National Institute of the Republic of Serbia

ὅδε οἶκος, ὦ ἑταῖρε, μνημεῖον ἐστιν ζωῶν τῶν σοφῶν ἀνδρῶν, καὶ τῶν ἔργων αὐτῶν

**Seminar for Geometry, education and visualization with applications **

**PROGRAM**

MATEMATIČKI INSTITUT SANU

Seminar geometriju, obrazovanje i vizualizaciju sa primenama

Plan rada za APRIL 2024.

Koncirkularna polu-simetrična metrička koneksija predstavlja specijalnu klasu polu-simetrične metričke koneksije i definisana je u radu [W. Slosarska,

U drugom delu predavanja će biti predstavljena primena ove koneksije na Lorencove mnogostrukosti.

Classical spectral analysis has been expanded to include tensors as generalizations of matrices. But tensors are approached in various manners, through four distinct perspectives. There exist numerous generalizations of the classical eigenproblem. The aim is to present most of these varieties, offering appropriate motivations, definitions, properties, and applicative aspects for each. The main subject is the geometrical eigenproblem concerning tensors within Riemannian space.

Every algebraic variety by the very definition can be equipped with a symplectic structure, given by the Kahler form of any positive polarization. Therefore a natural problem is to study lagrangian submanifolds - real submanifolds which posses the lagrangian property. This problem seems to be important itself; moreover certain modern approaches to Mirror Symmetry conjecture are based on a duality between complex and symplectic geometries of Kahler manifolds and therefore one needs to know which lagrangian submanifolds a given algebraic variety admits. At the same time not too much is known even for the simplest and basic cases: f.e. for the projective space CPn one has two natural possibilities - real part of the complex projective space RPn and Liouville tori Tn \subset CPn coming from the fact that one has here the complete set of first integrals.

In 2004 Andrey Mironov presented new construction which gives a number of new examples of lagrangian submanifolds in Cn and CPn and moreover he showed that these lagrangian submanifolds are Hamiltonian minimal. Leaving aside the minimality story we can present a natural generalization of the Mironov construction which can be applied for much broad case of algebraic variety. Namely suppose that a given compact algebraic variety X equipped with a symplectic form, given by the Kahler stucture, admits an TK - action by Kahler isometries (here k can be less that the complex dimension of X) on X which is compatible with a fixed anti holomorphic structure \sigma on X. Suppose that the real part X_R \subset X with respect to \sigma has the maximal possible dimension. Then for each l from 0 to k we can construct a lagrangian submanifold, smooth or with self intersections.

As an example of this generalized construction we present how it works for the case of complex grassmanian Gr(k,n).

Miroslava Antić

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Rukovodilac seminara

Djordje Kocić

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Sekretar seminara