Scalar Curvature in Manifold Topology and Conformal Geometry - IMS

Scalar Curvature in Manifold Topology and Conformal Geometry
(01 Nov - 31 Dec 2014)

~ Abstracts ~

Spherical T-duality
Peter Bouwknegt, Australian National University, Australia

T-duality is an equivalence of String Theories on manifolds which are circle (or more generally, torus) bundles equipped with a background flux. Mathematically it provides an isomorphism for certain twisted cohomologies and K-theories for these manifolds. In this talk I will briefly review T-duality for circle bundles (U(1)-bundles), and then discuss a recent generalization to 3-sphere bundles (SU(2)-bundles), with applications to 7-twisted cohomologies and K-theories and the homotopy groups of 3-spheres. Applications to manifolds of positive sectional curvature will be discussed as well. This talk is based on joint work with Jarah Evslin and Mathai Varghese. [arXiv:1405.5844/1409.1296].

The universal eta invariant
Ulrich Bunke, Universität Regensburg, Germany

The universal eta invariant is the universal bordism invariant obtained from the eta invariants of Dirac operators by a Kreck-Stolz type construction. I will explain its definition, topological interpretation and explain how several classical bordism invariants are special cases. I will also give a generalization to manifolds with boundary and an application to the construction of tertiary invariants.

Steady vortex solutions for Euler equation of two dimension
Daomin Cao, Chinese Academy of Sciences, China

In this talk, the speaker will talk about the existence of solutions with small vorticity set. He will first recall the two main methods used to obtain the steady solutions of Euler equation. Next we will explain the relation between the existence of critical points of Kirchhoff - Routh function and the existence of steady solutions of Euler equations. Lastly he will present some of the results obtained in his two recent papers with Zhongyuan Liu, Juncheng Wei and Shuangjie Peng, Shusen Yan respectively. The solutions are obtained by using Lyapunov - Schmidlt to construct some semilinear elliptic equations with nonlinearities like Heviside functions.

A flow approach to perturbation theorem in prescribed scalar curvature problem on $S^n$
Xuezhang Chen, Nanjing University, China

In these four lectures, we will sketch the strategy of the gradient flow used in prescribed scalar curvature problem on the standard sphere $S^n$, as well as some details of the proof of the main theorem. Recently, such flow approaches sound to be a very powerful tool in the prescribed curvature problems in conformal geometry. These lectures are based on the joint work with Professor Xingwang Xu:
X. Chen and X. Xu, The scalar curvature flow on $S^n$---perturbation theorem revisited, Invent. Math 187(2012), no. 2, 395-506. (with an erratum: Invent. Math. 187(2012), no. 2, 507-509.)

Some vanishing results of Witten genus
Qingtao Chen, International Centre for Theoretical Physics, Italy

The Witten genus is the loop space analogue of the Hirzebruch A-hat genus. On a string manifold, the Witten genus is a level 1 modular form over SL(2,Z). First I will discuss the vanishing of Witten genus on string complete intersection in a product of projective spaces which generalize Landweber-Stone's result. Their result is one of the major eveidences for the Hohn-Stolz conjecture that existence of a positive Ricci curvature metric on a string manifold implies the vanishing of the Witten genus.Then I will discuss a mod 2 extension of the original Witten genus. We will present Landweber-Stone type vanishing results for mod 2 Witten genera on string complete intersections and describe a mod 2 version of the Hohn-Stolz conjecture. This represents our joint work with Fei Han and Weiping Zhang.

Analytic torsion and modular forms
Xianzhe Dai, University of California, Santa Barbara, USA

The analytic torsion is introduced by Ray-Singer as an analytic analog of a topological invariant, the Reidemeister torsion. The analytic torsion for complex manifolds, or holomorphic torsion, depends explicitly on the metric structure, unlike its real counterpart. This salient feature gives rise to many interesting applications. In this talk, we will explain one aspect of it, namely the connection with modular forms on the moduli spaces. This is joint work with Ken-Ichi Yoshikawa.

Nonnegative curvature, cohomogeneity and cohomology
Anand Dessai, University of Fribourg, Switzerland

Gromov's Betti number theorem gives a severe restriction on the existence of metrics of nonnegative curvature. The question whether other finiteness conditions hold for the class of simply-connected n-dimension manifolds has been addressed and studied by various people including Grove, Fang, Rong and Totaro. In my talk I will describe an infinite family of $8$-dimensional manifolds of non-negative curvature and cohomogeneity one with pairwise non-isomorphic complex cohomology rings. This family is optimal in many respects and improves the 8-dimensional examples of Totaro.

The p-Laplacian and geometric structure of Riemannian manifolds
Nguyen Thac Dung, Vietnam Institute for Advanced Study in Mathematics, Vietnam

It is well-known that there are beautifull relationship between the theory of p-harmonic function, topology and geometric structure of Riemannian manifolds. In this talk, I will recall some results on this topic. Moreover, I show that if the first eigenvalue for the p-Laplacian achievies its maximal value on a Kahler manifold or a quaternionic Kahler manifold then such a manifold must be connected at infinity unless it is a topological cylinder with an explicit warped product metric. The same problem in the setting of smooth metric measure space is also discussed.

Rigidity results for elliptic PDEs
Mostafa Fazly, University of Alberta, Canada

This talk provides classification and symmetry results for certain local and nonlocal elliptic PDEs with power type nonlinearities. We start with a brief background on the standard methods and ideas developed over the past couple of decades and then we talk about the new challenges. Monotonicity Formulas, Liuoville theorems and pointwise estimates would be our main focus in this talk.

Alexandrov-Fenchel type inequalities in the hyperbolic space
Yuxin Ge, Université Paris-Est Créteil Val de Marne, France

In this talk, we describe the various Alexandrov-Fenchel inequalities and weighted Alexandrov-Fenchel inequalities in the hyperbolic space. As an application, we obtain an optimal Penrose type inequality for the new Gauss-Bonnet-Chern mass for a class of asymptotically hyperbolic manifolds. This talk is based on the joint works with Guofang Wang and Jie Wu.

Second order estimates for complex fully nonlinear elliptic equations
Bo Guan, Ohio State University, USA

We consider a wide class of fully nonlinear elliptic equations on Kahler or more general Hermitian manifolds. Under a set of structure conditions which are essential optimal, we are able to derive estimates for the second order derivatives. In the talks we shall outline the proof and discuss connections the the equations to problems in complex geometry.

Boundary expansions of minimal surfaces in the hyperbolic space
Qing Han, University of Notre Dame, USA

The minimal surface equation in the hyperbolic space is given by a quasilinear elliptic equation, which is non-uniformly elliptic and becomes singular on the boundary. The focus of this talk is to use the expansion near the boundary to discuss the regularity of solutions.

Projective families of Dirac type operators
Pedram Hekmati, University of Adelaide, Australia

Elements in twisted K-theory can be realised as projective families of Fredholm operators. In this talk I will briefly review the construction of twisted K-theory classes for compact Lie groups and discuss some progress towards generalising this construction to other spaces.
This is joint work with Jouko Mickelsson.

Results on curvature flow
Pak Tung Ho, Sogang University, Korea

In this talk, I will first talk about Q-curvature flow. Then I will talk about Yamabe flow and CR Yamabe flow, which were introduced to study the Yamabe problem and the CR Yamabe problem respectively. If time permits, I will also talk about the study of Nirenberg's problem by using the curvature flow method.

Quantising noncompact Spin-c manifolds
Peter Hochs, University of Adelaide, Australia

Geometric quantisation originates from physics, where it is a way to construct a quantum mechanical phase space, i.e. a Hilbert space, from a classical mechanical phase space, i.e. a symplectic (or Poisson) manifold. Geometric quantisation is especially interesting in the presence of a group action that leaves the underlying physical system invariant. Then the 'quantisation commutes with reduction' principle states that geometric quantisation is compatible with the ways such a group action can be used to simplify the description of the physical system, in classical and quantum mechanics. This principle was recently generalised from symplectic manifolds to Spin-c manifolds, which gives a potentially much greater scope for applications, possibly including obstructions to positive scalar curvature. In joint work with Mathai Varghese, we generalised this from compact groups and manifolds to noncompact ones.

Finite volume flows and Witten's deformation
Wenchuan Hu, Sichuan University, China

We will talk about an explicit connection between deformation approach to Morse theory by Witten and the finite volume flow technique to Morse theory by Harvey and Lawson, as an answer to a question proposed by the latter.

Recent developments around the Gromov-Lawson-Rosenberg conjecture
Michael Joachim, Universität Münster, Germany

The Gromov-Lawson-Rosenberg Conjecture for a group $\pi$ states that a closed connected spin manifold with fundamental group $\pi$ of dimension $n\ge 5$ admits a Riemannian metric of positive scalar curvature if and only if a specific index obstruction which takes values in the $K$-theory of the reduced group $C^*$-algebra $C_{r}^{*}\pi$ vanishes. In our talk we first briefly discuss the basic set-up which led to conjecture and mention some of the most important results. We then focus on various more recent developments. In particular we will discuss the state of affairs for finite fundamental groups as well as certain twisted analogs of the conjecture.

The Hirzebruch X_y-genus revisited
Ping Li, Tongji University, China

The Hirzebruch X_y-genus was introduced by Hirzebruch in 1950s and can be computed topologically via the celebrated Hirzebruch-Riemannian-Roch Theorem. I will in this talk review several important features of this genus and some applications. Along this line some of my own works related to this topic are also presented.

On symplectic critical surfaces
Jiayu Li, University of Science and Technology, China

In this talk, we introduce new functionals to study the existence of holomorphic curves in Kähler surfaces. We study the properties of the critical surfaces of the functionals.

Some recent works on Moser-Trudinger and Adams type inequalities: best constants and existence of extremals
Guozhen Lu, Wayne State University, USA

In this talk, I will describe some recent works on best constants and existence of externals for Moser-Trudinger and Adams inequalities in a number of geometric settings: Hyperbolic spaces, complete and non-compact Riemannian manifolds, Heisenberg groups and high order Sobolev spaces. The symmetrization property does not hold in those cases. These are joint works with Hanli Tang and Maochun Zhu (Beijing Normal University), N. Lam (University of Pittsburgh), Jungang Li (Wayne State University).

Analytic torsions and Toeplitz operators
Xiaonan Ma, Université Paris Diderot - Paris 7, France

Real analytic torsion is a spectral invariant of a compact Riemannian manifold equipped with a at Hermitian vector bundle, that was introduced by Ray-Singer in 1971. Ray and Singer conjectured that for unitarily flat vector bundles, this invariant coincides with the Reidemeister torsion, a topological invariant. This conjecture was established by Cheeger and Mueller, and by Bismut-Zhang to arbitrary flat vector bundles. In this talk, we explain first the theory of Toeplitz operators associated with a positive line bundle. We show then how Toeplitz operators appear naturally in the study of the asymptotic properties of cohomology and of the Ray-Singer analytic torsion associated with a family of flat vector bundles. In the case of arithmetic quotients, the asymptotics of the Ray-Singer analytic torsion give informations about the size of the torsion elements in the cohomology group.

Gradient estimates of mean curvature equations ad Hessian equations with Neumann boundary value problems
Xinan Ma, University of Science and Technology of China, China

In this paper, we use the maximum principle to get the gradient estimate for the solutions of the prescribed mean curvature equation with Neumann boundary value problem, which gives a positive answer for the question raised by Lieberman [16] in page 360. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations. We also got the gradient estimates for the hessian equation with Neumann boundary value problem.

Einstein constraint equations on Riemannian manifolds
Quoc Anh Ngo, Vietnam National University, Vietnam

Starting from the Einstein equation in general relativity, we carefully derive the Einstein constraint equations which specify initial data for the Cauchy problem for the Einstein equation. Then we show how to use the conformal method to study these constraint equations.

Infinitely many solutions for nonlinear Schrodinger equations involving electromagnetic fields and critical growth
Shuangjie Peng, Central China Normal University, China

In this talk, I will talk about the following nonlinear Schr\"{o}dinger equation $$ \Bigl(\frac{\nabla}{i}-A(|y'|,y'')\Bigr)^{2}u+V(|y'|,y'')u=|u|^{\frac{4}{N-2}}u,\,\,u\in
H^{1}(R^{N},\mathbb{C}), $$ where $A(|y'|,y'')$ is a bounded map from $\mathbb{R}^{+}\times \R^{N-2}$ to $\R^N$, and $V(|y'|,y'')$ is a bounded non-negative function in $\mathbb{R}^{+}\times \R^{N-2}$. We prove that if $N\geq 5$ and $r^{2}V(r,y'')$ has an isolated local minimum point or local maximum point $(r_0, y_0'')$ with $r_0>0$ and $r_0^2 V(r_0, y_0'')>0$, then the above problem has infinitely many complex-valued solutions, whose energy can be made arbitrarily large. This is based on the joint work with Prof. Shusen Yan and Dr. Chunhua Wang.

An introduction to the hypoelliptic Laplacian in the de Rham theory
Shu Shen, Max-Planck-Institut für Mathematik, Germany

The hypoelliptic Laplacian, constructed by Bismut, is a family of operators which interpolate between Hodge Laplacian and the Lie derivation along the generator of the geodesic flow. In this talk, we will give the construction of the hypoelliptic Laplaician, as well as some explicitly calculable examples. We will talk about the hypoelliptic versions of Hodge theory, Witten deformation and Cheeger-Müller Theorem.

Deforming conformal metrics with negative Bakry Emery Ricci Tensor on manifolds with boundary
Weimin Sheng, Zhejiang University, China

We consider the prescribing $k$-curvature problem of the Bakry-\'{E}mery Ricci tensor on a manifold with smooth boundary. For a $n$-dimensional manifold with negative $k$-curvature of Bakry-\'{E}mery Ricci tensor and non-positive mean curvature of the boundary with respect to its inward normal vector, $n\ge3$, we prove that there exists a conformal metric which solves this problem and the boundary of the manifold becomes minimal.

A higher index theorem for proper cocompact actions
Xiang Tang, Washington University, USA

In this talk I will desribe a cohomological formula for a higher index pairing between invariant elliptic differential operators and differentiable group cohomology classes. This index theorem generalizes the Connes-Moscovici L^2-index theorem and its variants. If time permits, I will explain the extension to groupoids. This is joint work with Markus Pflaum and Hessel Posthuma.

Stability and rigidity in positive scalar curvature
Wilderich Tuschmann, Karlsruher Institut für Technology, Germany

I will discuss several new stability and classification results for Riemannian and, in particular, Einstein manifolds with positive scalar curvature. This is joint work with M. Wiemeler.

Spin manifolds and proper group actions
Mathai Varghese, University of Adelaide, Australia

This is based on joint work in progress with Peter Hochs.

Index theory for virtual manifolds and virtual orbifolds
Bai-Ling Wang, Australian National University, Australia

In the study of moduli spaces from elliptic geometric PDEs arising from theoretical physics, the notion of virtual manifolds was proposed by Bohui Chen and Gang Tian, and further investigated by Chen-Li-Wang. In this talk, I will explain the general framework of index theory for virtual manifolds and virtual orbifolds, and discuss some applications to K-theoretical quantum invariants.

Noncommutative geometry and conformal geometry
Hang Wang, University of Adelaide, Australia

This talk concerns a series of my joint papers with Raphael Ponge. We use tools from noncommutative geometry (NCG) to study some aspects in conformal geometry. Spectral triples, invented by Connes, are the noncommutative analogue of manifolds and their differential geometry. In order to deal with some geometric situations where a manifold admits actions of a group of conformal diffeomorphisms, Connes and Moscovici introduced a counterpart, known as twisted spectral triples, to NCG. We will talk about index theory and Connes-Chern characters in the setting of twisted spectral triples and several applications of these concepts in NCG to conformal geometry: Vafa-Witten inequalities, a construction of a class of conformal invariants and local index formulas.

Multiple sign-changing solutions for nonlinear elliptic systems
Zhi-Qiang Wang, Utah State University, USA

We present recent work on multiplicity results on sign-changing solutions for a class of nonlinear Schrodinger type systems. For the repulsive case we construct proper invariant sets of a descending pseudo gradient flow and using minimax arguments we construct multiple sign-changing solutions. The method is built upon an abstract framework and is applicable to other problems.

Professor Ding's two ideas on the Liouville equation
Guofang Wang, Albert-Ludwigs-Universität, Germany

In this talk i will review two unpublished results of Professor Weiyue Ding about the Liouville equation in 80's. The ideas to prove these results are very interesting. I will present several applications

Introduction to gluing methods
Juncheng Wei, University of British Columbia, Canada

I will introduce the finite dimensional and infinite dimensional gluing methods. The two prototype equations are Allen-Cahn equation and nonlinear Schrodinger equation.
References:
1. M. Musso, Frank Pacard, Juncheng Wei.
Finite-energy sign-changing solutions with dihedral symmetry for the stationary non linear Schr\"{o}dinger equation Journal of European Mathematical Society 14(2012), no.6, 1923-1953.
2. M. del Pino, M. Kowalczyk, Juncheng Wei.
Entire Solutions of the Allen-Cahn Equation and Complete Embedded Minimal Surfaces of Finite Total Curvature Journal of Differential Geometry 83(2013), no.1, 67-131.

On Bombieri-De Giorgi-Giusti minimal graph and its applications
Juncheng Wei, University of British Columbia, Canada

We refine the estimates for the BDG minimal graph and then discuss several applications of it, including the De Giorgi Conjecture for Allen-Cahn equation, Serrin's overdetermined problem in unbounded domains, and translating graphs of mean curvature flow.

Index bundle gerbes and moduli spaces
Siye Wu, University of Hong Kong, Hong Kong

We construct the index bundle gerbe associated to a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we also give a geometric description which agrees with the above analytic construction. Finally, we apply the result to certain moduli spaces associated to Riemann surfaces. This is a joint work with P.Bouwknegt and V.Mathai.

Topological methods for prescribing the Webster scalar curvature on CR manifolds
Ridha Yacoub, Institut Préparatoire aux Etudes d'Ingénieurs de Monastir, Tunisia

This is to give an idea on topological methods used to solve the Webster scalar curvature problem that we consider in the case of the standard unit CR sphere S²ⁿ⁺¹.
Our basic tools come from generalized Morse theory in combination with the theory of critical points at infinity. Our method does not only give existence results, but also under generic conditions, a lower bound on the number of solutions, thanks to some "Morse Inequalities at Infinity".

Symmetry induced by concentration
Shusen Yan, University of New England, Australia

In this talk, we will discuss the symmetry induced by concentration for some elliptic problems such as the prescribed scalar curvature problem in $R^N$. These symmetry results can not be proved by the methods of moving plane. Instead, they are obtained from the local uniqueness of bubbling solutions.

Nonlinear elliptic equations with fractional Laplacian
Jianfu Yang, Jiangxi Normal University, China

In this talk, we present some existence results for nonlinear elliptic equations with fractional Laplacian.

Fourth order equations in conformal geometry
Paul Yang, Princeton University, USA

In each dimension greater than two, there is a conformally covariant operator of 4th order. Associated to this operator is a fourth order Q-curvature.
The natural questions are: when is this operator positive? How to solve the Q-curvature equation? Is there a maximum principle associated with this equation. I will discuss these issues in the four lectures.

Nodal and singular sets for solutions to some elliptic equations
Xiaoping Yang, Nanjing University of Science and Technology, China

In this talk, we will discuss some properties related to nodal sets and singular sets of solutions to certain elliptic equations. After defining frequency functions for solutions, the monotonicity formulae and doubling conditions are established. The measure estimates of nodal sets and geometric structure of singular sets of solutions are investigated.

Blow-up sequences of the prescribed scalar curvature equation on S^n
Feng Zhou, National University of Singapore

In this talk, we will present constructive results on blow-up sequences of infinite number of solutions for the prescribed scalar curvature equation on S^n. For n>=6, using the Lyapunov-Schmidt reduction method, we construct scalar curvature functions on S^n, so that each of them enables the prescribed scalar curvature equation to have an infinite number of positive solutions which form an aggregated blow-up sequence, or a towing blow-up sequence. This is joint work with Professor Leung Man Chun.

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