Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of $A_\infty$ structures. We extend and prove Fukaya's conjecture relating this Witten's deformed equivariant de Rham complexes, to a new Morse theoretical $A_\infty$ complexes defined by counting gradient trees with jumping which are closely related to the $\mathbb{S}^1$ equivariant symplectic cohomology proposed by Siedel.

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We propose some conjectures on the generating series of (equivariant) Euler characteristics of some vector bundles constructed from the tautological bundles on Hilbert schemes of points on affine $k$-spaces. We establish the surface case of these conjectures and present some verifications of the higher dimensional cases.

The path complex and its homology were defined in the previous papers of authors. The theory of path complexes is a natural discrete generalization of the theory of simplicial complexes and the homology of path complexes provide homotopy invariant homology theory of digraphs and (nondirected) graphs. In the paper we study the homology theory of path complexes. In particular, we describe functorial properties of paths complexes, introduce notion of homotopy for path complexes and prove the homotopy invariance of path homology groups. We prove also several theorems that are similar to the results of classical homology theory of simplicial complexes. Then we apply obtained results for construction homology theories on various categories of hypergraphs. We describe basic properties of these homology theories and relations between them. As a particular case, these results give new homology theories on the category of simplicial complexes.

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