# Witten deformation on noncompact manifolds

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Motivated by considerations from the mirror symmetry and

Landau-Ginzburg model, we consider Witten deformation on noncompact

manifolds.

Witten deformation is a deformation of the de Rham complex introduced by

Witten in an influential paper and has had many important applications,

mostly on compact manifolds. We will discuss some recent work with my

student Junrong Yan on the spectral theory of Witten Laplacian, the

cohomology of the deformation as well as its index theory.

A joint seminar with the Differential Geometry Seminar series.

# Comparing gauge theoretic invariants of homology S1 cross S3

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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.

# Skein algebras, elliptic curves, and Fukaya categories

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A ``skein relation'' can be viewed as a linear relation satisfied by the

R-matrix for a quantum group; one of the first uses of skein relations was

to give a combinatorial construction of Reshetikhin-Turaev invariants of

knots in S^3. The Hall algebra of an abelian (or triangulated) category

"counts extensions" in the category. We briefly describe how skein relations

appear in the Hall algebra of coherent sheaves of an elliptic curve, the

Hall algebra of the Fukaya category of a surface, and factorization homology

of a surface. No familiarity with the objects mentioned above will be

assumed for the talk.

# The cohomology rings of regular Hessenberg varieties

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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.

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# Fully nonlinear elliptic equations on real and complex manifolds

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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.

# Joint LA Topology Seminar

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Talks at UCLA. Please contact Li-Sheng Tseng if you plan to attend and would like to carpool.

**Peter Lambert-Cole (Georgia Tech): *** Bridge trisections and the Thom conjecture *

The classical degree-genus formula computes the genus of a nonsingular algebraic curve in the complex projective plane. The well-known Thom conjecture posits that this is a lower bound on the genus of smoothly embedded, oriented and connected surface in CP2. The conjecture was first proved twenty-five years ago by Kronheimer and Mrowka, using Seiberg-Witten invariants. In this talk, we will describe a new proof of the conjecture that combines contact geometry with the novel theory of bridge trisections of knotted surfaces. Notably, the proof completely avoids any gauge theory or pseudoholomorphic curve techniques.

**James Conway (UC Berkeley): *** Classifying contact structures on hyperbolic 3-manifolds*

Two of the most basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but with a notable absence of hyperbolic manifolds. In this talk, we will see a new classification of contact structures on an family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot, and see how it suggests some structural results about tight contact structures. This is joint work with Hyunki Min.

# The \'etale descent problem in algebraic K-theory

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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.

# Projective Geometry, Complex Hyperbolic Space, and Geometric Transitions.

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The natural analog of Teichmuller theory for hyperbolic manifolds in dimension 3 or greater is trivialized by Mostow Rigidity, so mathematicians have worked to understand more general deformations. Two well studied examples, convex real projective structures and complex hyperbolic structures, have been investigated extensively and provide independently developed deformation theories. Here we will discuss a surprising connection between the these, and construct a one parameter family of geometries deforming complex hyperbolic space into a new geometry built out of real projective space and its dual.