Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a Riemannian manifold with boundary? With a finite set of sources we can only hope to…
In order to explain the bi-concave shape of red blood cells, Helfrich proposed to study the minimisation of a bending energy amongst closed surfaces with given fixed area and volume. In the homogeneous case, the Helfrich functional simplifies to the scaling invariant Willmore functional. Thus, for the minimisation, the two constraints on area and volume…
I will give a brief introduction to Heegaard Floer homology and survey what's known about its "extended" structure via Lipshitz-Ozsvath-Thurston's bordered Floer homology and Douglas-Lipshitz-Manolescu's cornered Floer homology. Then I will sketch a connection between this extended structure and a more algebraic problem, the construction of tensor products for higher representations, arising from my recent work with…
In the 1980s Xavier proved that a complete non-planar minimal surface with bounded curvature of $\mathbb{R}^{3}$ can not lie in half-space. In 1990, Hoffman-Meeks proved that this half-space property holds for properly immersed non-planar minimal surfaces of $\mathbb{R}^{3}$ as well. And they went further, proving what is called "the strong half-space theorem" that states that…
In 2018, Khovanov and Robert introduced a version of Khovanov homology over a larger ground ring, termed U(1)xU(1)-equivariant Khovanov homology. This theory was also studied extensively by Taketo Sano. Ross Akhmechet was able to construct an equivariant annular Khovanov homology theory using the U(1)xU(1)-equivariant theory, while the existence of a U(2)-equivariant annular construction is still…
The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. I'll describe how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions as well as solutions to other elliptic equations. Recent results in this context will be presented, focusing on applications to…
Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin’s “sl(n)-like” Heegaard Floer knot invariants HFK_n recover both Alexander polynomial evaluations and sl(n) polynomial evaluations at certain roots of unity for links in S^3. We show that the equality of these evaluations can…
I will overview some important milestones in the development of the Invariant Theory from its classical times to modern days, leading into a discussion of the current progress in theory, computation, and applications. The highlights include Hilbert's basis theorem, geometric invariant theory, differential algebra of invariants, the moving frame approach, Lie's work on symmetries of…
We will prove lower bounds for the p-energies of mappings of real, complex and quaternionic projective spaces to arbitrary Riemannian manifolds. The equality cases of the results for real and complex projective space give strong characterizations of some families of energy-minimizing harmonic mappings of these spaces. If we have enough time, we will also describe…
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 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…