Moduli stacks of algebraic structures and deformation theory

We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's homotopical algebraic geometry. This includes simplicial moduli spaces of algebraic structures over a given object (for instance a cochain complex). When these algebraic structures are parametrised by properads, the tangent complexes give the known cohomology theory for such structures and there is an associated obstruction theory for infinitesimal, higher order and formal deformations. The methods are general enough to be adapted for more general kinds of algebraic structures.Comment: several corrections, especially in sections 6 and 7. Final version, to appear in the J. Noncommut. Geo

Persistence for Circle Valued Maps

We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real valued maps, circle valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle valued map on an input simplicial complex.Comment: A complete algorithm to compute barcodes and Jordan cells is provided in this version. The paper is accepted in in the journal Discrete & Computational Geometry. arXiv admin note: text overlap with arXiv:1210.3092 by other author

A spectral sequence for spaces of maps between operads

We construct a tower of fibrations approximating the derived mapping space between two simplicially enriched operads subject to mild conditions. The n-th stage of the tower is obtained by neglecting operations with more than n inputs. The main theorem describes the layers of this tower.Comment: v2: some typos corrected, some simplifications, bibliography improve