It is a smooth curve parameterized by its arclength satisfying ∇_γ' γ' = κ J γ'. When both γ((-∞,0]) and γ ([0,∞)) are unbounded, we can consider its limit points γ(-∞) = lim _{t → -∞} γ(t) and γ(∞) = lim_{t → ∞} γ(t) in the ideal boundary ∂M if they exist. In my talk, we study trajectories whose limit points coincide. For such a trajectory γ, we investigate the angle between the initial vector γ'(0) and the initial vector of the geodesic joining γ(0) and γ(t). By comparing this string-elevation to that on a complex hyperbolic space, we give a condition that such a trajectory to be holomorphic horocycle.

He studies Riemannian manifolds from curve theoretic point of view. He is also interested in graphs whose edges are colored by two colors. He is now a member of editors of Moroccan Journal of Algebra and Geometry with Applications. He is also a certified public psychologist in Japan.

The classical experiments by Alfred Yarbus shows that compensation of eye movements leads to loss of vision in 2-3 sec. The configuration space of the eye is described by the Listing's Law. We propose an interpretation of the Listing's Law in terms of the Hopf bundle χ : S³ → S²_e over the eye sphere ( the boundary of the eyeball). More precisely, the configuration space of the eye is the Listing's equatorial hemisphere (S²₊)_L of the 3- sphere S³ = H¹ of unit quaternions. Saccades are described as geodesic segments ab of the Listing's hemisphere. The Hopf projection χ(ab) = AB ⊂ S²_e is the segment AB of the circle S¹ = S²_e ∩ Π(A, B, −i) where −i is the center of the fovea. The segment AB is the trajectory of the gaze during the saccade. We study saccadic cycles, which are described by geodesic polygons a₀a₁ · · · a_n, an_n= a₀ in the Listing's hemisphere. It is carried out a set of sequentially implemented rotation of the eye around axes Ω₁, · · · , Ω_n. We prove that these axis determines the saccadic cycle. in particular, the angles of rotations. We discuss the role of saccades and drift in vision, in particular, in the solution of the vision stability problem, which had been formulated by Persian scholar Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham , latinised Ibn Al-Haytham or Alhazen,(965-1039) and, since then was discussed by many scientists, as R. Descartes, H. von Helmholtz, E. Mach, C. Sherrington and many others. Roughly speaking, vision is the result of the interaction of stochastic information about light, entering the retina during drift and recorded by photoreceptors, and oculomotor information about the eye position after remapping that accompanies a saccade. This may be considered as a discrete version of the E. Cartan moving frame method.

(1967-1976), as Senior Lecturer (1976-1987) and Associate Professor (1987-1990) at Moscow State Pedagogical University, and then had been Chief Researcher at Scientific Center "Sophus Lie " (1990-2000). He moved then to Hull University, where he had the position of Chair in Pure Mathematics (2000-2005).In 2005, he integrated Eidenburg University as visiting Professor, until 2009, when he obtained honourable fellow (2009-2014). He also had been a visiting Professor at Brno University (2011-2014). He is a Member of the Edinburgh Mathematical Society. He is also a member of Editorial Board of many journals (Differential Geometry and Applications, Transformation Groups, Annals of Global Analysis and Geometry, International J. of Geom. Methods in Modern Phys., Monatshefte für Mathematik). His research interests are: Differential Geometry, Lie Groups and their Applications to Mathematical, Physics and Neuroscience: Riemannian and pseudo-Riemannian Geometry, Spinor Geometry, Homogeneous Spaces, Transformation Groups, Differential Geometry and PDE, Geometric structures on Manifolds, Holonomy Groups and Special Geometry, Generalized Geometry, General Relativity, Yang-Mills Fields, Supergravity, Applications of Differential Geometry to Neuroscience (Neurogeometry )., where he published 5 monographs and survey articles, 41 articles for Encyclopaedia of Mathematics and 100 papers.

of the K¨ahler-Einstein condition to the non-Kähler setting; the convergence of the normalized Chern-Ricci flow on compact complex surfaces. The talk is based on joint collaborations with Simone Calamai, Francesco Pediconi, Cristiano Spotti, Valentino Tosatti and others.

Thanks to research grants by INdAM, an Italian government research institution, he visited the University of Zaragoza in Spain, and the University of Parma. Later, he was a Junior Visiting fellow at Centro di Ricerca Matematica "Ennio de Giorgi" of Scuola Normale Superiore in Pisa. In 2015, he was granted national funding for the independence of young researchers: his research group on "Analytic aspects in complex and hypercomplex geometry" was based at University of Firenze and five people participated in it. Since 2022, he's been professor in Geometry at University of Firenze. In research, he is interested on Complex non-Kähler Geometry, Cohomological properties of non-Kähler manifolds, Hermitian Geometry, Conformal Differential Geometry, Symplectic Geometry, Functions of complex and hypercomplex variables, in which he publishes over than 50 research papers and 3 books. He has/had several master and PhD students and postdocs, and taught several courses in differential and complex analysis and geometry.