Complex geometry and real geometry

There is a well-known way to generalize the Riemann-Roch operator for Kahler manifold to that for Hermitian manifold. In this paper we show a slightly different way to get a generalized Riemann-Roch operator, which is just the Dirac operator. The difference between the two operators is that the latter one enables the so-called Pythagoras equalities.     

Generalized symmetric spaces,Yu. P. Solovyov’s formula,and the generalized Hermitian geometry

We collect some basic results on canonical affinor structures of classical type on generalized symmetric spaces. Yu. P. Solovyov’s stimulating influence on this topic during its initial stages is illustrated. Using special canonical f-structures on homogeneous k-symmetric spaces, we also present a new collection of homogeneous Hermitian f-manifolds. Dedicated to the memory of Yu. P. Solovyov Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 43–60, 2007.     

Noncommutative Batalin–Vilkovisky geometry and matrix integrals

I study the new type of supersymmetric matrix models associated with any solution to the quantum master equation of the noncommutative Batalin–Vilkovisky geometry. The asymptotic expansion of the matrix integrals gives homology classes in the Kontsevich compactification of the moduli spaces, which I associated with the solutions to the quantum master equation in my previous paper. I associate with the Bernstein–Leites matrix superalgebra equipped with an odd differentiation, whose square is nonzero, the family of cohomology classes of the compactification. This family is the generating function for the products of the tautological classes. The simplest example of my matrix integrals in the case of dimension zero is a supersymmetric extension of the Kontsevich model of 2-dimensional gravity.     

The Fischer–Marsden conjecture and contact geometry

In this paper, we consider the Fischer–Marsden conjecture within the frame-work of K-contact manifolds and \((\kappa ,\mu )\)-contact manifolds. First, we prove that a complete K-contact metric satisfying \(\mathcal {L}^{*}_g(\lambda )=0\) is Einstein and is isometric to a unit sphere \(S^{2n+1}\). Next, we prove that if a non-Sasakian \((\kappa ,\mu )\)-contact metric satisfies \(\mathcal {L}^{*}_g(\lambda )=0\), then \( M^{3} \) is flat, and for \(n > 1\), \(M^{2n+1}\) is locally isometric to the product of a Euclidean space \(E^{n+1}\) and a sphere \(S^n(4)\) of constant curvature \(+\,4\).     

Multiplier ideal sheaves in complex and algebraic geometry

The application of the method of multiplier ideal sheaves to effective problems in algebraic geometry is briefly discussed. Then its application to the deformational invari-ance of plurigenera for general compact algebraic manifolds is presented and discussed. Finally its application to the conjecture of the finite generation of the canonical ring is explored, and the use of complex algebraic geometry in complex Neumann estimates is discussed.     

Hyper-Kähler geometry and invariants of three-manifolds

We study a 3-dimensional topological sigma-model, whose target space is a hyper-Kähler manifold X. A Feynman diagram calculation of its partition function demonstrates that it is a finite type invariant of 3-manifolds which is similar in structure to those appearing in the perturbative calculation of the Chern-Simons partition function. The sigma-model suggests a new system of weights for finite type invariants of 3-manifolds, described by trivalent graphs. The Riemann curvature of X plays the role of Lie algebra structure constants in Chern-Simons theory, and the Bianchi identity plays the role of the Jacobi identity in guaranteeing the so-called IHX relation among the weights. We argue that, for special choices of X, the partition function of the sigma-model yields the Casson-Walker invariant and its generalizations. We also derive Walker's surgery formula from the SL(2, Z) action on the finite-dimensional Hilbert space obtained by quantizing the sigma-model on a two-dimensional torus.     

Paths and the geometry of l'Hôpital's Rule

In this short note is presented an easy and instructive proof of l'Hôpital's Rule in both the 0/0 and ∞/ ∞ cases, obtained by translating the theorem into a simple geometric statement about paths in the plane.     

Complex geometry: Its brief history and its future

This paper reviews the brief history of complex geometry and looks into its future.     

Metric Möbius geometry and a characterization of spheres

We obtain a Möbius characterization of the n-dimensional spheres S ⁿ endowed with the chordal metric d ₀. We show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to (S ⁿ , d ₀) for some n ≥ 1.     

On the geometry,preemptions and complexity of multiprocessor and shop scheduling

In this paper we study multiprocessor and open shop scheduling problems from several points of view. We explore a tight dependence of the polynomial solvability/intractability on the number of allowed preemptions. For an exhaustive interrelation, we address the geometry of problems by means of a novel graphical representation. We use the so-called preemption and machine-dependency graphs for preemptive multiprocessor and shop scheduling problems, respectively. In a natural manner, we call a scheduling problem acyclic if the corresponding graph is acyclic. There is a substantial interrelation between the structure of these graphs and the complexity of the problems. Acyclic scheduling problems are quite restrictive; at the same time, many of them still remain NP-hard. We believe that an exhaustive study of acyclic scheduling problems can lead to a better understanding and give a better insight of general scheduling problems. We show that not only acyclic but also a special non-acyclic version of periodic job-shop scheduling can be solved in polynomial (linear) time. In that version, the corresponding machine dependency graph is allowed to have a special type of the so-called parti-colored cycles. We show that trivial extensions of this problem become NP-hard. Then we suggest a linear-time algorithm for the acyclic open-shop problem in which at most m−2 preemptions are allowed, where m is the number of machines. This result is also tight, as we show that if we allow one less preemption, then this strongly restricted version of the classical open-shop scheduling problem becomes NP-hard. In general, we show that very simple acyclic shop scheduling problems are NP-hard. As an example, any flow-shop problem with a single job with three operations and the rest of the jobs with a single non-zero length operation is NP-hard. We suggest linear-time approximation algorithm with the worst-case performance of ( , respectively) for acyclic job-shop (open-shop, respectively), where (‖ℳ‖, respectively) is the maximal job length (machine load, respectively). We show that no algorithm for scheduling acyclic job-shop can guarantee a better worst-case performance than . We consider two special cases of the acyclic job-shop with the so-called short jobs and short operations (restricting the maximal job and operation length) and solve them optimally in linear time. We show that scheduling m identical processors with at most m−2 preemptions is NP-hard, whereas a venerable early linear-time algorithm by McNaughton yields m−1 preemptions. Another multiprocessor scheduling problem we consider is that of scheduling m unrelated processors with an additional restriction that the processing time of any job on any machine is no more than the optimal schedule makespan C _max^*. We show that the (2m−3)-preemptive version of this problem is polynomially solvable, whereas the (2m−4)-preemptive version becomes NP-hard. For general unrelated processors, we guarantee near-optimal (2m−3)-preemptive schedules. The makespan of such a schedule is no more than either the corresponding non-preemptive schedule makespan or max {C _max^*,p _max}, where C _max^* is the optimal (preemptive) schedule makespan and p _max is the maximal job processing time. E.V. Shchepin was partially supported by the program “Algebraical and combinatorial methods of mathematical cybernetics” of the Russian Academy of Sciences. N. Vakhania was partially supported by CONACyT grant No. 48433.     

Noncommutative geometry,negative probabilities and Cantorian–Fractal spacetime

A straightforward explanation of the Young's two-slit experiment of a quantum particle is obtained within the framework of the Noncommutative Geometry associated with El Naschie's Cantorian–Fractal transfinite spacetime manifold.     

Differential geometry of grassmannians and the Plücker map

Using the Plücker map between grassmannians, we study basic aspects of classic grassmannian geometries. For ‘hyperbolic’ grassmannian geometries, we prove some facts (for instance, that the Plücker map is a minimal isometric embedding) that were previously known in the ‘elliptic’ case.     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.     

Conley pairs in geometry—Lusternik–Schnirelmann theory and more

Firstly, we wish to motivate that Conley pairs, realized via Salamon’s definition (Salamon, 1990), are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces (Weber, 2017). From this fell off quite naturally, firstly, an alternative proof (Weber, 2016) of the cell attachment theorem in Morse theory (Milnor, 1963) and, secondly, some ideas (Majer and Weber, 2015) how to try to organize the closures of the unstable manifolds of a Morse–Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use these Conley pairs to implement the gradient flow proof of the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934) proposed in Bott’s survey (Bott, 1982).Secondly, we shall use this opportunity to provide an exposition of Lusternik–Schnirelmann (LS) theory based on thickenings of unstable manifolds via Conley pairs. We shall cover the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934), cuplength, subordination, the LS refined minimax principle, and a variant of the LS category called ambient category.     

Lelong–Poincaré formula in symplectic and almost complex geometry

In this paper, we present two applications of the theory of singular connections developed by Harvey and Lawson (1993). The first one is a version of the Lelong–Poincaré formula with estimates for sections of vector bundles over an almost complex manifold. The second one is a convergence theorem for divisors associated to a general family of symplectic submanifolds constructed by Donaldson (1996) (the case of hypersurfaces) and by Auroux in (1997) (for arbitrary dimensional submanifolds).