# 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)

**title**

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**Fatma Terzioglu** (University of Chicago)

**title**

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**Brian Harvie** (UC Davis)

**title**

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**Teemu Saksala **(North Carolina State U)

**title**

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**Tracey Balehowski **(U Helsinki)

**title**

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