# TBA

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# CMC surfaces in Minkowski space

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In joint work with F. Bonsante and A. Seppi, we solve a

Dirichlet-type problem for entire constant mean curvature hypersurfaces in

Minkowski n+1-space, proving that such surfaces are essentially in bijection

with lower semicontinuous functions on the n-1-sphere. This builds off of

existence theorems by Treibergs and Choi-Treibergs, which themselves rely on

the foundational work of Cheng and Yau. I'll present their maximum principle

argument as well the extra tool that leads to our complete existence and

uniqueness theorem. Time permitting, I'll compare with the analogous problem

of constant Gaussian curvature and present a new result on their intrinsic

geometry.

Joint seminar with the Differential Geometry Seminar series.

# Classical and quantum traces coming from SL_n(C) and U_q(sl_n)

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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.

# Constructions of Lefschetz fibrations using cyclic group actions

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We construct families of Lefschetz fibrations over S2 using

finite order cyclic group actions on the product manifolds ΣgxΣg for g>0.

We also obtain more families of Lefschetz fibrations by applying the

rational blow-down operation to these Lefschetz fibrations. This is a joint

work with Anar Akhmedov and Mohan Bhupal.

# A twist on A-infinity algebras and its application on symplectic manifolds

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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.

# 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.