# Geometry & Analysis Seminar

**Fall 2020**

**G&A Zoominar**

**Wednesdays @ 3pm California Time**

**Schedule ***(click dates for title and abstract)*

10/7 | Richard Montgomery | UCSC |

10/14 | Alejandro Bravo-Doddoli | UCSC |

10/21 | Burak Hatinoglu | UCSC |

10/28 | Anna Zemlyanova | Kansas State University |

11/4 | Elijah Fender | UCSC |

11/11 |
VETERANS DAY | (holiday; no talk) |

11/18 | Fatma Terzioglu | University of Chicago |

11/25 | Brian Harvie | UC Davis |

12/2 | Teemu Saksala | North Carolina State U |

12/9 | Tracey Balehowski | U Helsinki |

**Richard Montgomery, UCSC**

**N-body scattering and billiards**

A solution to the Newtonian N-body problem is “hyperbolic” if all interbody distances diverge asymptotically linear with time, in both time directions. The subset of initial conditions leading to hyperbolic solutions forms a large open subset of phase space.

We begin with the original scattering problem, that of Rutherford concerning the case N = 2 and instrumental in the discovery of the nuclei of atoms.

In this study we define a “scattering map” to study the map from the distant past asymptotics to the distant future asymptotics by compacting the spatial infinity of phase space using the method developed by McGehee. Parts of the picture we end up with for this map are inspired by (1922) and Melrose (more recently) . Those hyperbolic solutions which never stray far from infinity limit onto a non-deterministic “flow” at infinity described via a “point billiard system”. This non-deterministic system is known in the Melrose world as ’time π broken geodesic flow’ – the ’breaks’ being intersections with the collision locus at infinity. In the talk I will expound on this picture of scattering, using animations and an email of Rick Moeckel’s as a launching pad.

The talk synthesizes two papers on which I am a co-author:

- arXiv:1910.05871 https://arxiv.org/abs/1910.05871

- arXiv:1606.01420 https://arxiv.org/abs/1606.01420

**Alejandro Bravo-Doddoli, UCSC**

*Metric Lines in the k-Jet space*

The space \(J^k = J^k(\mathbb{R}, \mathbb{R})\) admits a canonical rank 2 distribution of Goursat type. Its subRiemannian geodesics have a simple and beautiful characterization in terms of degree k polynomials of the independent variable x first described by Anzaldo-Meneses and Monroy-Peréz . Among these geodesics are candidate metric lines: geodesics defined on all of the real line which minimize between any two points. These special geodesics are always asymptotic to singular lines -- abnormal geodesics of \(J^k\), with the asymptotic singular line for \(s \to -\infty\) different from the asymptotic singular line for \(s \to +\infty\)

**Burak Hatinoglu, UCSC**

**A complex analytic approach to inverse spectral problems**

In this talk we will consider the Schroedinger operator on a finite interval with an L^1-potential. Borg’s two spectra theorem says that the potential can be uniquely recovered from two spectra. By another classical result of Marchenko, the potential can be uniquely recovered from the spectral measure or Weyl m-function. After a brief review of inverse spectral theory of one dimensional Schroedinger operators, we will discuss the following mixed spectral problem as a complex analysis problem: Can one spectrum together with subsets of another spectrum and norming constants uniquely recover the potential?

**Anna Zemlyanova, Kansas State University**

**Hollow vortex problem in a wedge and the associated Riemann-Hilbert problem on an elliptic Riemann surface**

Conformal mappings from canonical slit domains onto multiply-connected physical domains with free boundaries find applications in many problems arising in fluid mechanics. In this talk, a fluid flow in a wedge around a hollow vortex is studied. An exact formula for the conformal map from the exterior of two slits onto the doubly connected flow domain is obtained. The map is expressed in terms of a rational function on an elliptic surface topologically equivalent to a torus and the solution to a symmetric Riemann-Hilbert problem on a finite and a semi-infinite segments on the same Riemann surface. Owing to its special features, the Riemann-Hilbert problem requires a novel analogue of the Cauchy kernel on an elliptic surface. Such a kernel is proposed, its properties are studied, and it is employed to derive a closed-form solution to the Riemann-Hilbert problem.The solution procedure also includes the solution to the associated Jacobi inversion problem and a transcendental equation for the conformal mapping parameters. The final formula for the conformal map possesses a free parameter which allows to construct a one-parametric family of hollow vortices in a wedge. Numerical results are reported. This is a joint work with Yuri Antipov, Louisiana State University.

**Elijah Fender, UCSC**

**Relating Local Homology groups of Reeb orbits**

If one considers a neighborhood of a Reeb orbit which can be identified with _{}^{1} x *B*^{2n} in a contact manifold, then there are two useful ways one can associate Floer homology groups with said Reeb orbit. One is by taking a symplectization of the neighborhood of the Reeb orbit and computing either the _{}^{1}-equivariant Floer homology or the standard Hamiltonian Floer homology in this neighborhood. When doing so, a convex Hamiltonian is chosen in such a fashion that the Reeb orbit is an isolated family of one-periodic points for that Hamiltonian’s flow on the symplectization. On the other hand, the Poincaré return map associated with the orbit is the germ of a Hamiltonian diffeomorphism of Euclidean space. This return map also has an associated Hamiltonian Floer Homology. We will discuss isomorphisms which relate these homology groups and methods involved in proving those isomorphisms.

**Fatma Terzioglu, University of Chicago**

*Mathematics of Multi-Energy Computed Tomography*

Multi-Energy Computed Tomography (ME-CT) is a medical imaging modality aiming to reconstruct the spatial density of materials by using energy-dependent attenuation properties of probing x-rays. While conventional single-energy CT can only reveal the morphology of scanned objects, and hence is qualitative, ME-CT can provide absolute and quantitative information on the scanned object, e.g., its chemical composition.

ME-CT measurements are mathematically modeled by a nonlinear function that maps line integrals of the unknown densities of a finite number of materials (typically bone, water and contrast agents) to their energy-weighted integrals obtained using several x-ray source energy spectra. Image reconstruction in ME-CT may thus be achieved in two steps: first the reconstruction of line integrals of the material densities from their energy-weighted integrals, and then the reconstruction of material densities from their line integrals. The second step is the standard linear X-ray CT problem whose invertibility is well-known. The first step is however a nonlinear map, with no known analytical inverse.

Although developing numerical material reconstruction algorithms in ME-CT has attracted significant interest in the last decade, the uniqueness and stability of inversion has been less studied.

In this talk, I will introduce the mathematics of ME-CT and will present sufficient criteria that guarantee global uniqueness of ME-CT reconstructions as well as a stability estimate. The results of a reconstruction algorithm whose convergence is ensured by these criteria will also be provided.

**Brian Harvie, UC Davis**

**Singularities of the Inverse Mean Curvature Flow**

An extrinsic geometric flow is a rule for deforming a surface by its curvature over time. The behavior of the evolving surface is modelled by nonlinear parabolic PDE, and these nonlinearities cause the formation of singularities in the flow where certain geometric quantities become infinitely large in finite time. Much research in analysis and geometry focuses on determining necessary and sufficient conditions for singularity formation as well as characterizing the shape of the surface as it approaches the singular time. These questions have been studied most thoroughly in the context of Mean Curvature Flow (MCF), an extrinsic flow which contracts a surface through time. However, they have been largely unaddressed for Inverse Mean Curvature Flow (IMCF), the counterpart to MCF which expands a surface through time.

In this talk, I will present my recent work on the formation and characterization of singularities for IMCF. As we will see, the singular behavior of IMCF differs from the behavior of MCF and other flows in fascinating ways. This talk is meant to be accessible to graduate students with some background in analysis and differential geometry.

**Teemu Saksala, North Carolina State U**

*Probing an unknown elastic body with waves that scatter once: an inverse problem in anisotropic elasticity*

We consider a geometric inverse problem of recovering some material parameters of an unknown elastic object by probing with elastic waves that scatter once inside the body. That is we send elastic waves from the boundary of an open bounded domain. The waves propagate inside the domain and scatter from an unknown point scatterer. We measure the entering and exiting directions of the waves and their total travel times. Geometrically this is equivalent to knowing the broken scattering relation of the unknown wave speed. The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds satisfying a convex foliation condition have the same broken scattering relation, then they are isometric. The talk is based on a joint work with: M.V. de Hoop, J. Ilmavirta and M. Lassas.

**Tracey Balehowski, University of Helsinki**

**An inverse problem for the relativistic Boltzmann equation**

In this talk, we consider the following problem: Given the source-to-solution map for a relativistic Boltzmann equation on a neighbourhood V of an observer in a Lorentzian spacetime (M, g) and knowledge of g|V , can we determine (up to diffeomorphism) the spacetime metric g on the domain of causal influence for the set V ?

We will show that the answer is yes. The problem we consider is a so-called inverse problem. We will review results and techniques developed in the study of inverse problems similar to ours. We will also introduce the relativistic Boltzmann equation and comment on the existence of solutions to this PDE given some initial data. We then will sketch the key ideas of the proof of our result. One such key point is that the nonlinear term in the relativistic Boltzmann equation which describes the behaviour of particle collisions captures information about a source-to-solution map for a related linearized problem. We use this relationship together with an analysis of the behaviour of particle collisions by classical microlocal techniques to determine the set of locations in V where we first receive light particle signals from collisions in the unknown domain. From this data we are able to parametrize the unknown region and determine the metric.

The new results presented in this talk are joint work with Antti Kujanapää, Matti Lassas, and Tony Liimatainen, (University of Helsinki).

Preprint: https://arxiv.org/abs/2011.09312