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Algebraic quantum groupoids have been developed by two of the authors of this note (AVD and SHW) in a series of papers [A. Van Daele and S. Wang, Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples, Banach Center Publ. 98 (2012) 367–415; A. Van Daele and S. Wang, Weak multiplier Hopf algebras I. The main theory, J. Reine Angew. Math. 705 (2015) 155–209; A. Van Daele and S. Wang, Weak multiplier Hopf algebras II. The source and target algebras, preprint (2014), arXiv: 1403.7906v2 [math.RA]; A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], see also [A. Van Daele, Algebraic quantum groupoids. An example, preprint (2017), arXiv: 1702.04903 [math.RA]]. By an algebraic quantum groupoid, we understand a regular weak multiplier Hopf algebra with enough integrals. Regular multiplier Hopf algebroids are obtained also by two authors of this note (TT and AVD) in [T. Timmermann and A. Van Daele, Regular multiplier Hopf algebroids. Basic theory and examples, Commun. Algebra 46 (2017) 1926–1958]. Integral theory and duality for those have been studied by one author here (TT) in [T. Timmermann, Integration on algebraic quantum groupoids, Int. J. Math. 27 (2016) 1650014, arXiv: 1507.00660 [QA]; T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]. In these papers, the term algebraic quantum groupoid is used for a regular multiplier Hopf algebroid with a single faithful integral. Finally, again two authors of us (TT and AVD) have investigated the relation between weak multiplier Hopf algebras and multiplier Hopf algebroids in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. In the paper, Weak multiplier Hopf algebras III. Integrals and duality [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], one of the main results is that the dual of an algebraic quantum groupoid, admits a dual of the same type. In the paper, On duality of algebraic quantum groupoids [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]], a result of the same nature is obtained for regular multiplier Hopf algebroids with a single faithful integral. The duality of regular weak multiplier Hopf algebras with a single integral can be obtained from the duality of regular multiplier Hopf algebroids (see [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]). That is however not the obvious way to obtain this result. It is more difficult and less natural than the direct way followed in [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]]. We will discuss this statement further in the paper. Nevertheless, it is interesting to investigate the relation between the two approaches to duality in greater detail. This is what we do in this paper. We build further on the intimate relation between weak multiplier Hopf algebras and multiplier Hopf algebroids as studied in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. We now add the presence of integrals. That seems to be done best in a framework of dual pairs. It is in fact more general than the duality of these objects coming with integrals. We are convinced that the material we present in this paper will provide a deeper understanding of the duality of algebraic quantum groupoids, both within the framework of weak multiplier Hopf algebras, as well as more generally for multiplier Hopf algebroids. Finally, we feel it is also appropriate to include some historical comments on the development of these duality theories. [ABSTRACT FROM AUTHOR]

As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures concerning convex curves, Int. J. Math. 16(10) (2005) 1157–1173] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar N = 4 SYM amplitudes, J. High Energy Phys. 2021 (2021) 65]. [ABSTRACT FROM AUTHOR]

For a homogeneous space G / H of reductive type, we consider the tangential homogeneous space G / H . In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino's condition by direct calculations. In this paper, some easy-to-check necessary conditions (= obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister's theorem (the existence problem of common zero points of polynomials of odd degree). [ABSTRACT FROM AUTHOR]

This is a corrigendum to the authors paper [A reconstruction theorem for complex polynomials, Int. J. Math. 26(9) (2015) 1550073]. Here we present a completely rewritten and self-contained Sec. 4 of the aforementioned paper and provide the reader with the corrected proof of the main result that briefly states: "Given a generic polynomial P of degree at least 2 , there exist positive integers N , M such that Re (P k (z)) = Re (P k (w)) for all 0 ≤ k ≤ N implies P M (z) = P M (w) ". [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notion of rational singularities for any reflexive sheaf of rank 1, link our notion of rational singularities with the notion of rational singularities in, [S. Kovacs, Irrational centers, Pure Appl. Math. Q. 7(4) (2011) 1495–1515], and prove generalizations of standard facts about rational singularities. Moreover, by using a definition of non-rational locus, we introduce the notion of (Bq+1) as a dual notion of well-known Serre’s notion of (Sq+1), and prove a theorem about q-birational morphisms. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an L 2 -almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)]. [ABSTRACT FROM AUTHOR]

In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with Ric ≥ (n − 1) κ , κ = 0 or 1 , which extends the results in [A. Alvino, C. Nitsch and C. Trombetti, A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions, Comm. Pure Appl. Math. 76(3) (2023) 585–603]. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel–Daners inequality for Robin Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we study the Gromov–Witten theory of the Hilbert scheme X [ 2 ] of two points on an elliptic surface X. Assume that | K X | contains an element supported on the smooth fibers of X. By analyzing the degeneracy locus and localized virtual cycle arising from the cosection localization theory of Kiem and Li [Y. Kiem and J. Li, Gromov–Witten invariants of varieties with holomorphic 2-forms, preprint; Y. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025–1050], we determine the 1 -point genus- 0 Gromov–Witten invariant 〈 w 〉 0 , d (β f − 2 β 2) X [ 2 ] up to some rational number m (d , X) depending only on d and X , where w ∈ H 4 (X [ 2 ] , ℂ) , d ≥ 1 , f is a smooth fiber of X , β f = x 0 + f ⊂ X [ 2 ] with x 0 ∈ X − f being a fixed point, and β 2 = { ξ ∈ X [ 2 ] | Supp (ξ) = { x 0 } }. Moreover, we propose a conjecture regarding m (d , X) , and prove that the conjecture is true for X = C × E where E is an elliptic curve and C is a smooth curve. [ABSTRACT FROM AUTHOR]

The goal of this paper is to prove new upper bounds for the first positive eigenvalue of the p -Laplacian operator in terms of the mean curvature and constant sectional curvature on Riemannian manifolds. In particular, we provide various estimates of the first eigenvalue of the p -Laplacian operator on closed orientate n -dimensional Lagrangian submanifolds in a complex space form n (4) with constant holomorphic sectional curvature 4 . As applications of our main theorem, we generalize the Reilly-inequality for the Laplacian [R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52(4) (1977) 525–533] to the p -Laplacian for a Lagrangian submanifold in a complex Euclidean space and complex projective space for = 0 and = 1 , respectively. [ABSTRACT FROM AUTHOR]

Given a spectral triple on a C ∗ -algebra together with a unital injective endomorphism α , the problem of defining a suitable crossed product C ∗ -algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of α () in can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on and α (). [ABSTRACT FROM AUTHOR]

Various ample divisors on smooth del Pezzo surfaces of degree 2 have been verified to allow polar cylinders in [I. Cheltsov, J. Park and J. Won, Cylinders in del Pezzo surfaces, Int. Math. Res. Not.2017(4) (2015) 1179–1230]. We show that affine cones over smooth del Pezzo surfaces of degree 2 polarized by such ample divisors are flexible in codimension one. All varieties in this paper are assumed to be defined over an algebraically closed field of characteristic zero. [ABSTRACT FROM AUTHOR]

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot is minimal where or . [ABSTRACT FROM AUTHOR]

In this paper, we study the finite representation theory of the map Lie conformal algebra , where G is a finite simple Lie conformal algebra and A is a commutative associative algebra with unity over . In particular, we give a complete classification of nontrivial finite irreducible conformal modules of provided A is finite-dimensional. [ABSTRACT FROM AUTHOR]

Fix , let be the loop Lie algebra over with basis and relations , where . In this paper, a formal distribution Lie algebra of is constructed. Then the associated conformal algebra is studied, where has a -basis with -brackets and . In particular, we determine the conformal derivations and rank one conformal modules of this conformal algebra. Finally, we study the central extensions and extensions of conformal modules. [ABSTRACT FROM AUTHOR]

This paper is a continuation of [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part II, to appear in Internat. J. Math.]. We construct by-hand Legendrian surfaces for which specific properties of their gradient flow trees hold. These properties enable us to complete the proof in [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part II, to appear in Internat. J. Math.] that the Cellular DGA defined in [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part I, preprint (2016), arXiv:1608.02984] is stable tame isomorphic to the Legendrian contact homology DGA defined in [T. Ekholm, J. Etnyre and M. Sullivan, The contact homology of Legendrian submanifolds in ℝ 2 n + 1 , J. Differential Geom.71(2) (2005) 177–305]. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to survey some recent results on the Chern-Moser question concerning existence of umbilical points on three-dimensional CR submanifolds in . [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of non-weight modules for some algebras related to the Virasoro algebra: The algebra (a, b), the twisted deformative Schrödinger-Virasoro Lie algebras and the Schrödinger algebra. We study the modules whose restriction to the Cartan subalgebra (modulo center) are free of rank 1 for these algebras. Moreover, the simplicities of these modules are determined. [ABSTRACT FROM AUTHOR]

Finite dimensional irreducible modules for the affine quantum Schur algebra were classified in [B. Deng, J. Du and Q. Fu, A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory, London Mathematical Society Lecture Note Series, Vol. 401 (Cambridge University Press, Cambridge, 2012), Chapt. 4] when is not a root of unity. We will classify finite-dimensional irreducible modules for affine quantum Schur algebras at roots of unity and generalize [J. A. Green, Polynomial Representations of , 2nd edn., with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker, Lecture Notes in Mathematics, Vol. 830 (Springer-Verlag, Berlin, 2007), (6.5f) and (6.5g)] to the affine case in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Izumi's result on uniqueness of realization of finite C-tensor categories in the endomorphism category of the injective factor of type III1 for finitely generated strongly amenable C-tensor categories by applying Popa's classification theorem of strongly amenable subfactors of type III1. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Izumi's result on uniqueness of realization of finite C-tensor categories in the endomorphism category of the injective factor of type III1 for finitely generated strongly amenable C-tensor categories by applying Popa's classification theorem of strongly amenable subfactors of type III1. [ABSTRACT FROM AUTHOR]

Given a compact connected Riemann surface Σ of genus g Σ ≥ 2 , and an effective divisor D = ∑ i n i x i on Σ with degree (D) < 2 (g Σ − 1) , there is a unique cone metric on Σ of constant negative curvature − 4 such that the cone angle at each point x i is 2 π n i [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc.103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on Σ corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on Σ parametrized by a nonempty open subset of H 0 (Σ , K Σ ⊗ 2 ⊗ Σ (− 2 D)) that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin's results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case D = 0. [ABSTRACT FROM AUTHOR]

Motivated by Giaquinta and Hildebrandt's regularity result for harmonic mappings [M. Giaquinta and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 1982(336) (1982) 124–164, Theorems 3 and 4], we show a C α , α 2 -regularity result of the harmonic flow between two Riemannian manifolds when the image is in a regular geodesic ball. The proof is based on De Giorgi–Moser's iteration and Schauder estimate. [ABSTRACT FROM AUTHOR]

By using the integral method, we prove a rigidity theorem for spacelike self-shrinkers in pseudo-Euclidean space under a minor growth condition in terms of the mean curvature and the second fundamental form, which generalizes Theorem 1.1 in [H. Q. Liu and Y. L. Xin, Some Results on Space-Like Self-Shrinkers, Acta Math. Sin. (Engl. Ser.) 32(1) (2016) 69–82]. [ABSTRACT FROM AUTHOR]

We construct asymptotically flat, scalar flat extensions of Bartnik data (Σ , γ , H) , where γ is a metric of positive Gauss curvature on a two-sphere Σ , and H is a function that is either positive or identically zero on Σ , such that the mass of the extension can be made arbitrarily close to the half area radius of (Σ , γ). In the case of H ≡ 0 , the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav.32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon (Σ , γ) , for any metric γ with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom.62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys.292(1) (2009) 271–284]. [ABSTRACT FROM AUTHOR]

We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their étale coverings and their fundamental groups are also defined. These fundamental groups approximate the fundamental group of an appropriate affine curve. We also define vector bundles on these objects and the category of orbifold bundles on any smooth projective curve. Analogues of various statements about vector bundles which are true in characteristic zero are also proved. Some of these are positive characteristic avatars of notions which appear in the second author's work [A. J. Parmeswaran, Parabolic coverings I: Case of curves, J. Ramanujam Math. Soc.25(3) (2010) 233–251.] in characteristic zero. [ABSTRACT FROM AUTHOR]

Let C be a smooth irreducible projective curve and let (L , H 0 (L)) be a complete and generated linear series on C. Denote by M L the kernel of the evaluation map H 0 (L) ⊗ 𝒪 C → L. The exact sequence 0 → M L → H 0 (L) ⊗ 𝒪 C → L → 0 fits into a commutative diagram that we call the Butler's diagram. This diagram induces in a natural way a multiplication map on global sections m W : W ∨ ⊗ H 0 (K C) → H 0 (S ∨ ⊗ K C) , where W ⊆ H 0 (L) is a subspace and S ∨ is the dual of a subbundle S ⊂ M L . When the subbundle S is a stable bundle, we show that the map m W is surjective. When C is a Brill–Noether general curve, we use the surjectivity of m W to give another proof of the semistability of M L , moreover, we fill up a gap in some incomplete argument by Butler: With the surjectivity of m W we give conditions to determine the stability of M L , and such conditions imply the well-known stability conditions for M L stated precisely by Butler. Finally we obtain the equivalence between the (semi)stability of M L and the linear (semi)stability of (L , H 0 (L)) on k -gonal curves. [ABSTRACT FROM AUTHOR]

We show that for each integer n ≥ 3 any finite group G acts smoothly and freely on a connected sum (S n × S n + 1) # ⋯ # (S n × S n + 1) ︸ r for some r. Moreover, as a module over Z [ G ] , the middle dimensional homotopy group can be specified in advance to belong to the stable syzygy Ω n + 1 G (Z). [ABSTRACT FROM AUTHOR]

We describe all rigid algebras and all irreducible components in the variety of four-dimensional Leibniz algebras 𝔏𝔢𝔦𝔟4 over ℂ. In particular, we prove that the Grunewald–O'Halloran conjecture is not valid and the Vergne conjecture is valid for 𝔏𝔢𝔦𝔟4. [ABSTRACT FROM AUTHOR]

By using the Bergman representative coordinates, we give the necessary and sufficient condition for the degree of automorphisms of quasi-circular domains fixing the origin to be bounded by the resonance order, thus solving a conjecture of the author. [ABSTRACT FROM AUTHOR]

We give a bound on which singularities may appear on Kollár-Shepherd-Barron-Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with ) whose unique non-Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry. [ABSTRACT FROM AUTHOR]

We study knots of order in the grope filtration and the solvable filtration of the knot concordance group. We show that, for any integer , there are knots generating a subgroup of . Considering the solvable filtration, our knots generate a subgroup of distinct from the subgroup generated by the previously known -torsion knots of Cochran, Harvey, and Leidy. We also present a result on the -torsion part in the Cochran, Harvey, and Leidy's primary decomposition of the solvable filtration. [ABSTRACT FROM AUTHOR]

We derive a formula of the greatest lower bounds on Ricci curvature of Fano homogeneous toric bundles. A criteria for the ampleness of line bundles over general homogeneous toric bundles is also obtained. [ABSTRACT FROM AUTHOR]

Consider an inclusion of diffuse von Neumann algebras . We say that has the absorbing amenability property (AAP) if for any diffuse subalgebra and any amenable intermediate algebra we have that is contained in We prove that the cup subalgebra associated to any subfactor planar algebra has the AAP. [ABSTRACT FROM AUTHOR]