. We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset \Omega^\circ$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper
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Which Hessenberg varieties are GKM? Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Rebecca Goldin, Julianna Tymoczko
Hessenberg varieties $\mathcal{H}(X,H)$ form a class of subvarieties of the flag variety $G/B$, parameterized by an operator $X$ and certain subspaces $H$ of the Lie algebra of $G$. We identify several families of Hessenberg varieties in type $A_{n-1}$ that are $T$-stable subvarieties of $G/B$, as well as families that are invariant under a subtorus $K$ of $T$. In particular, these varieties are candidates
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Quantum Witten localization Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Eduardo González, Chris T. Woodward
We prove a quantum version of the localization formula of Witten $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1185834}{[31]}$, see also $[\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1792291}{28}$, $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1722000}{22}$, $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=2198772}{35}$], that relates invariants
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Almost invariant subspaces and operators Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 David Kazhdan, Alexander Polishchuk
We prove an efficient version of the Wagner’s theorem on almost invariant subspaces (see $\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1608090}{[5]}$) and deduce some consequences in the context of Galois extensions.
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Seismic imaging with generalized Radon transforms: stability of the Bolker condition Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Peer Christian Kunstmann, Eric Todd Quinto, Andreas Rieder
Generalized Radon transforms are Fourier integral operators which are used, for instance, as imaging models in geophysical exploration. They appear naturally when linearizing about a known background compression wave speed. In this work we first consider a linearly increasing background velocity in two spatial dimensions. We verify the Bolker condition for the zero-offset scanning geometry and provide
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Stability and bifurcations of symmetric tops Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Eugene Lerman
We study the stability and bifurcation of relative equilibria of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3) \times SO(2)$ invariant metric and an $SO(2) \times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at singular values of the $SO(2) \times SO(2)$ momentum map and study the stability of the equilibria of the reduced systems
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Symplectic reduction and a Darboux–Moser–Weinstein theorem for Lie algebroids Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Yi Lin, Yiannis Loizides, Reyer Sjamaar, Yanli Song
We extend the Marsden–Weinstein reduction theorem and the Darboux–Moser–Weinstein theorem to symplectic Lie algebroids. We also obtain a coisotropic embedding theorem for symplectic Lie algebroids.
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Singular Lie filtrations and weightings Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Yiannis Loizides, Eckhard Meinrenken
We study weightings (a.k.a. quasi-homogeneous structures) arising from manifolds with singular Lie filtrations. This generalizes constructions of Choi–Ponge, Van Erp–Yuncken, and Haj–Higson for (regular) Lie filtrations.
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Bohr–Sommerfeld quantization of $b$-symplectic toric manifolds Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Pau Mir, Eva Miranda, Jonathan Weitsman
We introduce a Bohr–Sommerfeld quantization for bsymplectic toric manifolds and show that it coincides with the formal geometric quantization of $\href{ https://mathscinet.ams.org/mathscinet/relay-station?mr=3804693}{[\textrm{GMW18b}]}$. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the torus action on the manifold.
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Internal symmetry of the $L_{\leqslant 3}$ algebra arising from a Lie pair Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-11-20 Dadi Ni, Jiahao Cheng, Zhuo Chen, Chen He
$\def\DerL{\operatorname{Der}(L)}$A Lie pair is an inclusion $A$ to $L$ of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical $L_{\leqslant 3}$ algebra $\Gamma (\wedge^\bullet A^\vee \otimes L/A)$ whose unary bracket is the Chevalley–Eilenberg differential arising from every Lie pair $(L,A)$. In this note, we prove
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Kerr stability for small angular momentum Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-07-20 Sergiu Klainerman, Jérémie Szeftel
This is our main paper in which we prove the full, unconditional, nonlinear stability of the Kerr family $Kerr(a,m)$ for small angular momentum, i.e. $\lvert a \rvert /m \ll 1$, in the context of asymptotically flat solutions of the Einstein vacuum equations (EVE). We rely on our GCM papers $\href{https://mathscinet.ams.org/mathscinet-getitem?mr=4462882}{[40]}$ and $\href{https://mathscinet.ams.or
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Hilbert reciprocity using $K$-theory localization Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Oliver Braunling
Usually the boundary map in $K$-theory localization only gives the tame symbol at $K_2$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially
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Metrics on twisted pluricanonical bundles and finite generation of twisted canonical rings Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Bojie He, Xiangyu Zhou
In this paper, we first introduce the notion of admissible Bergman metrics. Then we establish a connection between singularities of admissible Bergman metrics and finite generation of twisted pluricanonical rings with $m$-multiplier ideal sheaves on smooth projective pairs. It involves an analytic approach to Boucksom’s result about asymptotic multiplier ideal of a graded system of ideals. In the end
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Moment polytopes on Sasaki manifolds and volume minimization Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Akito Futaki
We show that transverse coupled Kähler–Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli–Sparks–Yau. This is done assuming the Calabi–Yau
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Non-density of stable mappings on non-compact manifolds Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Shunsuke Ichiki
Around 1970, Mather established a significant theory on the stability of $C^\infty$ mappings and gave a characterization of the density of proper stable mappings in the set of all proper mappings. The result yields a characterization of the density of stable mappings in the set of all mappings in the case where the source manifold is compact. The aim of this paper is to complement Mather’s result.
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Genericity on submanifolds and application to universal hitting time statistics Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Han Zhang
We investigate Birkhoff genericity on submanifolds of homogeneous space $X = SL_d (\mathbb{R}) \ltimes (\mathbb{R}^d)^k / SL_d (\mathbb{Z}) \ltimes (\mathbb{Z}^d)^k$, where $d \geq 2$ and $k \geq 1$ are fixed integers. The submanifolds we consider are parameterized by unstable horospherical subgroup $U$ of a diagonal flow $a_t$ in $SL_d (\mathbb{R})$. As long as the intersection of the submanifold
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Quantum complexity of permutations Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Andrew Yu
Quantum complexity of a unitary measures the runtime of quantum computers. In this article, we study the complexity of a special type of unitaries, permutations. Let $S_n$ be the symmetric group of all permutations of ${\lbrace 1, \dotsc , n \rbrace}$ with two generators: the transposition and the cyclic permutation (denoted by $\sigma$ and $\tau$). The permutations ${\lbrace \sigma, \tau, \tau^{-1}
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Sum expressions for $p$-adic Hecke $L$-functions of totally real fields Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Luochen Zhao
As a continuation of previous work, we establish sum expressions for $p$-adic Hecke $L$-functions of totally real fields in the sense of Delbourgo, assuming a totally real analog of Heegner hypothesis. This is done by finding explicit formulas of the periods of the corresponding $p$-adic measures. As an application, we extend the Ferrero–Greenberg formula of derivatives of $p$-adic $L$-functions to
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On the $\mu$ equals zero conjecture for fine Selmer groups in Iwasawa theory Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Shaunak V. Deo, Anwesh Ray, R. Sujatha
We study the Iwasawa theory of the fine Selmer groups associated to Galois representations arising from modular forms. The vanishing of the $\mu$-invariant is shown to follow in some cases from a natural property satisfied by Galois deformation rings. We outline conditions under which the $\mu = 0$ conjecture is shown to hold for various Galois representations of interest.
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Some inequalities for the dual $p$-quermassintegrals Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Weidong Wang, Yanping Zhou
Based on the definitions of dual quermassintegrals, dual affine quermassintegrals and dual harmonic quermassintegrals, we generalize them to the dual $p$-quermassintegrals, such that the cases $p = 1$ , $n$ and $-1$ just are the dual quermassintegrals, dual affine quermassintegrals and dual harmonic quermassintegrals, respectively. Further, we orderly establish the dual $L_q$ Brunn–Minkowski type inequality
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Künneth formulas for path homology Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Fang Li, Bin Yu
We study the path homology groups with coefficients in a general ring $R$, and show that such groups are always finitely generated. We further prove two versions of Eilenberg–Zilber theorem for the Cartesian product and the join of two regular path complexes over a commutative ring $R$. Hence Künneth formulas are derived for the two cases over a PID. Note that this generalizes the related results proved
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On special generic maps of rational homology spheres into Euclidean spaces Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 Dominik J. Wrazidlo
Special generic maps are smooth maps between smooth manifolds with only definite fold points as their singularities. The problem of whether a closed $n$-manifold admits a special generic map into Euclidean $p$-space for $1\leq p \leq n$ was studied by several authors including Burlet, de Rham, Porto, Furuya, Èliašberg, Saeki, and Sakuma. In this paper, we study rational homology nspheres that admit
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Noncommutative geometry of computational models and uniformization for framed quiver varieties Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-07 George Jeffreys, Siu-Cheong Lau
We formulate a mathematical setup for computational neural networks using noncommutative algebras and near-rings, in motivation of quantum automata. We study the moduli space of the corresponding framed quiver representations, and find moduli of Euclidean and non-compact types in light of uniformization.
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On $p$-integrality of instanton numbers Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-03 Frits Beukers, Masha Vlasenko
We show integrality of instanton numbers in several key examples of mirror symmetry. Our methods are essentially elementary, they are based on our previous work in the series of papers called Dwork crystals I, II and III.
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A proof of van der Waerden’s Conjecture on random Galois groups of polynomials Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-03 Manjul Bhargava
Of the $(2H+1)^n$ monic integer polynomials $f(x) =x^n + a_1 x^{n-1} + \dotsc + a_n$ with $\max{\lbrace \lvert a_1 \rvert, \dotsc , \lvert a_n \rvert \rbrace} \leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as may be obtained by setting $a_n = 0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should
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Span of restriction of Hilbert theta functions Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-03 Gabriele Bogo, Yingkun Li
In this paper, we study the diagonal restrictions of certain Hilbert theta series for a totally real field $F$, and prove that they span the corresponding space of elliptic modular forms when the $F$ is quadratic or cubic. Furthermore, we give evidence of this phenomenon when $F$ is quartic, quintic and sextic.
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Functional equations of polygonal type for multiple polylogarithms in weights $5$, $6$ and $7$ Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-03 Steven Charlton, Herbert Gangl, Danylo Radchenko
We present new functional equations of multiple polylogarithms in weights $5$, $6$ and $7$ and use them for explicit depth reduction. These identities generalize the crucial identity $\mathbf{Q}_4$ from the recent work of Goncharov and Rudenko that was used in their proof of the weight $4$ case of Zagier’s Polylogarithm Conjecture.
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Generating Picard modular forms by means of invariant theory Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-03 Fabien Cléry, Gerard van der Geer
We use the description of the Picard modular surface for discriminant $-3$ as a moduli space of curves of genus $3$ to generate all vector-valued Picard modular forms from bi-covariants for the action of $\mathrm{GL}_2$ on the space of pairs of binary forms of bi-degree $(4, 1)$. The universal binary forms of degree $4$ and $1$ correspond to a meromorphic modular form of weight $(4,-2)$ and a holomorphic
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Moufang patterns and geometry of information Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-03 Noemie Combe, Yuri I. Manin, Matilde Marcolli
Technology of data collection and information transmission is based on various mathematical models of encoding. The words “Geometry of information” refer to such models, whereas the words “Moufang patterns” refer to various sophisticated symmetries appearing naturally in such models. In this paper, we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian
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Indefinite theta series: the case of an $N$-gon Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-03 Jens Funke, Stephen Kudla
In this note, we provide a construction of the indefinite theta series attached to $N$-gons in the symmetric space of an indefinite inner product space of signature $(m-2,2)$ following the suggestions of Section C in the recent paper of Alexandrov, Banerjee, Manschot, and Pioline, [2]. We prove the termwise absolute convergence of the holomorphic mock modular part of these series and also obtain an
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On finite multiple zeta values of level two Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-03 Masanobu Kaneko, Takuya Murakami, Amane Yoshihara
We introduce and study a “level two” analogue of finite multiple zeta values.We give conjectural bases of the space of finite Euler sums as well as that of usual finite multiple zeta values in terms of these newly defined elements. A kind of “parity result” and certain sum formulas are also presented.
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Unitary matrix models, free fermions, and the giant graviton expansion Pure Appl. Math. Q. (IF 0.7) Pub Date : 2023-04-03 Sameer Murthy
We consider a class of matrix integrals over the unitary group $U(N)$ with an infinite set of couplings characterized by a series $f(q) = \sum_{n \geq 1} a_n \, q^n$, with $a_n \in \mathbb{Z}$. Such integrals arise in physics as the partition functions of free four-dimensional gauge theories on $S^3$ and, in particular, as the superconformal index of super Yang–Mills theory. We show that any such model