Mathematics

A. Mironov proposed a construction of lagrangian submanifolds in C n and C P n ; there he was mostly motivated by the fact that these lagrangian submanifolds (which can have in general self intersections, therefore below we call them lagrangian cycles) present new example of minimal or Hamiltonian minimal lagrangian submanifolds. However the Mironov construction of lagrangian cycles itself can be directly extended to much wider class of compact algrebraic varieties: namely it works in the case when algebraic variety X of complex dimension n admits T k - action and an anti - holomorphic involution such that the real part X R ⊂X has real dimension n and is transversal to the torus action. For this case one has families of lagrangian submanifolds and cycles. In the present small text we show how the construction of Mironov cycles works for the complex Grassmannians, resulting in simple examples of smooth lagrangian submanifolds in Gr(k,n+1) , equipped with a standard Kahler form under the Plücker embedding. For sure the text is not complete but in the new reality we would like to fix it, hoping to continue the investigations and to present in a future complete list of Mironov cycles in Gr(k,n+1) .

We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.

We identify a family of torus representations such that the corresponding singular symplectic quotients at the 0 -level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may have graded isomorphic algebras of real regular functions and graded Poisson isomorphic complex symplectic quotients yet not be graded regularly diffeomorphic nor graded regularly symplectomorphic.

An Engel manifold is a 4-manifold with a completely non-integrable 2-distribution called Engel structure. I research the functorial relation between Engel manifolds and Contact 3-orbifolds. And I construct an Engel manifold that the automorphism group is trivial.

Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known "elementary" building blocks for Lagrangian cobordisms that are smoothly the attachment of 0 - and 1 -handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is "decomposable" into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms.

We prove some contact analogs of smooth embedding theorems for closed π -manifolds. We show that a closed, k -connected, π -manifold of dimension (2n + 1) that bounds a π -manifold, contact embeds in the (4n−2k+3) -dimensional Euclidean space with the standard contact structure. We also prove some isocontact embedding results for π -manifolds and parallelizable manifolds.

If a contact form on a (2n+1) -dimensional closed contact manifold admits closed Reeb orbits, then its systolic ration is defined to be the quotient of (n+1) -th power of the shortest period of Reeb orbits by the contact volume. We prove that every co-oriented contact structure on any closed contact manifold admits a contact form with arbitrarily large systolic ratio. This statement generalizes the result of Abbondandolo et al. in dimension three to higher dimensions. The proof is inductive and uses the three dimensional result as its basis step and relies on the Giroux correspondence for the inductive step. The proof does not require any plug construction that is used by Abbondandolo et al. and by the author in the previous version of the proof.

It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation; the contact struture is universally tight; and in the case of the three-sphere, the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid.

We establish the continuous functoriality of wrapped Fukaya categories with respect to Liouville automorphisms, yielding a way to probe the homotopy type of the automorphism group of a Liouville sector. These methods prove Liouville and monotone cases of a conjecture of Teleman from the 2014 ICM. In the case of a cotangent bundle, we show that the Abouzaid equivalence between the wrapped category and the infinity-category of local systems intertwines our action with the action of diffeomorphisms of the zero section. In particular, our methods yield a typically non-trivial map from the rational homotopy groups of Liouville automorphisms to the rational string topology algebra of the zero section.

We lay the foundations of convex hypersurface theory (CHT) in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be C 0 -approximated by a convex one. We also prove that a C 0 -generic family of mutually disjoint closed hypersurfaces parametrized by t∈[0,1] is convex except at finitely many times t 1 ,…, t N , and that crossing each t i corresponds to a bypass attachment. As applications of CHT, we prove the existence of compatible (relative) open book decompositions for contact manifolds and an existence h-principle for codimension 2 contact submanifolds.