Sabin Cautis
The higher structure of vertex operators
We will sketch a 2-category which captures the higher structure
of the Heisenberg algebra associated to an affine Dynkin diagram.
One can then define complexes of 1-morphisms in this 2-category
which satisfy the relations in the corresponding quantum affine
Lie algebra. This lifts the "vertex operator" construction
of Frenkel-Kac-Segal from vector spaces to categories. (joint work
with Tony Licata)
Ben Cooper
Generalized Hecke Algebras
:(Work in preparation). Hecke algebras have been closely tied to
invariants of knots and links since the foundational work of Jones.
I will describe new, geometrically defined, families of these algebras
and explain how they can be used to enrich the story surrounding
quantum invariants.
Ben Elias
Categorical actions of Coxeter groups and Hecke algebras
Actions of Coxeter groups and their braid groups by functorial equivalences
are quite common in geometry and representation theory. Such an
action is "strict" if it can be equipped with a compatible
system of natural transformations corresponding to composition of
functors. We present here an alternative way to view/check strictness,
by constructing a presentation of the Coxeter group or braid group
(as a monoidal category!) by generators and relations. This should
be thought of as a "higher presentation" of a Coxeter
group, and it is related to the topology of the (dual) Coxeter complex.
We then continue this approach to discuss categorical actions of
Hecke algebras, and a presentation of the Hecke category by generators
and relations. This is joint work with Geordie Williamson.
David Hill
Categorification of Kac-Moody Superalgebras
We categorify one half of a quantum Kac-Moody superalgebra with
non-isotropic odd roots.
David Jordan
Quantized multiplicative quiver varieties
We introduce a new class of algebras $D_q(Mat_d(Q))$ associated
to a quiver $Q$ and dimension vector $d$, which yield a flat (PBW)
$q$-deformation of the algebra of differential operators on the
space of matrices associated to $Q$. This algebra admits a $q$-deformed
moment map from the quantum group $U_q(gl_d)$, acting by base change
at each vertex. The quantum Hamiltonian reduction, $A^\xi_d(Q)$,
of $D_q$ by $\mu_q$ at the character $\xi$, is simultaneously a
quantization of the Crawley-Boevey and Shaw's multiplicative quiver
variety, and a $q$-deformation of Gan and Ginzburg's quantized quiver
variety.
Specific instances of the data $(Q,d,\xi)$ yield $q$-deformations
of familiar algebras in representation theory: for example, the
spherical DAHA's of type $A$ arise from Calogero-Moser quivers,
quantizations of parabolic character varieties (Deligne-Simpson
moduli spaces) arise from comet-shaped quivers, and algebras of
difference operators on Kleinian singularities arise from affine
Dynkin quivers.
Carl Mautner
An Auto-Equivalence of the Equivariant Derived Category of
a Nilpotent Cone
Let G be a complex reductive group and N the nilpotent cone of its
Lie algebra. For any field k, one can consider the G-equivariant
derived category of sheaves of k-vector spaces. This category is
known to encode much information about the representation theory
of the Weyl group and the finite groups of Lie type. The subject
of this talk will be joint work with P. Achar in which we exhibit
an interesting auto-equivalence of this equivariant derived category
and in the case of the general linear group recover a type of Ringel
duality.
Jaime Thind
Quantum McKay Correspondence and Equivariant Sheaves on $\mathbb{P}^{1}_{q}$
The McKay correspondence gives a bijection between finite subgroups
of SU(2) and affine A,D,E Dynkin diagrams. There is a quantum version
of this statement (due to Kirillov Jr and Ostrik) which relates
"finite subgroups" of $U_{q} (sl_{2})$ and finite A,D,E
Dynkin diagrams. We use this correspondence to construct the category
of "equivariant" coherent sheaves on $\mathbb{P}^{1}_{q}$
- the quantum projective line. This is done by defining analogues
of the symmetric algebra and the structure sheaf, and using them
to define a triangulated category which is a natural analogue of
the derived category of equivariant sheaves on $\mathbb{P}^{1}$.
We then produce natural objects in this triangulated category, which
provide tilting objects relating our category to the derived category
of representations of the corresponding A,D,E quiver. This can be
thought of as a quantum analogue of the projective McKay correspondence
of Kirillov Jr.
Ben Webster
3 talks on symplectic singularities and their representation
theory:
Talk 1: The flag variety: why is this cotangent bundle different
from all other cotangent bundles?
Abstract: I'll give an introduction to the structures that interest
me in the case of the flag variety. This includes the classical
theory of category O, Harish-Chandra bimodules, the geometric construction
of the Hecke algebra, left and right cells, Soergel bimodules, primitive
ideals and all that jazz.
Talk 2:Flavors of conic symplectic resolutions: waffle, sugar or
cake?
Abstract: The next step is to try to reconstruct this representation
theory in other contexts, provided by the geometry of symplectic
singularities such as quiver varieties, Hilbert schemes, hypertoric
varieties and Slodowy slices. I'll introduce this cast of characters
and what is known about generalizing classical representation theory
to these contexts, with a light dusting of symplectic reflection
algebras, and sprinkling of the theory of categorical representations.
Talk 3: How one (might) take the dual of a symplectic singularity:
did you try reversing the polarity?
Abstract: I'll try to lay out the evidence that there is a duality
operation on symplectic singularities. At the moment there is no
definition as such for this operation, but there are a lot of surprising
coincidences pointing in that direction, coming from Koszul duality
in representation theory, mirror symmetry in physics and the geometry
of the varieties themselves.
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