Holomorphic disks and knot invariants

'Operads are powerful tools, and this is the book in which to read about them' - ""Bulletin of the London Mathematical Society"". Operads are mathematical devices that describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of 'homotopy', where they play a key role in organizing hierarchies of higher homotopies. Significant examples from algebraic topology first appeared in the sixties, although the formal definition and appropriate generality were not forged until the seventies. In the nineties, a renaissance and further development of the theory were inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, symplectic and contact geometry, combinatorics, knot theory, moduli spaces, cyclic cohomology, and, last but not least, theoretical physics, especially string field theory and deformation quantization. The book contains a detailed and comprehensive historical introduction describing the development of operad theory from the initial period when it was a rather specialized tool in homotopy theory to the present when operads have a wide range of applications in algebra, topology, and mathematical physics. Many results and applications currently scattered in the literature are brought together here along with new results and insights. The basic definitions and constructions are carefully explained and include many details not found in any of the standard literature.

Operads in algebra, topology, and physics

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is M. Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim $X\geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

/pdf/braid-group-actions-on-derived-categories-of-coherent-3ys299b3e8.pdf

Braid group actions on derived categories of coherent sheaves

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel [29], whose triangulated structure goes back to Auslander-Reiten’s work [6], [44], [7].

/pdf/on-triangulated-orbit-categories-4k4m3u3a5b.pdf

On triangulated orbit categories.

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.

Mutation in triangulated categories and rigid Cohen–Macaulay modules

Fukaya Categories and Picard-Lefschetz Theory

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when $\dim X \geq 2$, our braid group actions are always faithful. 
We describe conjectural mirror symmetries between smoothings and resolutions of singularities that lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

La cohomologie de Floer symplectique est un groupe abelien gradue, mais seulement dans un sens relatif. La notion de sous-variete Lagrangienne graduee sert a lever cette ambiguite. On expose cette notion en details, avec plusieurs applications.

/pdf/graded-lagrangian-submanifolds-549fv81dsr.pdf

Graded lagrangian submanifolds

We consider the derived categories of modules over a certain family A_m of graded
 rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which
 are the Milnor fibres of simple singularities of type A_m. We show that each of these two
 rather different objects encodes the topology of curves on an (m+1)-punctured disc. We
 prove that the braid group B_{m+1} acts faithfully on the derived category of A_m-modules
 and that it injects into the symplectic mapping class group of Milnor fibers. The
 philosophy behind our results is as follows. Using Floer cohomology, one should be able to
 associate to the Milnor fibre a triangulated category (its construction has not been
 carried out in detail). This triangulated category should contain a full subcategory which
 is equivalent, up to a slight difference in the grading, to the derived category of
 A_m-modules. The full embedding would connect the two occurrences of the braid group, thus
 explaining the similarity between them.

/pdf/quivers-floer-cohomology-and-braid-group-actions-2d2t6bbd0j.pdf

Paul Seidel

Papers

Fukaya Categories and Picard-Lefschetz Theory

Braid group actions on derived categories of coherent sheaves

Braid group actions on derived categories of coherent sheaves

Graded lagrangian submanifolds

Quivers, Floer cohomology, and braid group actions