This is a report on the joint work with Matthew Levy. We use surjection operations on integral cochains tof a topological space X (described by McClure-Smith and Berger-Fresse) to describe…
The Ricci curvature Ric(g) is a symmetric 2-tensor on a Riemannian manifold (M,g) that encodes curvature information. It features in several interesting geometric PDEs such as the Ricci flow and the Einstein equation.…
When is a function in the spacetime uniquely determined by its integrals over all light rays? I will introduce the problem, discuss why we might care about it, and how…
The question of measuring "handedness" is of some significance in both mathematics and in the real world. Propellors and screws, proteins and DNA, in fact *almost everything* is chiral. Can…
One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular, one can study exact Lagrangian fillings of Legendrian links. There are…
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…
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.…
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…
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…
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…
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…
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…
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,…
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…
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…
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.…
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…
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…
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…
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…