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Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for…

A constructive procedure to obtain superintegrable deformations of the classical Smorodinsky-Winternitz Hamiltonian by using quantum deformations of its underlying Poisson sl(2) coalgebra symmetry is introduced. Through this example, the…

We use spectral invariants in Lagrangian Floer theory in order to show that there exist \emph{isometric} embeddings of normed linear spaces (finite or infinite dimensional, depending on the case) into the space of Hamiltonian deformations…

We use a decomposition of the tensor of the fundamental representation of the quantum group $U_q(\mathfrak{sl}_N)$ and the Rosso-Jones formula to establish a peculiar ``panhandle'' shape of the HOMFLY-PT polynomial of the reverse parallel…

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…

It is shown that the $\mathfrak{gl}(3)$ polynomial integrable system, introduced by Sokolov-Turbiner in [arXiv:1409.7439], is equivalent to the $\mathfrak{gl}(3)$ quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian as…

The symmetric group on 4 letters has the reflection group $D_{3}$ as an isomorphic image. This fact follows from the coincidence of the root systems $A_{3}$ and $D_{3}$. The isomorphism is used to construct an orthogonal basis of…

We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic…

The anomaly polynomial of a theory can involve not only curvature two-forms of the flavor symmetry background but also two-forms on the space of coupling constants. As an example, we point out that there is a mixed anomaly between the…

By using invariant theory we show that a (higher-dimensional) Lorentzian metric that is not characterised by its invariants must be of aligned type II; i.e., there exists a frame such that all the curvature tensors are simultaneously of…

In this paper we are concerned with the existence of invariant tori in nearly integrable Hamiltonian systems \begin{equation*} H=h(y)+f(x,y,t), \end{equation*} where $y\in D\subseteq\mathbb{R}^n$ with $D$ being a closed bounded domain,…

A polynomial automorphism of $\mathbb{A}^n$ over a field of characteristic zero is called co-tame if, together with the affine subgroup, it generates the entire tame subgroup. We prove some new classes of automorphisms, including…

We derive an inductive, combinatorial definition of a polynomial-valued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the Reshetikhin-Turaev invariant corresponding to the exceptional simple Lie…

The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a…

We collect the recent results on invariant f-structures in the generalized Hermitian geometry. Here the canonical f-structures on homogeneous k-symmetric spaces play a remarkable role. Specifically, these structures provide a wealth of…

This paper is a sequel to "Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians", in which we gave a new construction of resonant normal forms with an exponentially small remainder for near-integrable Gevrey…

A $\gamma$-deformed version of $\mathfrak{su}(2)$ algebra has been obtained from a bi-orthogonal system of vectors in $\bf{C^2}$. Fusion of Jordan-Schwinger realization of complexified $\mathfrak{su}(2)$ with Dyson-Maleev representation…

Motivated by the shape invariance condition in supersymmetric quantum mechanics, we develop an algebraic framework for shape invariant Hamiltonians with a general change of parameters. This approach involves nonlinear generalizations of Lie…

We consider manifolds of oriented flags SO(n)/SO(2)xSO(n-3) (n>=4) as 4- and 6-symmetric spaces and indicate characteristic conditions for invariant Riemannian metrics under which the canonical f-structures on these homogeneous…

Let T be a tilting object in a triangulated category equivalent to the bounded derived category of a hereditary abelian category with finite dimensional homomorphism spaces and split idempotents. This text investigates the strong global…