University of Connecticut

PDE and Differential Geometry Seminar

    [Semester] => Fall
    [Subject] => pde
    [Year] => 2015

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Title: A Shi-type estimate for Ricci curvature along the Ricci flow
Speaker: Chih-Wei Chen (National Taiwan University)
Time: Monday, August 31, 2015 at 3:30 pm
Place: MSB 109AAbstract: Along the Ricci flow, once the curvature is controlled, the derivatives of it can be controlled simultaneously. This classical fact is called Shi’s estimate. Suppose only the Ricci curvature is bounded, we aim to derive boundedness results for the derivative of Ricci curvature and the curvature operator. We will see that this goal is partially achieved by assuming a “Bianchi inequality” hold.

Title: On the Morse index in the min-max theory of minimal surfaces
Speaker: Xin Zhou (Massachusetts Institute of Technology)
Time: Monday, September 14, 2015 at 3:00 pm
Place: MSB 109AAbstract: The min-max theory, developed by F. Almgren and J. Pitts, is a variational theory aimed to construct unstable minimal surfaces in a closed manifold. Recently, the min-max theory has had seminal achievements lead by F. Marques and A. Neves. However, the understanding of this theory is still very limited. In this talk, I will focus on the geometric properties of the min-max surface. In particular, I will discuss several recent progresses on the Morse index of the min-max surfaces.

Title: Curvature dimension inequalities on Riemannian manifolds and sub-Riemannian manifolds
Speaker: Bumsik Kim (University of Connecticut)
Time: Monday, September 21, 2015 at 3:00 pm
Place: MSB 109AAbstract: In Riemannian manifolds, via Bochner's identity, the curvature dimension inequality is an equivalent condition for Ricci tensor lower bound. By Bakry Emery's Gamma_2 analysis, this notion of inequality can be naturally extended to more general metric measure spaces. Among the consequences of this curvature bounds in a class of sub Riemannian manifolds, such as heat kernel bounds, Harnack type inequality, Poincare-Sobolev inequality, we will mainly discuss the optimal lower bound of the first eigenvalue. Our class of manifolds - Riemannian foliations with totally geodesic leaves - contains Heisenberg group, CR Sasakian manifolds, Carnot group (stratified nilpotent Lie group) of step 2, and more. By the end of the talk, more recent results on Weitzenbock formula for differential forms, and some open questions will be introduced.

Title: A Tikhonov regularization for the inverse nodal
Speaker: C.K. Law (National Sun Yat-sen University)
Time: Monday, October 5, 2015 at 3:00 pm
Place: MSB 109AAbstract: In many applications, certain nodal set associated with a potential of a Dirichlet Sturm-Liouville problem can be measured. Hence it is desirable to recover the potential with this nodal set. The inverse nodal problem was first defined by McLaughlin. She showed that knowledge of the nodal points alone can determine the potential function in L2(0,1) up to a constant. Up till now the issues of uniqueness, reconstruction, smoothness and stability are all solved for q ∈ L1(0,1). Recently we apply the Tikhonov regularization method to reconstruct potentials of a Sturm-Liouville problem as well as a p-Laplacian eigenvalue problem using only zeros of one eigenfunction. This method helps to study the accuracy requirement on the measurement of the nodal sets on the reconstruction error. This is joint work with Xinfu Chen and Yan-Hsiou Cheng.

Title: Ricci flow and the moduli spaces of positive isotropic curvature metrics on four manifolds
Speaker: BingLong Chen (Sun Yat-sen University)
Time: Monday, October 12, 2015 at 3:00 pm
Place: MSB 109AAbstract: It is well-known that in lower dimensions, the Ricci flow can often be used to give “canonical geometric structures” on the manifolds. We observe that in the same time, the Ricci flow provides a deformation of the moduli spaces of Riemannian metrics, and it is possible that Ricci flow can be used to reveal the topological structures of the moduli spaces of Riemannian metrics with certain curvature restrictions. Our main result of the talk is to prove that the moduli spaces of positive isotropic curvature metrics on many 4-manifolds, including 4-sphere etc., are path-connected. This talk is based on a joint work with XianTao Huang.

Title: Minimizing Closed Geodesics
Speaker: Ian Adelstein (Trinity College)
Time: Monday, October 19, 2015 at 3:00 pm
Place: MSB 109AAbstract: 1/k-geodesics, introduced by Christina Sormani, are closed geodesics which minimize on any subinterval of length L/k, where L is the length of the geodesic. These curves arise as critical points of the uniform energy, a finite dimensional approximation to the Morse energy function. I'll discuss the existence and behavior of these curves on compact Riemannian manifolds.

Title: Singularities of Minimal Graphs
Speaker: Spencer Hughes (Massachusetts Institute of Technology)
Time: Monday, November 9, 2015 at 3:00 pm
Place: MSB 109AAbstract: One of the questions that motivated the study of the regularity of weak solutions to non-linear PDE is: Can a codimension one minimal graph have singularities? (By “minimal” we just mean “critical point of the area functional”). The ideas used by De Giorgi to answer this question in the 50's (no: such a graph must be smooth) were later developed by De Giorgi himself and then by Reifenberg, Federer, Almgren and Allard into a robust philosophy (the so-called “blow-up method”) capable of proving certain regularity results for very general classes of minimal submanifolds. In many of these classes (e.g. area-minimizing currents), singularities can occur and in some such cases the blow-up method can be used to get results about the structure and regularity of the singular set. With particular reference to the recent work of the speaker on two-valued (i.e. 'two-sheeted') minimal graphs, I will attempt to outline some of the ideas behind the blow-up method as it is used today.

Title: Sharp log-Sobolev inequalities on closed manifolds
Speaker: Marcos Montenegro (Federal University of Minas Gerais)
Time: Monday, November 16, 2015 at 3:00 pm
Place: MSB 109AAbstract: We show a sharp $L^p$ log-Sobolev inequality on closed manifolds as a limiting case of sharp Nash inequalities. This inequality is a Riemannian version of the Gross's log-Sobolev inequality. Our method consists in controlling two best Nash constants and relies mainly on the study of concentration of minimizers to related energy functionals. This is joint work with Jurandir Ceccon.

Title: TBA
Speaker: Georgios Daskalopoulos (Brown University)
Time: Monday, November 30, 2015 at 3:00 pm
Place: MSB 109AAbstract: TBA

Title: Shedding Light on Dark Matters
Speaker: Alan Parry (University of Connecticut)
Time: Monday, December 7, 2015 at 3:00 pm
Place: MSB 109AAbstract: General relativity lies at one of the most interesting and exciting intersections of mathematics and physics. This theory describes the nature of gravity, the driving force of all dynamics in the cosmos, and is explained in the language of differential geometry. The main idea of general relativity is the idea that gravity is an effect of the curvature of the spacetime. A current mystery of the universe is the presence and nature of unexplained but observed extra gravitational effects in galaxies and galaxy clusters. This phenomenon is called dark matter. This name reflects the possibility that if matter is causing the extra gravity, then that is all it is doing; it is not emitting light and is undetectable by any other means. However, since all we are detecting is unexplained gravitational effects, from a mathematics perspective, what we are really seeing is simply unexplained curvature in the spacetime. In this talk, we will discuss this physics problem and use it to motivate the study of a class of geometric PDEs called the Einstein-Klein-Gordon equations and their Newtonian analogue, the Schrodinger-Newton equations. We will describe some recent results about these equations in spherical symmetry and their applications to the dark matter problem. We also describe how some of these projects have motivated us to study some other interesting open problems in spherically symmetric general relativity. These results represent joint work with Hubert Bray and Andrew Goetz.

Title: On the instability of the Riemannian Penrose inequality in higher dimensions
Speaker: Armando Cabrera (University of Miami)
Time: Thursday, December 10, 2015 at 2:00 pm
Place: MSB 203Abstract: Given an asymptotically flat Riemannian manifold (M,g) with non-negative scalar curvature and containing a horizon boundary, the Riemannian Penrose inequality bounds the total mass of (M,g) from below in terms of the area of its horizon. Recently, Mantoulidis and Schoen constructed 3-dimensional asymptotically flat initial data sets with horizon boundary, whose mass can be arranged to be arbitrarily close to the optimal value in the Riemannian Penrose inequality, while the intrinsic geometry of the horizon can be far away from being rotationally symmetric; their result can be interpreted as an statement establishing the instability of the Riemannian Penrose inequality. In this talk, we will describe how to adapt their construction to provide examples of higher dimensional black hole initial data with analogous properties.

Organizer: Lihan Wang