
Scalar Curvature in Manifold Topology and Conformal Geometry
(01 Nov  31 Dec 2014)
~ Abstracts ~
Spherical Tduality Peter Bouwknegt, Australian National University, Australia
Tduality 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 Ktheories for these manifolds. In this talk I will briefly review Tduality for circle bundles (U(1)bundles), and then discuss a recent generalization to 3sphere bundles (SU(2)bundles), with applications to 7twisted cohomologies and Ktheories and the homotopy groups of 3spheres. 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].
« Back... 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 KreckStolz 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.
« Back... 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.
« Back... 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, 395506. (with an erratum: Invent. Math. 187(2012), no. 2, 507509.)
« Back... 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 Ahat 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 LandweberStone's result. Their result is one of the major eveidences for the HohnStolz 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 LandweberStone type vanishing results for mod 2 Witten genera on string complete intersections and describe a mod 2 version of the HohnStolz conjecture. This represents our joint work with Fei Han and Weiping Zhang.
« Back... Analytic torsion and modular forms Xianzhe Dai, University of California, Santa Barbara, USA
The analytic torsion is introduced by RaySinger 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 KenIchi Yoshikawa.
« Back... 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 simplyconnected ndimension 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 nonnegative curvature and cohomogeneity one with pairwise nonisomorphic complex cohomology rings. This family is optimal in many respects and improves the 8dimensional examples of Totaro.
« Back... The pLaplacian and geometric structure of Riemannian manifolds Nguyen Thac Dung, Vietnam Institute for Advanced Study in Mathematics, Vietnam
It is wellknown that there are beautifull relationship between the theory of pharmonic 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 pLaplacian 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.
« Back... 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.
« Back... AlexandrovFenchel type inequalities in the hyperbolic space Yuxin Ge, Université ParisEst Créteil Val de Marne, France
In this talk, we describe the various AlexandrovFenchel inequalities and weighted AlexandrovFenchel inequalities in the hyperbolic space. As an application, we obtain an optimal Penrose type inequality for the new GaussBonnetChern mass for a class of asymptotically hyperbolic manifolds. This talk is based on the joint works with Guofang Wang and Jie Wu.
« Back... 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.
« Back... 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 nonuniformly 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.
« Back... Projective families of Dirac type operators Pedram Hekmati, University of Adelaide, Australia
Elements in twisted Ktheory can be realised as projective families of Fredholm operators. In this talk I will briefly review the construction of twisted Ktheory classes for compact Lie groups and discuss some progress towards generalising this construction to other spaces.
This is joint work with Jouko Mickelsson.
« Back... Results on curvature flow Pak Tung Ho, Sogang University, Korea
In this talk, I will first talk about Qcurvature 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.
« Back... Quantising noncompact Spinc 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 Spinc 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.
« Back... 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.
« Back... Recent developments around the GromovLawsonRosenberg conjecture Michael Joachim, Universität Münster, Germany
The GromovLawsonRosenberg 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 setup 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.
« Back... 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 HirzebruchRiemannianRoch 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.
« Back... 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.
« Back... Some recent works on MoserTrudinger 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 MoserTrudinger and Adams inequalities in a number of geometric settings: Hyperbolic spaces, complete and noncompact 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).
« Back... 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 RaySinger 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 BismutZhang 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 RaySinger analytic torsion associated with a family of flat vector bundles. In the case of arithmetic quotients, the asymptotics of the RaySinger analytic torsion give informations about the size of the torsion elements in the cohomology group.
« Back... 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.
« Back... 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.
« Back... 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 $$ \Bigl(\frac{\nabla}{i}A(y',y'')\Bigr)^{2}u+V(y',y'')u=u^{\frac{4}{N2}}u,\,\,u\in
H^{1}(R^{N},\mathbb{C}), $$ where $A(y',y'')$ is a bounded map from $\mathbb{R}^{+}\times \R^{N2}$ to $\R^N$, and $V(y',y'')$ is a bounded nonnegative function in $\mathbb{R}^{+}\times \R^{N2}$. 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 complexvalued solutions, whose energy can be made arbitrarily large. This is based on the joint work with Prof. Shusen Yan and Dr. Chunhua Wang.
« Back... An introduction to the hypoelliptic Laplacian in the de Rham theory Shu Shen, MaxPlanckInstitut 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 CheegerMüller Theorem.
« Back... 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 nonpositive 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.
« Back... 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 ConnesMoscovici L^2index theorem and its variants. If time permits, I will explain the extension to groupoids. This is joint work with Markus Pflaum and Hessel Posthuma.
« Back... 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.
« Back... Spin manifolds and proper group actions Mathai Varghese, University of Adelaide, Australia
This is based on joint work in progress with Peter Hochs.
« Back... Index theory for virtual manifolds and virtual orbifolds BaiLing 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 ChenLiWang. In this talk, I will explain the general framework of index theory for virtual manifolds and virtual orbifolds, and discuss some applications to Ktheoretical quantum invariants.
« Back... 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 ConnesChern characters in the setting of twisted spectral triples and several applications of these concepts in NCG to conformal geometry: VafaWitten inequalities, a construction of a class of conformal invariants and local index formulas.
« Back... Multiple signchanging solutions for nonlinear elliptic systems ZhiQiang Wang, Utah State University, USA
We present recent work on multiplicity results on signchanging 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 signchanging solutions. The method is built upon an abstract framework and is applicable to other problems.
« Back... Professor Ding's two ideas on the Liouville equation Guofang Wang, AlbertLudwigsUniversität, 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
« Back... 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 AllenCahn equation and nonlinear Schrodinger equation.
References:
1. M. Musso, Frank Pacard, Juncheng Wei.
Finiteenergy signchanging solutions with dihedral symmetry for the stationary non linear Schr\"{o}dinger equation Journal of European Mathematical Society 14(2012), no.6, 19231953.
2. M. del Pino, M. Kowalczyk, Juncheng Wei.
Entire Solutions of the AllenCahn Equation and Complete Embedded Minimal Surfaces of Finite Total Curvature Journal of Differential Geometry 83(2013), no.1, 67131.
« Back... On BombieriDe GiorgiGiusti 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 AllenCahn equation, Serrin's overdetermined problem in unbounded domains, and translating graphs of mean curvature flow.
« Back... 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 selfadjoint Diractype 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.
« Back... 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 S^{2n+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".
« Back... 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.
« Back... 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.
« Back... 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 Qcurvature.
The natural questions are: when is this operator positive? How to solve the Qcurvature equation? Is there a maximum principle associated with this equation. I will discuss these issues in the four lectures.
« Back... 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.
« Back... Blowup 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 blowup sequences of infinite number of solutions for the prescribed scalar curvature equation on S^n. For n>=6, using the LyapunovSchmidt 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 blowup sequence, or a towing blowup sequence. This is joint work with Professor Leung Man Chun.
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