On Operadic Actions on Spaces of Knots and 2-Links

In the present work, we realize the space of string 2-links $\mathcal{L}$ as a free algebra over a colored operad denoted $\mathcal{SCL}$ (for "Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of $\mathcal{L}$ by its center and is compatible with Budney's freeness theorem for long knots. From an algebraic point of view, our main result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy classes of string links. Topologically, it expresses the homotopy type of the isotopy class of a string 2-link in terms of the homotopy types of the classes of its prime factors.Comment: Comments are welcom

Superpotential algebras and manifolds

In this paper we study a special class of Calabi-Yau algebras (in the sense of Ginzburg): those arising as the fundamental group algebras of acyclic manifolds. Motivated partly by the usefulness of `superpotential descriptions' in motivic Donaldson-Thomas theory, we investigate the question of whether these algebras admit superpotential presentations. We establish that the fundamental group algebras of a wide class of acyclic manifolds, including all hyperbolic manifolds, do not admit such descriptions, disproving Ginzburg's conjecture regarding them. We also describe a class of manifolds that do admit such descriptions, and discuss a little their motivic Donaldson-Thomas theory. Finally, some links with topological field theory are described.Comment: 31 pages, 2 figures, final version. Thanks to M. Kontsevich, V. Ginzburg, M, Van den Bergh and B. Keller for helpful comments and corrections. I've added some examples e.g. Klein bottl

The Tilting Theory of Contraction Algebras

To every minimal model of a complete local isolated cDV singularity Donovan--Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these algebras and then use the structure of an associated hyperplane arrangement to control the compositions, obtaining a faithful group action on the bounded derived category. Further, we determine precisely those standard equivalences which are induced by two-term tilting complexes and show that any standard equivalence between contraction algebras (up to algebra isomorphism) can be viewed as the composition of our constructed functors. Thus, for a contraction algebra, we obtain a complete picture of its derived equivalence class and, in particular, of its derived autoequivalence group.Comment: 36 pages, proof of Lemma 4.11 corrected and other minor change

On properties of modeling control software for embedded control applications with CSP/CT framework

This PROGRESS project (TES.5224) traces a design framework for implementing embedded real-time software for control applications by exploiting its natural concurrency. The paper illustrates the stage of yielded automation in the process of structuring complex control software architectures, modeling controlled mechatronic systems and designing corresponding control laws, simulating them, generating control code out of simulated control strategy and implementing the software system on a (embedded) computer. The gap between the development of control strategies and the procedures of implementing them on chosen hardware targets is going to be overcome