We propose a generalization of a conjecture of D. Quillen, on the vanishing of André-Quillen homology, to simplicial commutative rings. This conjecture characterizes a notion of local complete intersection, extended to the simplicial setting, under a suitable hypothesis on the local characteristic. Further, under the condition of finite-type homology, we then prove the conjecture in the case of...

Dynamic programming [1] has full sensitivity, but too slow for large scale homology search. FASTA / BLAST type of heuristics [2] trade sensitivity for speed. Can we have both sensitivity and speed? We present the mathematical theory of optimized spaced seeds which allows modern homology search to achieve high sensitivity and high speed simultaneously. The spaced seed methodology is implemented ...

Given a Heegaard splitting of a 3-manifold, we use Lagrangian Floer homology to construct a relatively Z/8Z-graded abelian group, which we conjecture to be a 3-manifold invariant. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary 3-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU(2)-connectio...

Abstract. This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2–dimensional cobordisms by taking into account their embedding into R. Secondly, we extend the underlying cobordism category to a 2–category, where the usual relations hold up to ...

In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is weighted and signed count bars module, in which bar form [a,b) degree d counted with weight (e−a−e−b) sign (−1)d. Persistent magnitude has good formal properties, such as additivity respect to exact sequences compatibility tensor products, interpretations terms...