An operad (this paper deals with non-symmetric operads)may be conceived as a partial algebra with a family of insertion operations, Gerstenhaber's circle-i products, which satisfy two kinds of associativity, one of them involving commutativity. A Cat-operad is an operad enriched over the category Cat of small categories, as a 2-category with small hom-categories is a category enriched over Cat. The notion of weak Cat-operad is to the notion of Cat-operad what the notion of bicategory is to the notion of 2-category. The equations of operads like associativity of insertions are replaced by isomorphisms in a category. The goal of this paper is to formulate conditions concerning these isomorphisms that ensure coherence, in the sense that all diagrams of canonical arrows commute. This is the sense in which the notions of monoidal category and bicategory are coherent. The coherence proof in the paper is much simplified by indexing the insertion operations in a context-independent way, and not in the usual manner. This proof, which is in the style of term rewriting, involves an argument with normal forms that generalizes what is established with the completeness proof for the standard presentation of symmetric groups. This generalization may be of an independent interest, and related to matters other than those studied in this paper. Some of the coherence conditions for weak Cat-operads lead to the hemiassociahedron, which is a polyhedron related to, but different from, the three-dimensional associahedron and permutohedron.

We consider a modification of GR with a special type of a non-local f(R). The structure of the non-local operators is motivated by the string field theory and p-adic string theory. The spectrum is derived explicitly and the ghost-free condition for the model is formulated. We pay special attention to the classical stability of the de Sitter solution in our model and formulate the conditions on the model parameters to have a stable configuration. Relevance of unstable configurations for the description of the de Sitter phase during inflation is specifically discussed.

Here, we study the genesis and evolution of geometric ideas and techniques in investigations of movable singularities of algebraic ordinary differential equations. This leads us to the work of Mihailo Petrovic on algebraic differential equations and in particular his geometric ideas captured in his polygon method from the last years of the XIXth century, which have been left completely unnoticed by the experts. This concept, also developed in a bit a different direction and independently by Henry Fine, generalizes the famous Newton-Puiseux polygonal method and applies to algebraic ODEs rather than algebraic equations. Although remarkable, the Petrovic legacy has been practically neglected in the modern literature, while the situation is less severe in the case of results of Fine. Thus, we study the development of the ideas of Petrovic and Fine and their places in contemporary mathematics.

We study triangles and quadrilaterals which are inscribed in a circle and circumscribed about a parabola. Although these are particular cases of the celebrated Poncelet's Theorem, in this paper we {\it do not assume} the theorem but prove it along the way. Similarly, our arguments here \emph{are logically independent} from Cayley's condition, describing points of a finite order on an elliptic curve or any other use of the theory of elliptic curves. Instead, we use purely planimetric methods, including the Joachimsthal notation, to fully describe such polygons and associated circles and parabolas. We prove that a circle contains the focus of a parabola if and only if there is a triangle inscribed in the circle and circumscribed about the parabola. We prove that if the center of a circle coincides with the focus of a parabola, then there exists a quadrilateral inscribed in the circle and circumscribed about the parabola. We further prove that the quadrilaterals obtained in such a way are antiparallelograms. If the center of a circle does not coincide with the focus of a parabola, then a quadrilateral inscribed in the circle and circumscribed about the parabola exists if and only if the directrix of the parabola contains the point of intersection of the polar of the focus with respect to the circle with the line determined by the center and the focus. That point coincides with the intersection of the diagonals of any quadrilateral inscribed in the circle and circumscribed about the parabola. In particular, for a given circle and a confocal pencil of parabolas with the focus different than the center of the circle, there is a unique parabola for which there exists a quadrilateral circumscribed about it and inscribed in the circle.

We solve the Dirichlet problem $\left.u\right|_{\mathbb{B}^n}=\varphi,$ for hyperbolic Poisson's equation $\Delta_h u=\mu$ where $\varphi\in L_1(\partial \mathbb{B}^n)$and$\mu$ is a measure that satisfies a growth condition. Next we present a short proof for Lipschitz continuity of solutions of certain hyperbolic Poisson's equations, previously established at \cite{ChenRas}. In addition, we investigate some alternative assumptions on hyperbolic Laplacian, which are connected with Riesz's potential. Also, local H\"{o}lder continuity is proved for solution of certain hyperbolic Poisson's equations. We show that, if $u$ is hyperbolic harmonic in the upper half-space, then $\frac{\partial u}{\partial y}(x_0,y)\to 0, y\to 0^+$, when boundary function $f$ of the functions $u$ is differentiable at the boundary point $x_0$. As a corollary, we show $C^1(\overline{\mathbb{H}^n})$ smoothness of a hyperbolic harmonic function, which is reproduced from the $C_c^1(\mathbb{R}^{n-1})$ boundary values.

We propose a generalized version of knots-quivers correspondence, where the quiver series variables specialize to arbitrary powers of the knot HOMFLY-PT polynomial series variable. We explicitely compute quivers for large classes of knots, as well as many homologically thick 9- and 10-crossings knots, including the ones with the super-exponential growth property of colored HOMFLY-PT polyomials. In addition, we propose a new, compact, quiver-like form for the colored HOMFLY-PT polynomials, where the structure of colored differentials is manifest. In particular, this form partially explains the non-uniqueness of quivers corresponding to a given knot via knots-quivers correspondence.

The generalized knots-quivers correspondence extends the original knots-quivers correspondence, by allowing higher level generators of quiver generating series. In this paper we explore the underlined combinatorics of such generating series, relationship with the BPS numbers of a corresponding knot, and new combinatorial interpretations of the coefficients of generating series.

In this work, we introduce a novel concept of magic billiards, which can be seen as an umbrella, unifying several well-known generalisations of mathematical billiards. We analyse properties of magic billiards in the case of elliptical boundaries. We provide explicit conditions for periodicity in algebro-geometric, analytic, and polynomial forms. A topological description of those billiards is given using Fomenko graphs.

We study pairs of conics $(\mathcal{D},\mathcal{P})$, called \textit{$n$-Poncelet pairs}, such that an $n$-gon, called an \textit{$n$-Poncelet polygon}, can be inscribed into $\mathcal{D}$ and circumscribed about $\mathcal{P}$. Here $\mathcal{D}$ is a circle and $\mathcal{P}$ is a parabola from a confocal pencil $\mathcal{F}$ with the focus $F$. We prove that the circle contains $F$ if and only if every parabola $\mathcal{P}\in\mathcal{F}$ forms a $3$-Poncelet pair with the circle. We prove that the center of $\mathcal{D}$ coincides with $F$ if and only if every parabola $\mathcal{P}\in \mathcal{F}$ forms a $4$-Poncelet pair with the circle. We refer to such property, observed for $n=3$ and $n=4$, as \textit{$n$-isoperiodicity}. We prove that $\mathcal{F}$ is not $n$-isoperiodic with any circle $\mathcal{D}$ for $n$ different from $3$ and $4$. Using isoperiodicity, we construct explicit algebraic solutions to Painlevé VI equations.

The celebrated Poncelet porism is usually studied for a pair of smooth conics that are in a general position. Here we discuss Poncelet porism in the real plane - affine or projective, when that is not the case, i.e. the conics have at least one point of tangency or at least one of the conics is not smooth. In all such cases, we find necessary and sufficient conditions for the existence of an n-gon inscribed in one of the conics and circumscribed about the other.

We prove the integrability of magnetic geodesic flows of $SO(n)$--invariant Riemannian metrics on the rank two Stefel variety $V_{n,2}$ with respect to the magnetic field $\eta\, d\alpha$, where $\alpha$ is the standard contact form on $V_{n,2}$ and $\eta$ is a real parameter. Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for $SO(n)$-invariant sub-Riemannian structures on $V_{n,2}$. All statements in the limit $\eta=0$ imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by $SO(n)\times SO(2)$--invariant Riemannian metrics. For $n=3$, using the isomorphism $V_{3,2}\cong SO(3)$, the obtained integrable magnetic models reduce to integrable cases of a motion of a heavy rigid body with a gyrostat around a fixed point: Zhukovskiy--Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevski top with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrange gyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).

We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schröder paths. We determine generating functions of such paths, located in a region determined by a type of a torus knot under consideration, and show that they encode colored HOMFLY-PT polynomials of this knot. The generators of uncolored HOMFLY-PT homology correspond to a basic set of such paths. Invoking the knots-quivers correspondence, we express generating functions of such paths as quiver generating series, and also relate them to quadruply-graded knot homology. Furthermore, we determine corresponding A-polynomials, which provide algebraic equations and recursion relations for generating functions of generalized Schröder paths. The lattice paths of our interest explicitly enumerate BPS states associated to knots via brane constructions.

This paper is a review of recent and classical results on integrable geodesic flows on Riemannian manifolds and topological obstructions to integrability. We also discuss some open problems.

We consider a family of genus $g$ hyperelliptic curves as double ramified coverings over the Riemann sphere with the set of branch points of the form $\{0, \infty, x_1, \dots, x_g, u_1, \dots, u_g\}$. The branch point at infinity $P_\infty$ is selected to be a marked point on the Riemann surfaces. A meromorphic differential $\Omega$ with a unique pole being of order two at $P_\infty$, is completely defined by the values of half of its periods, the $a$-periods. Fixing values of $a$-periods of $\Omega$, we then find a continuous subfamily in the considered family of hyperelliptic curves along which all the periods of $\Omega$ are constant. This subfamily is defined by the functions $u_j(x_1, \dots, x_g)$, while $x_1, \dots, x_g$ are independent parameters. We derive a system of differential equations for the functions $u_j(x_1, \dots, x_g)$, which, remarkably, has rational coefficients. We call this subfamily the isoperiodic deformations of the hyperelliptic curves relative to the given differential of the second kind $\Omega.$ We deduce necessary and sufficient conditions for the existence and uniqueness of isoperiodic deformations. We discuss reality conditions as well. Using the obtained results, we solve the following problem for the Korteweg-de Vries and sine-Gordon equations: starting from an algebro-geometric data which generate a real periodic solution of a period $T$, how to deform the data, so that the associated solutions remain periodic with the same period $T$.

Positive modalities in systems in the vicinity of S4 and S5 are investigated in terms of categorial proof theory. Coherence and maximality results are demonstrated, and connections with mixed distributive laws and Frobenius algebras are exhibited.

A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of $\sigma$-functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of $n$-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were "virtual", while in the present case the resulting bundles are effectively realizable.

We generalize the $F_K$ invariant, i.e. $\widehat{Z}$ for the complement of a knot $K$ in the 3-sphere, the knots-quivers correspondence, and $A$-polynomials of knots, and find several interconnections between them. We associate an $F_K$ invariant to any branch of the $A$-polynomial of $K$ and we work out explicit expressions for several simple knots. We show that these $F_K$ invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using $R$-matrices. We generalize the quantum $a$-deformed $A$-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter $a$, and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide $t$-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d $\mathcal{N}=2$ theory $T[M_3]$ and to the data of the associated modular tensor category $\text{MTC} [M_3]$.