42 research outputs found
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Matrix Factorizations and Khovanov Homology
No full textIn this thesis we develop a geometric interpretation for Rasmussen's spectral sequences using a construction for Khovanov-Rozansky link homology developed by Oblomkov and Rozansky. In the special case of Khovanov homology, we provide a proof for the geometric construction of Rasmussen's differentials by examining the relationship between matrix factorizations and Soergel bimodules. Finally we leverage the techniques developed in order to provide an alternative method for computing the Khovanov homology of knots and links.Doctor of Philosophy (Ph.D.
Double affine Hecke algebras and noncommutative geometry
No full textThesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 93-96).In the first part we study Double Affine Hecke algebra of type An-1 which is important tool in the theory of orthogonal polynomials. We prove that the spherical subalgebra eH(t, 1)e of the Double Affine Hecke algebra H(t, 1) of type An-1 is an integral Cohen-Macaulay algebra isomorphic to the center Z of H(t, 1), and H(t, 1)e is a Cohen-Macaulay eH(t, 1)e-module with the property H(t, 1) = EndeH(t,tl)(H(t, 1)e). This implies the classification of the finite dimensional representations of the algebras. In the second part we study the algebraic properties of the five-parameter family H(tl, t2, t3, t4; q) of double affine Hecke algebras of type CVC1, which control Askey- Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. We prove that the only fiat de- formations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials. We also prove several results on the universality of the five-parameter family H(tl, t2, t3, t4; q) of algebras.by Alexei Oblomkov.Ph.D
Geometric Representation Theory and Gauge Theory
No full textThis book offers a review of the vibrant areas of geometric representation theory and gauge theory, which are characterized by a merging of traditional techniques in representation theory with the use of powerful tools from algebraic geometry, and with strong inputs from physics. The notes are based on lectures delivered at the CIME school "Geometric Representation Theory and Gauge Theory" held in Cetraro, Italy, in June 2018. They comprise three contributions, due to Alexander Braverman and Michael Finkelberg, Andrei Negut, and Alexei Oblomkov, respectively. Braverman and Finkelberg’s notes review the mathematical theory of the Coulomb branch of 3D N=4 quantum gauge theories. The purpose of Negut’s notes is to study moduli spaces of sheaves on a surface, as well as Hecke correspondences between them. Oblomkov's notes concern matrix factorizations and knot homology. This book will appeal to both mathematicians and theoretical physicists and will be a source of inspiration for PhD students and researchers
HOMFLY homology vs homology
No full textNon UBCUnreviewedAuthor affiliation: University of MassachusettsFacult
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Skein Theory and Algebraic Geometry for the Two-Variable Kauffman Invariant of Links
No full textWe conjecture a relationship between the Hilbert schemes of points on a singular plane curve and the Kauffman invariant of the link associated to the singularity. Specifcally, we conjecture that the generating function of certain weighted Euler characteristics of the Hilbert schemes is given by a normalized specialization of the difference between the Kauffman and HOMFLY polynomials of the link. We prove the conjecture for torus knots. We also develop some skein theory for computing the Kauffman polynomial of links associated to singular points on plane curves.MathematicsDoctor of Philosophy (Ph.D.
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Topology of the Affine Springer Fiber in Type A
No full textWe develop algorithms for describing elements of the affine Springer fiber in type A for certain 2 g(C[[t]]). For these , which are equivalued, integral, and regular, it is known that the affine Springer fiber, X, has a paving by affines resulting from the intersection of Schubert cells with X. Our description of the elements of Xallow us to understand these affine spaces and write down explicit dimension formulae. We also explore some closure relations between the affine spaces and begin to describe the moment map for the both the regular and extended torus action.MathematicsDoctor of Philosophy (PhD
3D TQFT and HOMFLYPT homology
No full textWe describe a family of 3d topological B-models whose target spaces are
Hilbert schemes of points in . The interfaces separating theories
with different numbers of points correspond to braid strands. The Hilbert space
of the picture of a closed braid is the HOMFLY-PT homology of the corresponding
link.Comment: 57 pages, many figures, section 2, that discusses physics theories,
is expande
Categorical Chern character and braid groups
No full textTo a braid we associate a complex of sheaves on
such that the previously defined triply graded link homology of
the closure is isomorphic to the homology of .
The construction of relies on the Chern functor
defined in the paper
together with its adjoint functor
.
We prove a
formula for the closure of sufficiently positive elements of the Jucys-Murphy
algebra previously conjectured
by Gorsky, Negut and Rasmussen.Comment: 55 pages, no figures; many proofs and definitions are expanded; the
introductory sections on matrix factorizations are adde
