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## Fundamental Group of the Complement to Certain Special Divisors on the Moduli Space of Points on CP^2

P. 171-200.

Alekseeva D., Shevchishin V.

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.

Language:
English

Keywords: Symplectic Geometry

M.: National Research University Higher School of Economics, 2017

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 ...

Added: December 11, 2019

Pushkar P. E., Chekanov-type theorem for spherized cotangent bundles / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.08743.

We prove a Chekanov-type theorem for the spherization of the cotangent bundle ST∗B of a closed manifold B. It claims that for Legendrian submanifolds in ST∗B the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies. ...

Added: December 7, 2016

Slavnov S. A., Theoretical Computer Science 2006 Vol. 357 No. 1-3 P. 215-229

Added: March 4, 2013

Slavnov S. A., Annals of Pure and Applied Logic 2005 Vol. 131 No. 1-3 P. 177-225

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Abouzaid M., Auroux D., Efimov A. et al., Journal of the American Mathematical Society 2013 Vol. 26 No. 4 P. 1051-1083

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 ...

Added: October 31, 2013

Shevchishin V., Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds / Cornell University. Series math "arxiv.org". 2010.

We introduce the secondary Stiefel-Whitney class $\tilde w_2$ of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class $\tilde w_2$ vanishes.
Using this, we give a detailed description of the combinatorial structure of the diffeotopy group of ruled symplectic 4-manifolds ...

Added: March 18, 2013

Shevchishin V., Izvestiya. Mathematics 2009 Vol. 73 No. 4 P. 797-859

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle K in R4 and CP2. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr:X→S2 and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results ...

Added: March 18, 2013

Galkin S., Golyshev V., Iritani H., Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures / Cornell University. Series math "arxiv.org". 2014. No. 1404.6407.

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

Added: May 4, 2014

Alekseeva D., Shevchishin V., , in: Research Perspectives CRM Barcelona. .: Basel: Birkhauser/Springer, 2020..

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. ...

Added: April 20, 2020

Entov M., Verbitsky M., Full symplectic packing for tori and hyperkahler manifolds / Cornell University. Series math "arxiv.org". 2014.

Let M be a closed symplectic manifold of volume V. We say that M admits a full symplectic packing by balls if any collection of symplectic balls of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in ...

Added: February 5, 2015

Lekili Y., Polishchuk A., Compositio Mathematica 2020 Vol. 156 No. 7 P. 1310-1347

Using Auroux's description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of k+1 generic hyperplanes in ℂℙ^n, for k≥n, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In ...

Added: August 12, 2020

Galkin S., Golyshev V., Iritani H., Duke Mathematical Journal 2016 Vol. 165 No. 11 P. 2005-2077

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

Added: November 18, 2014

Verbitsky M., Communications in Mathematical Physics 2013 Vol. 324 No. 1 P. 173-177

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 ...

Added: February 16, 2013

Springer, 2013

Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry. ...

Added: February 20, 2013

Pushkar P. E., On Hamiltonian and contact isotopy liftings / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.07948.

We construct counterexamples to lifting properties of Hamiltonian and contact isotopies ...

Added: December 7, 2016

Bondal A. I., Zhdanovskiy I., , in: Primitive Forms and Related Subjects — Kavli IPMU 2014. .: Tokyo: Mathematical Society of Japan, 2019.. P. 1-18.

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 ...

Added: October 22, 2015

Basel: Birkhauser/Springer, 2020

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. ...

Added: December 11, 2019