# CMC surfaces in Minkowski space

Peter Smillie

Caltech

## Time:

Tuesday, January 14, 2020 - 4:00pm

## Location:

RH 306

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)

Daniel Douglas

USC

## Time:

Monday, January 27, 2020 - 4:00pm

## Location:

RH 340P

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.

# TBA

Bahar Acu

## Institution:

Northwestern University

## Time:

Monday, May 4, 2020 - 4:00pm

RH 340P

# TBA

Nur Saglam

Virginia Tech

## Time:

Monday, March 9, 2020 - 4:00pm

RH 340P

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

Jiawei Zhou

## Institution:

Tsinghua University

## Time:

Monday, January 13, 2020 - 4:00pm

## Location:

RH 340P

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

Xianzhe Dai

UC Santa Barbara

## Time:

Monday, October 28, 2019 - 4:00pm

## Location:

RH 340P

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

Jianfeng Lin

UC San Diego

## Time:

Tuesday, October 1, 2019 - 4:00pm

## Location:

RH 306

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

Peter Samuelson

UC Riverside

## Time:

Monday, October 21, 2019 - 4:00pm

## Location:

RH 340P

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

Peter Crooks

## Institution:

Northeastern University

## Time:

Monday, November 18, 2019 - 4:00pm

## Location:

RH 340P

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.

# How to count constant maps?

Si Li

## Institution:

Tsinghua University

## Time:

Monday, November 25, 2019 - 4:00pm

## Location:

RH 340P

The art of using quantum field theory to derive mathematical
results often lies in a mysterious transition between infinite dimensional
geometry and finite dimensional geometry. In this talk we describe a general
mathematical framework to study the quantum geometry of sigma-models when
they are effectively localized to small fluctuations around constant maps.
We illustrate how to turn the physics idea of exact semi-classical
approximation into a geometric set-up in this framework, using Gauss-Manin
connection. This leads to a theory of “counting constant maps” in a
nontrivial way.  We explain this program by a concrete example of
topological quantum mechanics and show how “counting constant loops”  leads
to a simple proof of the algebraic index theorem.