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Mikhail Khovanov

Bio: Mikhail Khovanov is an academic researcher from Columbia University. The author has contributed to research in topics: Categorification & Functor. The author has an hindex of 44, co-authored 119 publications receiving 8334 citations. Previous affiliations of Mikhail Khovanov include University of California & University of California, Davis.


Papers
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TL;DR: In this article, Khovanov et al. constructed a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial, and proved that it is the case for all links.
Abstract: Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.

1,123 citations

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TL;DR: Khovanov et al. as mentioned in this paper constructed a doubly-graded homology theory of links with the Euler characteristic, which is based on matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.
Abstract: Author(s): Khovanov, Mikhail; Rozansky, Lev | Abstract: For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.

715 citations

Journal ArticleDOI
TL;DR: In this article, the derived categories of modules over a certain family of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which contain the Milnor fibres of simple singularities of type A_m were considered, and it was shown that each of these two objects encodes the topology of curves on an (m+1)-punctured disc.
Abstract: 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.

351 citations

Journal ArticleDOI
TL;DR: In this article, a complex of big-raded vector spaces is assigned to an oriented link as the closure of a braid and the Euler characteristic of this complex is the HOMFLYPT polynomial of the link.
Abstract: To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.

302 citations


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TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

1,479 citations

Journal ArticleDOI
TL;DR: In this article, Khovanov et al. constructed a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial, and proved that it is the case for all links.
Abstract: Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.

1,123 citations

Journal ArticleDOI
TL;DR: In this article, a Floer-homology invariant for knots in an oriented three-manifold, closely related to the Heegaard Floer homologies for threemanifolds defined in an earlier paper, was defined.

915 citations