We construct a stable homotopy refinement of quantum annular homology, link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an $L$ naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic the as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on equivariant version Burnside category approach Laws...

We give a new proof of the non-triviality wheel graph homology classes using higher operations on Lie and derived version Koszul duality for modular operads.

A standard combinatorial construction, due to Kontsevich, associates to any A∞-algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We propose an alternative version of this construction based on noncommutative geometry and use it to prove that homotopy equivalent algebras give rise to the same cohomology ...

We introduce a new recombination operator, the Maximum Homologous Crossover for Linear Genetic Programming. In contrast to standard crossover, it attempts to preserve similar structures from parents, by aligning them according to their homology, thanks to an algorithm used in Bio-Informatics. To highlight disruptive effects of crossover operators, we introduce the Royal Road landscapes and the ...

A standard combinatorial construction, due to Kontsevich, associates to any A∞-algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We propose an alternative version of this construction based on noncommutative geometry and use it to prove that homotopy equivalent algebras give rise to the same cohomology ...

Seidel-Smith and Manolescu constructed knot homology theories using symplectic fibrations whose total spaces were certain varieties of matrices. These knot homology theories were associated to SL(n) and tensor products of the standard and dual representations. In this paper, we place their geometric setups in a natural, general framework. For any complex reductive group and any sequence of minu...

Annotating the tremendous amount of sequence information being generated requires accurate automated methods for recognizing homology. Although sequence similarity is only one of many indicators of evolutionary homology, it is often the only one used. Here we find that supplementing sequence similarity with information from biomedical literature is successful in increasing the accuracy of homol...

The difficulty in annotating the vast amounts of biological information poses one of the greatest current challenges in biological research. The number of genomic, proteomic, and metabolomic datasets has increased dramatically over the last two decades, far outstripping the pace of curation efforts. Here, we tackle the challenge of curating metabolic network reconstructions. We predict organism...