Titles and abstracts for the summer school and workshop talks can be found below.
Titles and abstracts for the summer school and workshop talks can be found below.
I will define the category of Soergel bimodules and its application to the construction of Khovanov-Rozansky homology. I will then describe a relation between this category and algebraic geometry of the Hilbert scheme of points on the plane. All notions will be introduced in lectures, no preliminary knowledge is assumed. The lectures are based on joint works with Hogancamp, Negut, Rasmussen and others.
The Witten-Reshetikhin-Turaev (WRT) invariants of 3-manifolds are constructed by using quantum groups at roots of unity. There is a unified versions of WRT invariants for integral homology 3-spheres which take values in a completion of the 1-variable integral Laurent polynomial ring such that evaluation at each root of unity gives the WRT invariant. I plan to talk about this invariant and some of its variants. I also plan to discuss some ideas towards categorification of these invariants.
In my two lectures I will review some physical approaches to categorification of WRT invariant of closed 3-manifolds. In particular I will talk about analytic continuation of WRT invariant away from roots of unity, the role of resurgence theory, relation to invariants of 3-manifold valued in q-series, and their categorification in terms of Hilbert space of a certain physical system.
In this lecture series, I'll sketch an algebraic approach to categorify quantum structures at a prime root of unity, which is aimed at eventually lifting the Witten-Reshetikhin-Turaev 3D TQFT to a 4D TQFT. The talks will be mostly introductory, with examples arising from quantum topology emphasized.
Witten-Reshetikhin-Turaev quantum invariants use only partially the representation category of quantum sl(2) at roots of unity. Hennings invariants of 3-manifolds are constructed within the quantum group itself. Unfortunately they often vanish and do not extend to full TQFT. Modifed trace theory allows construction of invariants using non trivially the non semisimple structure.
We show that for finite dimensonal unimodular Hopf algebra, modified trace is obtained from (right/left) integral (joined work with Anna Beliakova and Azat Gainutdinov).
We discuss the invariants which are obtained from modified trace in case of restricted quantum sl(2) (joined work with Anna Beliakova and Nathan Geer).
WRT invaritants are often sums of false/mock modular forms, whose modular transforms reproduce the asymptotic (1/k) expansion of the former. In this talk, we observe a ``modularity dictionary'' between modular forms and WRT invariants of various plumbed manifolds. As an example, we reverse-engineer the asymptotics of WRT invariant of a 4-singularly fibered, non-spherical Seifert manifold. This is a joint work (in progress) with Sergei Gukov, Miranda Cheng, Sarah Harrison, and Francesca Ferrari.
We know what it means to diagonalize an operator in linear algebra. What might it mean to diagonalize a functor?
Suppose you have an operator f and a collection of distinct scalars kappa_i such that Prod(f - kappa_i) = 0. Then Lagrange interpolation gives a method to construct idempotent operators p_i which project to the kappa_i-eigenspaces of f. We think of this process as diagonalization, and we categorify it: given a functor F with some additional data (akin to the collection of scalars), we construct a complete system of orthogonal idempotent functors P_i. We will give some simple but interesting examples involving modules over the group algebra of Z/2Z. The categorification of Lagrange interpolation is related to the technology of Koszul duality.
Diagonalization is incredibly important in every field of mathematics. I am a representation theorist, so I will briefly indicate some of the important applications of categorical diagonalization to representation theory and knot theory. Significantly, the "Okounkov-Vershik approach" to the representation theory of the symmetric group can be categorified in this manner. This is all joint work with Matt Hogancamp.
The Boson-Fermion correspondence is a fundamental relationship in mathematical physics relating the Fock spaces of two elementary particles, bosons and fermions. Given that bosonic and fermionic Fock spaces are irreducible representations of the Heisenberg and Clifford algebras, the correspondence yields a way of relating these two actions via the theory of vertex operators. In this talk we will explain this correspondence from an algebraic and combinatorial perspective and discuss a categorification of this relationship.
We will sketch stable homotopy refinements of Chen-Khovanov / Stroppel’s platform algebras and bimodules. This is joint work with Tyler Lawson and Sucharit Sarkar.
Tensor products of 2-representations, studied in unpublished work of Rouquier, are an essential step in categorifying the Reshetikhin-Turaev approach to 3d TQFT to obtain 4d TQFT. For 2-representations of the positive half of U_q(gl(1|1)), the tensor product has intriguing connections with extended-TQFT constructions in Heegaard Floer homology due to Douglas-Manolescu. Time permitting, we will discuss the relevant 2-representations, their tensor products, and the relationship with Heegaard Floer homology. This project is joint with Raphaël Rouquier.
Gukov, Putrov and Vafa postulated the existence of some 3-manifold invariants, called homological blocks, that take the form of power series converging in the unit disk, and whose radial limits at the roots of unity give the WRT invariants. Furthermore, these power series have integer coefficients, and should admit a categorification. An explicit formula for the homological blocks exists for negative definite plumbings. In this talk I will explain what should be the analogue of the homological blocks for manifolds with torus boundary (such as knot complements), and propose a Dehn surgery formula for these invariants. The formula is conjectural, but it can be made explicit in the case of knots given by negative definite plumbings with an unframed vertex. This is joint work (in progress) with Sergei Gukov.
I will talk about string theoretic framework relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants. This construction leads to a conjectural colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.
String theory and quantum field theory predicts the existence of a family of two-dimensional topological quantum field theories (2D TQFTs) labelled by 3-manifolds. I will survey what is known about this correspondence and discuss possible ways to explicitly construct such 2D TQFTs.
A conjecture of Dunfield-Gukov-Rasmussen predicts a family of differentials on reduced HOMFLYPT homology, indexed by the integers, that give rise to a corresponding family of reduced link homologies. We'll discuss the proof of a variant of this conjecture, constructing an unreduced link homology theory categorifying the quantum gl_n link invariant for all non-zero values of n (including negative values!). To do so, we employ the technique of annular evaluation, which uses categorical traces to define and characterize type A link homology theories in terms of simple data assigned to the unknot. Of particular interest is the case of negative n, which gives a categorification of the "symmetric webs" presentation of the type A Reshetikhin-Turaev invariant, and which produces novel categorifications thereof (i.e. distinct from the Khovanov-Rozansky theory).
This is a joint work with A. Oblomkov. Motivated by a family of B-models related to U(r) instanton moduli spaces, we construct a `categorified' representation of an affine braid group, that is, define a homomorphism from the braid group to a monoidal category of endomorphisms of a special object in the 2-category of the B-model. The category of endomorphisms is a category of equivariant matrix factorizations over an algebraic variety. When r=1, the homomorphism factors through the ordinary braid group and the `character' of the representation yields link homology.
The Boson-Fermion correspondence is a fundamental relationship in mathematical physics relating the Fock spaces of two elementary particles, bosons and fermions. Given that bosonic and fermionic Fock spaces are irreducible representations of the Heisenberg and Clifford algebras, the correspondence yields a way of relating these two actions via the theory of vertex operators. In this talk we will explain this correspondence from an algebraic and combinatorial perspective and discuss a categorification of this relationship.
Various versions of Khovanov chain complexes are built from functors from the cube to Abelian groups. By lifting these functors to the Burnside category, one can construct CW complexes whose cellular chain complexes agree with the Khovanov complexes. I will present a general outline of this construction, focusing specifically on the even and odd Khovanov homology. The even theory is joint with Tyler Lawson and Robert Lipshitz, and the odd theory is joint with Chris Scaduto and Matt Stoffregen.
The subject of hopfological algebra was introduced as a means to categorifying quantum groups and their representations at a root of unity. In particular, p-DG structures on nilHecke algebras and Lauda's 2-category have led to a categorification of quantum sl(2) at a root of unity. In this talk, we will report on some progress towards categorification of tensor products of representations of quantum sl(2) at a root of unity. This is joint work with Mikhail Khovanov and You Qi.
The celebrated boson-fermion correspondence is an isomorphism between the bosonic Fock space and the fermionic Fock space. We present categorification of the Heisenberg algebra as a modification of Khovanov's Heisenberg category. The categorifcation of the fermionic Fock space is based on Honda's category studying contact topology in dimension three.