### Book

## Research Perspectives CRM Barcelona

We study the problem of description of the symplectic mapping class groups π0 Symp(X, ω) (SyMCG) of rational 4-manifolds X = CP #lCP . We specify certain class of symplectic forms ω on such X for which we give a finite presentation of the SyMCG with generators symplectic Dehn twists along Lagrangian spheres.

We study the problem of description of the symplectic mapping class groups π0 Symp(X, ω) (SyMCG) of rational 4-manifolds X = CP #lCP . We specify certain class of symplectic forms ω on such X for which we give a finite presentation of the SyMCG with generators symplectic Dehn twists along Lagrangian spheres.

This is a joint work with my scientific advisor Vsevolod Shevchishin.

The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential functions, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of double stochastic matrices. Mirror symmetry interprets the polynomial as a Landau-Ginzburg potential for corresponding Fano variety and relates the symplectic geometry of the variety with systems of unbiased projectors

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kahler manifold. We prove that any connected component of the moduli space of pseudoholomorphic curves on M is compact. This can be used to study pseudoholomorphic curves on a 6-dimensional sphere with the standard (G_2-invariant) almost complex structure.

The goal of this thesis is to understand symplectic mapping class group of rational 4-manifolds and how it changes, when the cohomology class of the symplectic form varies. For this aims, we will show that in some cases this group admits a realization as the fundamental group of the complement to certain divisor on Hilbert scheme of points on СP2. This divisor is the locus of special constellations of points which parameterize rational complex surfaces X with rational (-2)-curves on X. In the paper we will describe irreducible components of this divisor and show that the loops around the components of the divisor are natural generators of the symplectic mapping class group.

The goal of this diploma thesis is to understand symplectic mapping class group of rational 4-manifolds and how it changes, when the cohomology class of the symplectic form varies. For this aims, we will show that in some cases this group admits a realization as the fundamental group of the complement to certain divisor on Hilbert scheme of points on CP^2. This divisor is the locus of special constellations of points which parameterize rational complex surfaces with rational -curves on . In the paper we will describe irreducible components of this divisor and show that the loops around the components of the divisor are natural generators of the symplectic mapping class group.

We study the problem of description of the symplectic mapping class groups π0 Symp(X, ω) (SyMCG) of rational 4-manifolds X = CP #lCP . We specify certain class of symplectic forms ω on such X for which we give a finite presentation of the SyMCG with generators symplectic Dehn twists along Lagrangian spheres.

This is a joint work with my scientific advisor Vsevolod Shevchishin.

We prove that the wrapped Fukaya category of a punctured sphere ($ S^{2}$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.