We explain the essence of perturbation problems. The key to understanding is the structure of chain homotopy equivalence – the standard one must be replaced by a finer notion which we call a strong chain homotopy equivalence. We formulate an Ideal Perturbation Lemma and show how both new and classical (including the Basic Perturbation Lemma) results follow from this ideal statement.

A smooth, projective surface S is called a standard isotrivial fibration if there exists a finite group G which acts faithfully on two smooth projective curves C1 and C2, so that S is isomorphic to the minimal desingularization of T := (C1 × C2)/G, where G acts diagonally on the product. When the action of G is free, then S = T is called a quasi-bundle. In this paper we analyse several numerica...

We comment on the new trend in mathematical physics that consists of obtaining Taylor series for fabricated linear and nonlinear unphysical models by means of homotopy perturbation method (HPM), homotopy analysis method (HAM) and Adomian decomposition method (ADM). As an illustrative example we choose a recent application of the HPM to a dynamic system of anisotropic elasticity.

We outline the main features of the definitions and applications of crossed complexes and cubical ωgroupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types.

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the approximate analytical solutions of fourth-order integro-differential equations. The homotopy analysis method (HAM) is one of the most effective method to obtain the exact and approximate solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ħ, ...

We present some constructions of limits and colimits in pro-categories. These are critical tools in several applications. In particular, certain technical arguments concerning strict pro-maps are essential for a theorem about étale homotopy types. Also, we show that cofiltered limits in pro-categories commute with finite colimits.

Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschet...

(1.1) H = {(x, y) ∈ (C) × C : y1y2 + p(x) = z}, Here, p : (C) → C is the superpotential mirror to Y (following [7] or [9]), and z is any regular value of p. H is an affine threefold with trivial canonical bundle. Hence, it has a Fukaya category Fuk(H), whose objects are compact exact Lagrangian submanifolds equipped with gradings and Spin structures. This is an A∞-category over C. Consider the ...

We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R3. This ‘homotopy cardinality’ is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.

This paper is the extended introduction of a series of papers about modelling T-homotopy by refinement of observation. The notion of T-homotopy equivalence is discussed. A new one is proposed and its behaviour with respect to other construction in dihomotopy theory is explained.