Geometry & Analysis Seminar Winter 2020

Wednesday, January 8th, 2020

no talk

Wednesday, January 15th, 2020

no talk

Wednesday, January 22nd, 2020

Kamran Sadiq, Johann Radon Institute (RICAM)

The Bukhgeim-Beltrami Equation With Applications To Optical Molecular Imaging

We revisit the inverse source problem in a two-dimensional absorbing and scattering medium and present a direct reconstruction method, which does not require iterative solvability of the forward problem, using measurements of the radiating flux at the boundary. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequence valued maps, and extends the original ideas of A. Bukhgeim from the non-scattering to the scattering media. Of novelty here, the medium has an anisotropic scattering property that is neither negligible nor large enough for the diffusion approximation to hold. The numerical realization of the proposed reconstruction method is also presented, which is amenable for such scattering media. The feasibility of the proposed algorithm is demonstrated in several numerical experiments, including simulated scenarios for parameters meaningful in Optical Molecular Imaging. This is joint work with Alexandru Tamasan and Hiroshi Fujiwara.

Wednesday, January 29th, 2020

Eric Chen, UC Santa Barbara

Critical power integral pinching for the Ricci flow on asymptotically flat manifolds

I will discuss a condition based on a scale-invariant integral norm of curvature for initial metrics on asymptotically flat manifolds which guarantees the Ricci flow's long-time existence and convergence, and in particular implies that the initial manifold must have been diffeomorphic to Euclidean space. Unlike in the compact setting in which many curvature-pinching results have developed since Hamilton's work, in the noncompact setting there have been few previous results in this direction.

Wednesday, February 5th, 2020

no talk

Wednesday, February 12th, 2020

Sean Curry, UC Davis

The Embeddability Problem for Deformations of the Unit Sphere in ℂ^2

Abstract deformations of the unit sphere in ℂ^2 are encoded by complex functions on the sphere 𝕊^3. In sharp contrast with the higher dimensional case, for deformations of 𝕊^3 the natural integrability condition is vacuous, and generic abstract deformations are not embeddable even in ℂ^N for any N. It is therefore a very difficult problem to characterize when a complex function on 𝕊^3 gives rise to an actual deformation of 𝕊^3 inside ℂ^2. I will discuss some recent work in this direction; this is joint work with Peter Ebenfelt.

Wednesday, February 19th, 2020

Brian Harvie, UC Davis

Singularity Formation and Self-Intersection for the Inverse Mean Curvature Flow

Mean Curvature Flow (MCF) is the canonical geometric flow for contracting hypersurfaces in Euclidean Space, and as such geometers have extensively studied its properties for decades. Inverse Mean Curvature Flow (IMCF) is the canonical expanding geometric flow, and it has garnered increasing interest in recent decades thanks to its applications in mathematical physics. However, contrasting sharply with MCF, little is understood about the singularities of IMCF. Finite-time singularities may or may not form for IMCF, so one naturally asks what conditions on initial data guarantee singularity formation, and whether or not the singularities that do occur do so in some prescribed time interval as with MCF.

In this talk, I will begin with some background on mean curvature and extrinsic flows before turning my attention to new results for IMCF. I will show that weak solutions to the flow originally introduced by Huisken and Ilmanen are smooth with non-vanishing gradient outside any ball containing the initial surface. To connect this result back to the classical flow, I will show that consequently every initial surface without spherical topology must either become singular or self-intersect under IMCF within a time interval depending only on initial data.

Wednesday, February 26th, 2020

MATH Workshop on Software Techniques, presented by Francois Monard

Writing Math, Writing About Math: Q&As on TeX and research

I will share a couple of setups for typing LateX (mostly on Windows; future sessions on other OS may come later), addressing any questions you may have on the topic. This is in fact meant to open the floor to any practical research-related questions, in particular related to efficiency. For example, I plan to address: how to build and maintain a bib file with *minimal* effort; how to draw figures; reasons to love "vim-latex"... If you have specific questions in mind, feel free to email me in advance.

If you want to bring your laptop and try things out as we chat, here are the apps I'll use. Install them and try to get them running before coming. 

- https://miktex.org/ (tex compiler + editor)

- http://vim-latex.sourceforge.net/ (another TeX editor)

- https://www.jabref.org/ (GUI for handling bib files)

- http://ipe.otfried.org/ (app for drawing figures)

- also, if you haven't one yet, open an account at https://www.overleaf.com/ (online tex editor/compiler, easy to get started on TeX if you have never done it, and amenable to collaborative work)

Wednesday, March 4th, 2020

Antoine Song, UC Berkeley

Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces

The development of min-max theory in recent years led to the construction of infinitely many minimal hypersurfaces in any given closed Riemannian manifold of dimension at most 7. However, the available analytical constructions do not reveal much about the geometry and topology of these minimal hypersurfaces. In order to better understand these objects, we need to study the relationship between different measures of complexities. We will explain how, for general minimal hypersurfaces with uniformly bounded area, the topology and the singular set can be controlled by the Morse index, and how we can use that to study the sequence of minimal surfaces constructed by min-max theory in a given closed 3-manifold. Some natural open questions will be introduced.

Wednesday, March 11th, 2020

Alejandro Bravo, UCSC

Higher Elastica: Geodesics in the Jet Space

Carnot groups are ℛ manifolds.  As such they admit geodesic flows, which are left-invariant Hamiltonian flows on their cotangent bundles.  Some of these flows are integrable.  Some are not.  The space of k-jets for real-valued functions on the real line forms a Carnot group of dimension k+2.  We show that its geodesic flow is integrable and that its geodesics generalize Euler's elastica, with the case k=2 corresponding to the elastica.