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Mikhail Khovanov, Columbia University, Categorifications of natural numbers, integers and fractions
February 8, 2018 | 4:30 pm - 5:30 pm EST
Categorification lifts natural numbers to vector spaces and integers to complexes. Natural number n becomes a vector space of dimension n, and an integer becomes the Euler characteristic of a complex of vector spaces. A well-known example of categorification is lifting the Euler characteristic of a topological space to its homology or cohomology groups. The talk will discuss this and other examples of categorification and will be accessible to graduate students. At the end we’ll consider the challenges and progress in extending the framework of categorification to rational numbers.
Short Bio
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Mikhail Khovanov is a recognized leader of the “categorification” program, which plays an important role in modern mathematics and physics. Khovanov earned his Ph.D. in 1997 under the supervision of Professor Igor Frenkel at Yale University. Shortly thereafter he came up with his famous idea of categorifying the Kauffman bracket, which is a version of the celebrated Jones polynomial of links in a 3-sphere. This was the first example of the categorification which interprets polynomial invariants as Poincare polynomials of new homology theories. The construction of Khovanov homology was amazingly fruitful and very unexpected.
A further categorification of the HOMFLY-PT polynomial of links and a categorification of quantum groups are other major achievments of Khovanov which have now important implications in low-dimensional topology, algebraic and symplectic geometry, geometric representation theory and string theory.
Mikhail Khovanov is a recognized leader of the “categorification” program, which plays an important role in modern mathematics and physics. Khovanov earned his Ph.D. in 1997 under the supervision of Professor Igor Frenkel at Yale University. Shortly thereafter he came up with his famous idea of categorifying the Kauffman bracket, which is a version of the celebrated Jones polynomial of links in a 3-sphere. This was the first example of the categorification which interprets polynomial invariants as Poincare polynomials of new homology theories. The construction of Khovanov homology was amazingly fruitful and very unexpected.
A further categorification of the HOMFLY-PT polynomial of links and a categorification of quantum groups are other major achievments of Khovanov which have now important implications in low-dimensional topology, algebraic and symplectic geometry, geometric representation theory and string theory.