The foam and the matrix factorization sl3 link homologies are equivalent

We prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.Comment: We have filled a gap in the proof of Lemma 5.2. 28 page

The Economics of Life ++ - Reflections on the Term of Copyright

Copyright, and indeed all intellectual property, reflects a compromise between the need for reward on creations we see – by reserving them to the creator – and the need to let information freely flow so as to permit further creations to emerge with as few encumbrances as possible. Over the past quarter century or so, all parameters of copyright have been moved towards more protection, disturbing the underlying compromise. The term of protection extends well beyond what is practically useful for the vast majority of creators, much as it may serve the needs of a small number of large players who hold important older copyrights still producing revenue. This paradoxical situation results from a few founding principles considered untouchable in the countries members of the Berne Convention: it is automatically obtained, without formality and for a uniform and rather lengthy term. If we want to redress the balance underlying copyright, we may have to call these principles into question and lead creators individually to reveal the value they attach to their right by renewing it, allowing it to lapse into the public domain when they no longer value it. Whilst this would reintroduce formalities into the structure of copyright, technological advances may make these less of a burden than they were at the time of their abolition. Alternatively, one might consider an interpretation of equitable exceptions to copyright (such as fair use and fair dealing) so as to expand them gradually as the copyright in question ages. Such approaches would have the fortunate effect of avoiding that lobbying by the happy few needlessly locks up culture for most of us. Le droit d'auteur, et à vrai dire tous les droits intellectuels, reflète un compromis entre la nécessité de faire miroiter au créateur une rémunération pour les créations que l'on voit, et la nécessité de laisser l'information circuler librement de manière à permettre à de nouvelles créations d'émerger avec aussi peu d'obstacles que possible. Au cours du dernier quart de siècle ou à peu près, tous les paramètres du droit d'auteur ont été déplacés vers plus de protection, perturbant l'équilibre sous-jacent. La durée de protection s'étend bien au-delà de ce qui est nécessaire en pratique pour la très vaste majorité des créateurs, même si elle sert bien les besoins d'une infime minorité de grands joueurs détenant des droits d'auteur qui ont un certain âge mais continuent à produire des revenus. Cette situation résulte des principes tenus pour immuables dans les pays membres de l'Union de Berne : le droit est obtenu automatiquement, sans formalité et pour une période uniforme et de longue durée. Pour redresser l'équilibre sous-jacent au droit d'auteur, il faudra remettre en question ces principes et amener les créateurs individuellement à révéler la valeur qu'ils attachent à leur droit en le renouvelant, permettant que le droit glisse dans le domaine public s'ils n'y attachent plus de valeur suffisante. S'il est vrai qu'une telle approche réintroduirait des formalités dans le droit d'auteur, les avances techniques intervenues depuis leur abolition rendent l'accomplissement de ces formalités moins onéreux que dans le temps. Alternativement, on pourrait envisager une interprétation des exceptions équitables au droit d'auteur, comme le fair use ou l'utilisation équitable, de manière à les étendre à mesure que le droit d'auteur en question prend de l'âge. De telles approches auraient l'heureux effet d'éviter que le lobbying par les happy few entrainerait le verrouillage inutile de beaucoup de culture pour le commun des mortels.Intellectual property, copyright, term, fair dealing, Propriété intellectuelle, droit d'auteur, durée, utilisation équitable.

Categorified skew Howe duality and comparison of knot homologies

In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin-Turaev invariants for $\mathfrak{sl}_n$. Over the past decade, such invariants have been constructed in a variety of different ways, using matrix factorizations, category $\mathcal{O}$, affine Grassmannians, and diagrammatic categorifications of tensor products. While the definitions of these theories are quite different, there is a key commonality between them which makes it possible to prove that they are all isomorphic: they arise from a skew Howe dual action of $\mathfrak{gl}_\ell$ for some $\ell$. In this paper, we show that the construction of knot homology based on categorifying tensor products (from earlier work of the second author) fits into this framework, and thus agrees with other such homologies, such as Khovanov-Rozansky homology. We accomplish this by categorifying the action of $\mathfrak{gl}_\ell\times \mathfrak{gl}_n$ on $\bigwedge\nolimits^{\!p}(\mathbb{C}^\ell\otimes \mathbb{C}^n)$ using diagrammatic bimodules. In this action, the functors corresponding to $\mathfrak{gl}_\ell$ and $\mathfrak{gl}_n$ are quite different in nature, but they will switch roles under Koszul duality.Comment: 62 pages. preliminary version, comments welcom

Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory

Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential.Comment: 31 pages, final version, added one reference. To apper in CM

A diagrammatic categorification of the q-Schur algebra

In this paper we categorify the q-Schur algebra S(n,d) as a quotient of Khovanov and Lauda's diagrammatic 2-category U(sln). We also show that our 2-category contains Soergel's monoidal category of bimodules of type A, which categorifies the Hecke algebra H(d), as a full sub-2-category if d does not exceed n. For the latter result we use Elias and Khovanov's diagrammatic presentation of Soergel's monoidal category of type A.Comment: 60 pages, lots of figures. v3: Substantial changes. To appear in Quantum Topolog