Scalar Curvature and Intrinsic Flat Convergence

Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic Flat convergence, developed jointly with Wenger. This notion has been applied successfully to study sequences that arise in General Relativity. Gromov has suggested it should be applied in other settings as well. We first review intrinsic flat convergence, its properties, and its compactness theorems, before presenting the applications and the open problems.Comment: 53 pages, 4 figures, Geometric Analysis on Riemannian and singular metric spaces, Lake Como School of Advnaced Studies, 11-15 July 2016, v2: minor fixes as requested by referee, to appear as a chapter in the deGruyter book series on PDE and Measure Theory edited by Gigl

Local Rigidity of group actions: Past, Present, Future

This survey aims to cover the motivation for and history of the study of local rigidity of group actions. There is a particularly detailed discussion of recent results, including outlines of some proofs. The article ends with a large number of conjectures and open questions and aims to point to interesting directions for future research. {\em To Anatole Katok on the occasion of his 60th birthday.}Comment: Survey article, final version for publication, see also http://mypage.iu.edu/~fisherdm

Day's fixed point theorem, Group cohomology and Quasi-isometric rigidity

In this note we explain how Day's fixed point theorem can be used to conjugate certain groups of biLipschitz maps of a metric space into special subgroups like similarity groups. In particular, we use Day's theorem to establish Tukia-type theorems and to give new proofs of quasi-isometric rigidity results.Comment: 20 page

Journey to the Center of the Earth

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case.Comment: Survey article for Proceedings of the International Congress of Mathematical Physic

Partial generalizations of some Conjectures in locally symmetric Lorentz spaces

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_{1}^{n+1}$. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying some curvature conditions. By modifying Cheng-Yau's operator $\square$ given in {\cite{ChengYau77}}, we introduce a modified operator $L$ and give new estimates of $L(nH)$ and $\square(nH)$ of such spacelike hypersurfaces. Finally, we give partial generalizations of some conjectures in locally symmetric Lorentz spaces $L_{1}^{n+1}$

Manifolds with A Lower Ricci Curvature Bound

This paper is a survey on the structure of manifolds with a lower Ricci curvature bound.Comment: 22 page

Quasi-morphisms and quasi-states in symplectic topology

This is a survey about certain "almost homomorphisms" and "almost linear" functionals (called quasi-morphisms and quasi-states) in symplectic topology and their applications to Hamiltonian dynamics, functional-theoretic properties of Poisson brackets and algebraic and metric properties of symplectomorphism groups.Comment: Minor correction

Symmetry and rigidity: Only one kind of symmetry allow non-zero real symmetric solution

Leray guessed that, a blow-up solution should have similar structure as its initial data and proposed to consider self-similar solution. But Necas-Ruzicka-Sverak proved in 1996 that such solution should be zero. That is to say, Navier-Stokes equations have rigidity for self-similar structure. Recently, Yang-Yang-Wu found that the symmetry property plays an important role in the proof of ill-posedness result. Further, Yang applied Fourier transformation to consider symmetric solutions. He has shown that a party of symmetric solution should be zero and there exists some symmetric property can result in symmetric solution. In this paper, we consider the symmetry related to the independent variables of initial data and we analyze the symmetric structure of non-linear term. (i) We have found out what kinds of symmetric properties can generate symmetric solutions and we have also proved that the rest symmetric properties allow only zero solutions in some sense. For real initial data, we prove there exists only one kind of symmetry can generate non-zero symmetric solution. (ii) Further, to understand the structure of $B(u,v)$, we show it is sufficient to consider all the symmetric cases. (iii) Thirdly, we establish the well-posedness for some big initial values. (iv) Lastly, we apply such symmetric result to the Navier-Stokes equations on the domain and we prove the existence of smooth solution with energy conservation.Comment: 22page

Quasi-actions on trees: research announcement

We develop a battery of tools for studying quasi-isometric rigidity and classification problems for splittings of groups. The techniques work best for finite graphs of groups where all edge and vertex groups are coarse PD groups. For example, if Gamma is a graph of coarse PD(n) groups for a fixed n, if the Bass-Serre tree of Gamma has infinitely many ends, and if H is a finitely generated group quasi-isometric to pi_1(Gamma), then we prove that H is the fundamental group of a graph of coarse PD(n) groups, with vertex and edge groups quasi-isometric to those of Gamma. We also have quasi-isometric rigidity theorems for graphs of coarse PD groups of nonconstant dimension, under various assumptions on the edge-to-vertex group inclusions.Comment: 19 page

Actions of mapping class groups

This paper has three parts. The first part is a general introduction to rigidity and to rigid actions of mapping class group actions on various spaces. In the second part, we describe in detail four rigidity results that concern actions of mapping class groups on spaces of foliations and of laminations, namely, Thurston's sphere of projective foliations equipped with its projective piecewise-linear structure, the space of unmeasured foliations equipped with the quotient topology, the reduced Bers boundary, and the space of geodesic laminations equipped with the Thurston topology. In the third part, we present some perspectives and open problems on other actions of mapping class groups. The paper will appear in the Handbook of Group actions, vol. I (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press.Comment: To appear in the Handbook of Group actions, vol. I (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Pres