We discuss work-in-progress constructing a quantum trace map for
the special linear group SL_n.  This is a kind of Reshetikhin-Turaev
invariant for knots in thickened punctured surfaces, coming from an
interaction between higher Teichmüller theory and quantum groups.

Let S be a punctured surface of finite genus.  The SL_2-skein algebra of S
is a non-commutative algebra whose elements are represented by framed links
K in the thickened surface S x [0,1] subject to certain relations.  The
skein algebra is a quantization of the SL_2(C)-character variety of S, where
the deformation depends on a complex parameter q.  Bonahon and Wong
constructed an injective algebra map, called the quantum trace, from the
skein algebra of S into a simpler non-commutative algebra which can be
thought of as a quantum Teichmüller space of S. This map associates to a
link K in S x [0,1] a Laurent q-polynomial in non-commuting variables X_i,
which in the specialization q=1 recovers the classical trace polynomial
expressing the trace of monodromies of hyperbolic structures on S when
written in Thurston's shear-bend coordinates for Teichmüller space.  In the
early 2000s, Fock and Goncharov, among others, developed a higher
Teichmüller theory, which should lead to a SL_n-version of this invariant.

We will first review an algebra of special differential forms on sympectic manifolds, constructed by Tsai, Tseng and Yau. Then we introduce a twist on this algebra, which leads to a flatness condition. This twist is motivated by considering the connections on fiber bundles, and we can generalize it to A-infinity algebras, together with a generalized flatness condition.

While classical gauge theoretic invariants for 4-manifolds are usually
defined in the setting that the intersection form has nontrivial positive
part, there are several invariants for a 4-manifold X with the homology S1
cross S3. The first one is the Casson-Seiberg-Witten invariant LSW(X)
defined by Mrowka-Ruberman-Saveliev; the second one is the Fruta-Ohta
invariant LFO(X). It is conjectured that these two invariants are equal to
each other (This is an analogue of Witten’s conjecture relating Donaldson and
Seiberg-Witten invariants.)

In this talk, I will recall the definition of these two invariants, give
some applications of them (including a new obstruction for metric with
positive scalar curvature), and sketch a prove of this conjecture for
finite-order mapping tori. This is based on a joint work with Danny Ruberman
and Nikolai Saveliev.

A joint seminar with the Differential Geometry Seminar series.

   Hessenberg varieties form a distinguished class of subvarieties in
the flag variety, and their study is central to themes at the interface of
combinatorics and geometric representation theory. Such themes include the
Stanley-Stembridge and Shareshian-Wachs conjectures, in which the cohomology
rings of Hessenberg varieties feature prominently.

   I will provide a Lie-theoretic description of the cohomology rings of
regular Hessenberg varieties, emphasizing the role played by a certain
monodromy action and Deligne's local invariant cycle theorem. Our results
build on upon those of Brosnan-Chow, Abe-Harada-Horiguchi-Masuda, and
Abe-Horiguchi-Masuda-Murai-Sato. This represents joint work with Ana
Balibanu.