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Vardan Oganesyan

Publications and source records attributed to Vardan Oganesyan.

13 recordsLinked to original sources
arXiv2024

The first step towards symplectic homotopy theory

We consider two categories related to symplectic manifolds: 1. Objects are symplectic manifolds and morphisms are symplectic embeddings. 2. Objects are symplectic manifolds endowed with compatible almost complex structure and morphisms are pseudoholomorphic maps. We define new homotopy theories for these categories. In particular, we give definitions of homotopy equivalent symplectic manifolds and define new cohomology theories. Theses cohomology theories are functorial, homotopy invariant and have other interesting properties. We also construct triangulated persistence category of symplectic manifolds. This allows to apply machinery developed Biran, Cornea, and Zhang and define distances between symplectic manifolds.

math.SG↗
arXiv2021

Zoo of monotone Lagrangians in $\mathbb{C}P^n$

Let $P \subset \mathbb{R}^m$ be a polytope of dimension $m$ with $n$ facets. Assume that $P$ is Delzant and Fano. We associate a monotone embedded Lagrangian $L \subset \mathbb{C}P^{n-1}$ to $P$. As an abstract manifold, the Lagrangian $L$ fibers over some torus with fiber $\mathcal{R}_P$, where $\mathcal{R}_P$ is defined by a system of quadrics in $\mathbb{R}P^{n-1}$. We find an effective method for computing the Lagrangian quantum cohomology groups of the mentioned Lagrangians. Then we construct explicitly some rich set of wide and narrow Lagrangians. Our method yields many different monotone Lagrangians with rich topological properties, including non-trivial Massey products, complicated fundamental group and complicated singular cohomology ring. Interestingly, not only the methods of toric topology can be used to construct monotone Lagrangians, but the converse is also true: the symplectic topology of Lagrangians can be used to study the topology of $\mathcal{R}_P$. General formulas for the rings $H^{*}(\mathcal{R}_P, \mathbb{Z})$, $H^{*}(\mathcal{R}_P, \mathbb{Z}_2)$ are not known. Since we have a method for constructing narrow Lagrangians, the spectral sequence of Oh can be used to study the singular cohomology ring of $\mathcal{R}_P$.

math.SG↗
arXiv2020

Monotone Lagrangian submanifolds of $\mathbb{C}^n$ and toric topology

Mironov, Panov and Kotelskiy studied Hamiltonian-minimal Lagrangians inside $\mathbb{C}^n$. They associated a closed embedded Lagrangian $L$ to each Delzant polytope $P$. In this paper we develop their ideas and prove that $L$ is monotone if and only if the polytope $P$ is Fano. In some examples, we further compute the minimal Maslov numbers. Namely, let $\mathcal{N}\to T^k$ be some fibration over the $k$-dimensional torus with a fiber equal to either $S^k \times S^l$, or $S^k \times S^l \times S^m$, or $\#_5(S^{2p-1} \times S^{n-2p-2})$. We construct monotone Lagrangian embeddings $\mathcal{N} \subset \mathbb{C}^n$ with different minimal Maslov number, and therefore distinct up to Lagrangian isotopy. Moreover, we show that some of our embeddings are smoothly isotopic but not Lagrangian isotopic.

math.SG↗
arXiv2020

Non-isotopic monotone Lagrangian submanifolds of $\mathbb{C}^n$

Let $P$ be a Delzant polytope in $\mathbb{R}^k$ with $n+k$ facets. We associate a closed Lagrangian submanifold $L$ of $\mathbb{C}^n$ to each Delzant polytope. We prove that $L$ is monotone if and only if and only if the polytope $P$ is Fano. We pose the "Lagrangian version of Delzant Theorem". Then for even $p$ and $n$ we construct $\frac{p}{2}$ monotone Lagrangian embeddings of $S^{p-1} \times S^{n-p-1} \times T^2$ into $\mathbb{C}^n$, no two of which are related by Hamiltonian isotopies. Some of these embeddings are smoothly isotopic and have equal minimal Maslov numbers, but they are not Hamiltonian isotopic. Also, we construct infinitely many non-monotone Lagrangian embeddings of $S^{2p-1} \times S^{2p-1} \times T^2$ into $\mathbb{C}^{4p}$, no two of which are related by Hamiltonian isotopies.

math.SG↗
arXiv2016

Matrix commuting differential operators of rank 2

In this paper we propose a very effective method for constructing matrix commuting differential operators of rank 2 and vector rank (2,2). We find new matrix commuting differential operators L, M of orders 2 and 2g respectively.

math-ph↗
arXiv2016

Commuting differential operators of rank 2 with rational coefficients

In this paper we find new self-adjoint commuting operators of rank 2 with rational coefficients and prove that any elliptic and hyperelliptic curves of genus 2 are spectral curves of commuting operators with rational coefficients. Also the case when curves of genus 3 are spectral curves of commuting operators with rational coefficients is studied.

nlin.SI↗
arXiv2016

AKNS hierarchy and finite-gap Schrodinger potentials

In this paper we study AKNS hierarchy. We find explicit necessary conditions for functions $p$ and $q$ to be solution of some equation of AKNS hierarchy. Then we construct finite-gap Schrodinger potential using functions $p$ and $q$.

math-ph↗
arXiv2016

Explicit characterization of some commuting differential operators of rank 2

In this paper we consider differential opeartor L=d^4_x + u(x). We find the commutativity condition for operator L with a differential operator M of order 4g+2, where L and M are operators of rank 2. Some examples are constructed. These examples don't commute with differential opeartors of odd order.

math.CA↗