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.

Joint with Analysis Seminar.

Abstract: Fully nonlinear elliptic and parabolic equations on manifolds play central roles in some important problems in real and complex geometry. A key ingredient in solving such equations is to establish apriori  estimates up to second order. For general Riemannian manifolds, or Kaehler/Hermitian manifolds in the  complex case, one encounters difficulties caused by the curvature (as well as torsion in the Hermian case) of the manifolds. 

In this talk we report some results in our effort to overcome these obstacles over the past  few years. We shall emphasize on understanding the roles of subsolutions and concavity of the equation based on which our techniques were developed. We are interested both in equations on closed manifolds, and the Dirichlet problem for equations on manifolds with boundary of arbitrary geometry. 

For the Dirichlet problem on manifolds with boundary, we prove that under some fundamental structure conditions which were first proposed by Caffarelli-Nirenberg-Spruck and are now standard in the literature, there exist a smooth solution provided that there is a C2 subsolution. 

For equations on closed manifolds, there have appeared two different notations of weak subsolutions, the C-subsolution introduced by Gabor Szekelyshidi (JDG, 2018) and "tangent cone at infinity" condition by myself (Duke J Math, 2014). We show for type I cones the two notations coincide. We also construct examples showing for the Dirichlet problem that the subsolution condition can not be replaced by the weaker versions.

Algebraic K-theory is an invariant of rings (or algebraic varieties) that sees deep geometric and arithmetic information (ranging from Chow rings to special values of L-functions), but is generally difficult to compute. One reason for the complexity of algebraic K-theory is that it fails to satisfy \'etale descent. A general principle in algebraic K-theory (going to Lichtenbaum-Quillen, and proved in the work of Voevodsky-Rost on the Bloch-Kato conjecture) is that it is not too far off from doing so. I will explain this principle and some new extensions of this (joint with Dustin Clausen) in p-adic settings.