Representation Theory
and Combinatorics of Torus Links
University of Massachusetts, Amherst • July 8, 2017
Schedule and abstracts of talks
All talks will be held in Lederle Graduate Research Tower (LGRT) 1634 (16th floor).Saturday, July 8
8:30am–9:25am
Registration
9:25am
Welcoming remarks
9:30am–10:30am
Erik Carlsson
(UC Davis)
(UC Davis)
Geometry behind the shuffle conjecture
I will explain the construction of the algebra introduced recently by Anton Mellit and myself in our proof of the shuffle conjecture, which has many elements in common with DAHA's, and has recently been finding applications to knot invariants. I will then describe a current project with Eugene Gorsky and Mellit, in which we have discovered a geometric description of this algebra via the torus-equivariant K-theory of a certain smooth subscheme of the flag Hilbert scheme, which parametrizes flags of ideals of finite codimension in C[x,y].
I will explain the construction of the algebra introduced recently by Anton Mellit and myself in our proof of the shuffle conjecture, which has many elements in common with DAHA's, and has recently been finding applications to knot invariants. I will then describe a current project with Eugene Gorsky and Mellit, in which we have discovered a geometric description of this algebra via the torus-equivariant K-theory of a certain smooth subscheme of the flag Hilbert scheme, which parametrizes flags of ideals of finite codimension in C[x,y].
10:30am
Coffee break
11:00am–12:00pm
Matthew Hogencamp (USC)
How to compute some link homologies
I will discuss a technique, introduced in joint work with Ben Elias, for computing triply graded Khovanov-Rozansky homology. Using this technique, Anton Mellit recently computed the homology of all positive torus knots, extending earlier work of myself and Elias.
I will discuss a technique, introduced in joint work with Ben Elias, for computing triply graded Khovanov-Rozansky homology. Using this technique, Anton Mellit recently computed the homology of all positive torus knots, extending earlier work of myself and Elias.
12:00pm–2:00pm
Lunch
2:00pm–3:00pm
Lev Rozansky (UNC)
HOMFLY-PT link homology and Langlands-related 3-category.
This is a joint work with A. Oblomkov. First, I will explain how we use two categories of equivariant matrix factorizations in order to categorify ordinary and affine Hecke algebras and Ocneanu trace. Second, I will explain that these are categories of endomorphisms of special objects within 2-categories related to the commuting variety and to the Hilbert scheme of points on C2. Third, I will argue that these 2-categories are 2-categories associated to a circle (with marked points) on the B-side of the geometric Langlands duality. Fourth, I will argue that our categories of matrix factorizations are (almost) centers of G-equivariant 2-categories related to the product of two cotangent bundles to flag varieties, and the equivalence of two descriptions is a manifestation of the twice categorified Riemann-Roch-type relation.
This is a joint work with A. Oblomkov. First, I will explain how we use two categories of equivariant matrix factorizations in order to categorify ordinary and affine Hecke algebras and Ocneanu trace. Second, I will explain that these are categories of endomorphisms of special objects within 2-categories related to the commuting variety and to the Hilbert scheme of points on C2. Third, I will argue that these 2-categories are 2-categories associated to a circle (with marked points) on the B-side of the geometric Langlands duality. Fourth, I will argue that our categories of matrix factorizations are (almost) centers of G-equivariant 2-categories related to the product of two cotangent bundles to flag varieties, and the equivalence of two descriptions is a manifestation of the twice categorified Riemann-Roch-type relation.
3:00pm–4:00pm
Paul Wedrich
(Imperial College London)
(Imperial College London)
Link homologies in thickened surfaces and categorification of skein algebras
Khovanov-Rozansky link homologies can be extended to give functorial invariants of links in thickened surfaces, which categorify the evaluation of links in type A skein algebras. An interesting question is whether such link homology functors can be made monoidal, so as to categorify the skein algebra multiplication as well. Evidence that (and hints on how) this should be possible exists in the form of certain positivity phenomena in skein algebras. I will talk about this story and report on work in progress with Hoel Queffelec that aims towards categorifying the gl(2) skein algebra of the torus.
Khovanov-Rozansky link homologies can be extended to give functorial invariants of links in thickened surfaces, which categorify the evaluation of links in type A skein algebras. An interesting question is whether such link homology functors can be made monoidal, so as to categorify the skein algebra multiplication as well. Evidence that (and hints on how) this should be possible exists in the form of certain positivity phenomena in skein algebras. I will talk about this story and report on work in progress with Hoel Queffelec that aims towards categorifying the gl(2) skein algebra of the torus.
4:00pm
Coffee break
4:30pm–5:30pm
Anton Mellit
(IST Austria)
(IST Austria)
Motivic knot invariants and corrected HOMFLY-positivity
I will study knot invariants which arise naturally from convolutions of sheaves on GLn and whose values are algebraic varieties whose cohomology conjecturally coincides with Khovanov-Rozansky homology (up to a certain inclusion-exclusion summation in the link case). These invariants are interesting partly because they can be directly related to character varieties and conjectures of Hausel-Letellier-Villegas. Experiments show that for arbitrary products of Jucys-Murphy elements these invariants turn out to be computable by an extension of Elias-Hogancamp method. In particular, the cohomology turns out to be concentrated only in even degrees, and consequently certain linear combinations of HOMFLY polynomials are positive.
I will study knot invariants which arise naturally from convolutions of sheaves on GLn and whose values are algebraic varieties whose cohomology conjecturally coincides with Khovanov-Rozansky homology (up to a certain inclusion-exclusion summation in the link case). These invariants are interesting partly because they can be directly related to character varieties and conjectures of Hausel-Letellier-Villegas. Experiments show that for arbitrary products of Jucys-Murphy elements these invariants turn out to be computable by an extension of Elias-Hogancamp method. In particular, the cohomology turns out to be concentrated only in even degrees, and consequently certain linear combinations of HOMFLY polynomials are positive.
List of Participants
Coming soon.
