In general relativity, in the absence of special symmetries, there is no reasonable, nontrivial notion of mass-energy density accounting not only for all source fields but also for gravity itself. Nevertheless there are good definitions, such as the ADM mass, of the mass of an entire isolated system, modelled as an asymptotically flat space-time. Numerous…

The talk will begin with the reconstruction of three-dimensional bodies from their two-dimensional projections.  Then I analyze the induced action of the Euclidean group on the body's projected outlines using moving frames, leading to a complete classification of the outline differential invariants and the associated outline signature of the body. Zoom invitation is sent to…

I am going to report on recent work on the numerical optimization of tangent-point energies of curves and surfaces. After a motivation and brief introduction to the central computational tools (construction of suitable Riemannian metrics on the space of embedded manifolds, a polyhedral discretization of the energies, and fast multipole techniques), I am going to…

In this talk, I will discuss the Bernstein problem for minimal surfaces, and the recent solution to the stable Bernstein problem for minimal hypersurfaces in R^4. Precisely, we show that a complete, two-sided, stable minimal hypersurface in R^4 is flat. Corollaries include curvature estimates for stable minimal hypersurfaces in 4-dimensional Riemannian manifolds, and a structural…

We show that the moduli space of all smooth fibrations of a 3-sphere by oriented simple closed curves has the homotopy type of a disjoint union of a pair of 2-spheres, which coincides with the homotopy type of the finite-dimensional subspace of Hopf fibrations. In the course of the proof, we present a pair of…

Broadly speaking, there are two classes of inverse problems — those that are concerned with the analysis of PDEs, and those that are geometric in nature. In this talk, I will introduce the audience to these classes by highlighting two classical examples: Calderón’s problem for the PDE setting, and the boundary rigidity problem in the…

In an inverse boundary problem, one seeks to determine the coefficients of a PDE inside a domain, describing internal properties, from the knowledge of boundary values of solutions of the PDE, encoding boundary measurements. Applications of such problems range from medical imaging to non-destructive testing. In this talk, starting with the fundamental Calderon inverse conductivity…

In this talk I will introduce a geometric inverse problem that is motivated by geophysical imaging and seismology. Specifically, I will reconstruct a compact Riemannian manifold with strictly convex boundary from wave-based data on the boundary. The given data assumes the knowledge of an open measurement region on the boundary, and that for every point…

Waists and widths measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels. Zoom invitation is sent to the geometry and topology seminar list. If you are not on the list, please, contact Peter McGrath…

The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of optimal upper bounds for its eigenvalues is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. It turns out that the optimal isoperimetric inequalities for Laplacian eigenvalues are closely related to…

As described in the previous week's talk by Mikhail Karpukhin, there is a rich interplay between isoperimetric problems for Laplace eigenvalues on surfaces and the study of harmonic maps and minimal surfaces in spheres. Over the last 10-15 years, a program initiated by Fraser and Schoen has revealed a similar relationship between isoperimetric problems for the…

Numerical simulations on infinite domains are challenging. In this talk, we will take geometric approaches to analyze the problems and provide new solutions.  One problem we tackle is the perfectly matched layer (PML) problem for computational waves on infinite domains. PML is a theoretical wave-absorbing medium attached to the truncated domain that generates no reflection…

In this talk we provide new Myers–Steenrod theorem based proofs to confirm that: travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to a boundary fixing diffeomorphism, which is the natural gauge for all of these problems. The method of the proof leads…

Neutrinos are very weakly interacting particles that have unique properties that allow indirect measurements that cannot be realized with any other particle or field. I will give an introduction to neutrino physics and describe two inverse problems with neutrinos. This is based on joint work with Gunther Uhlmann. https://ncsu.zoom.us/j/94955574178?pwd=cFpRUlZScXRmdGFscE5FZEhIb1NWdz09 Meeting ID: 949 5557 4178 Passcode: 301451

Shape-shifting materials from 2D thin sheets to 3D shapes are attractive for broad applications in programmable machines and robots, functional biomedical devices, and four-dimensional printing. Kirigami, the art of paper cutting, has recently emerged as a promising approach for shape morphing structures and materials due to its new properties such as auxeticity, stretchability, conformability, multistability, and optical chirality. In this talk,…

In this talk, we will discuss bordered aspects of the Heegaard Floer surgery formulas of  Ozsvath--Szabo and Ozsvath--Manolescu. In particular, we will explain how their theories naturally define bordered invariants for manifolds with toroidal boundary components. Time permitting, we will discuss applications of the theory. One application is a proof of the equivalence of lattice…

I will talk about the new construction of genus-zero free boundary minimal surfaces embedded in the unit ball in the Euclidean three-space which are compact and lie arbitrarily close to the boundary unit sphere with an arbitrarily large number of connected boundary components. The construction is by PDE gluing methods and the surfaces are desingularizations…

I will discuss the travel time tomography problem for elastic media in the transversely isotropic setting. The mathematical study of this problem relates to X-ray tomography and boundary rigidity problems studied by de Hoop, Stefanov, Uhlmann, Vasy, et al., which reduce the inverse problems to the microlocal analysis of certain operators obtained from a pseudolinearization…

We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of smooth, compactly supported perturbations of the metric, whereas all…

A free boundary minimal surface (FBMS) in a given three-dimensional Riemannian manifold is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ambient manifold. We will talk about the existence of FBMS, with particular emphasis on an equivariant min-max method and on the computation…