arXiv subjects
Publications and source records attributed to Timur Kenzhaev.
We give a full and detailed solution of eighth Arnold's trivium problem. We find critical points of a smooth function on a given one-parametric two-dimensional surface with Lagrange multipliers method. Basic Morse theory and Poincare-Hopf theorem makes it possible to determine a genus of this surface and gives a beautiful training example of Morse surgery on a two-dimensional surface of genus $g = 6$.
We construct the Feigin-Stoyanovsky (combinatorial) basis in case of one-dimensional lattice vertex superalgebras $V_{\sqrt{N}\,\mathbb{Z}}$. Our proof is based on invariance of semi-infinite monomials linear span under action of corresponding Heisenberg algebra. Semi-infinite monomials are parametrized by natural generalization of Maya diagrams $\unicode{x2013}$ Fibonacci configurations on $\mathbb{Z}$, which allows us to construct a desired basis with character considerations. We also discuss some related questions such as functional realization of basic subspace's dual and representational proof of Feigin-Stoyanovsky construction in case of $V_{\sqrt{2}\,\mathbb{Z}}$.
We introduce a natural generalization of Maya diagrams -- the space of infinite Fibonacci configurations, which are specified functions on $\mathbb{Z}$ with values $1$ and $0$. Infinite Fibonacci configurations are particularly interesting as soon as they parametrize Feigin-Stoyanovsky type bases in lattice vertex superalgebras $V_{\sqrt{N}\mathbb{Z}}$ and their irreducible modules. We calculate the character of such configurations space by two different ways and obtain series of combinatorial identities. These identities turn out to be Durfee rectangle identities with shifts in base and height.
B. Feigin and A. Stoyanovsky found the basis of semi-infinite monomials in standard $\widehat{\mathfrak{sl}}_2'$-module $L_{(0, 1)}$ with Lefschetz formula for the corresponding flag variety. These semi-infinite monomials are constructed by modes of the current $e(z) = \sum\limits_{n\in\mathbb{Z}} e_n\,z^{- n - 1}$. We give an alternative proof of this fact using explicit fermionic construction of this module. Namely, we realize $L_{(0, 1)}$ inside of the zero-charge subspace of Fermionic Fock space and show linear independence of vectors corresponding to semi-infinite monomials.