Related papers: Filtered A-infinity structures in complex geometry
In an earlier paper (math.SG/0110169), we introduced absolute gradings on the three-manifold invariants developed in math.SG/0101206 and math.SG/0105202. Coupled with the surgery long exact sequences, we obtain a number of three- and…
Given any finite quiver, we consider a complete flag of vector spaces over each vertex. Consider the unipotent invariant subalgebra of the coordinate ring of the filtered quiver representation subspace. We prove that the dimension of the…
The Hodge theory of complex algebraic varieties is at heart a transcendental comparison of two algebraic structures. We survey the recent advances bounding this transcendence, mainly due to the introduction of o- minimal geometry as a…
For a one-parameter degeneration of reduced compact complex analytic spaces of dimension $n$, we prove the invariance of the frontier Hodge numbers $h^{p,q}$ (that is, with $pq(n{-}p)(n{-}q)=0$) for the intersection cohomology of the fibers…
We consider random arrays indexed by the leaves of an infinitary rooted tree of finite depth, with the distribution invariant under the rearrangements that preserve the tree structure. We call such arrays hierarchically exchangeable and…
Explicit representations of complex structures on closed manifolds are valuable, but relatively rare in the literature. Using isoparametric theory, we construct complex structures on isoparametric hypersurfaces with $g=4, m=1$ in the unit…
We provide an internal characterization of those finite algebras (i.e., algebraic structures) $\mathbf A$ such that the number of homomorphisms from any finite algebra $\mathbf X$ to $\mathbf A$ is bounded from above by a polynomial in the…
We establish basic facts about the varieties of homogeneous polynomials divisible by powers of linear forms, and explain consequences for geometric complexity theory. This includes quadratic set-theoretic equations, a description of the…
We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an…
$G_\infty$-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a $B_\infty$-algebra by an associative algebra. Actions of $B_\infty$-algebras on…
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…
Arguably, the first bridge between vast, ancient, but disjoint domains of mathematical knowledge, - topology and number theory, - was built only during the last fifty years. This bridge is the theory of spectra in stable homotopy theory.…
We show that for a complete complex algebraic variety the pure component of homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce…
We give three new proofs of a theorem of C. Sabbah asserting that the weight filtration of the limit mixed Hodge structure at infinity of cohomologically tame polynomials coincides with the monodromy filtration up to a certain shift…
In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial $f$ in $\C^{n+1}$, where $f$ satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm…
We develop an obstruction theory for the existence of gauge equivalences in complete differential graded Lie algebras. Specifically, this theory provides a characterization of homotopy equivalences between differential graded algebras…
We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is…
We study the homological algebra of an R = Q/I module M using A-infinity structures on Q-projective resolutions of R and M. We use these higher homotopies to construct an R-projective bar resolution of M, Q-projective resolutions for all…
Motivated by amplitude calculations in string theory we establish basic properties of homotopy invariant iterated integrals on affine curves.
We characterize $A_\infty$-structures that are transfers over a chain homotopy equivalence or a quasi-isomorphism, answering a question posed by D. Sullivan. Along the way, we present an obstruction theory for weak $A_\infty$-morphisms over…