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Floer Lectures 2019

June 27-28, Bochum

June 27-28, Bochum

We are delighted to have** Daniel Peralta-Salas** (ICMAT, Madrid) and **Gunther Uhlman****n** (University of Washington) as our two speakers. They will each give two talks.

This is a workshop hosted by the Floer Center of Geometry and funded by the Bochum-Cologne SFB collaboration program **CRC/TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics”**.

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**Schedule:**

**Thursday**

15:15-16:15 G. Uhlmann -* **"Harry Potter's Cloak via Transformation Optics"*

16:15-17:00 Coffee break

17:00-18:00 D. Peralta-Salas - "*Emergence of topological structures in PDE: stationary equations"*

**Friday**

15:15-16:15 D. Peralta-Salas - "*Emergence of topological structures in PDE: evolution equations"*

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16:15-17:00 Coffee break

17:00-18:00 G. Uhlmann -* **"Journey to the Center of the Earth"*

**Talks** will take place on Thursday in room HNC 10 and Friday in HNC 30, which are located between the buildings NC and ND.

**Coffee breaks** will take place at the reception of the main entrance building IB, floor 01.

**Abstracts:**

D. Peralta-Salas - "*Emergence of topological structures in PDE: stationary equations"* & "*Emergence of topological structures in PDE: evolution equations"*

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* In recent years, new experimental methods and designs have enabled the observation and study of topological structures emerging from a wide range of physical phenomena, from fluid mechanics to electromagnetic theory. These structures take the form, for example, of knotted vortex tubes in fluid flows,

or knotted topological dislocations in optics and condensed matter theory. They provide a powerful visual tool to gain understanding of complex physical systems.

Mathematically, these physical processes are described by a vector or scalar field which satisfies a system of partial differential equations, and the emerging topological structures are instances of invariant sets of these fields. The understanding of which topological structures are compatible with the PDE constraint is a subtle problem which involves the interplay between dynamical systems, partial differential equations and differential geometry. The goal

of these lectures is to introduce the theory recently developed by Alberto Enciso and the speaker to address the study of these problems. Some of the questions that I will consider are: does there exist a steady Euler flow in Euclidean space having stream lines of all knot and link types? Can the nodal components of

a monochromatic wave exhibit arbitrary topology? Is there a solution to the heat equation with hot spots travelling along any prescribed spatial curve? How does the topology of a quantum vortex change accross reconnections? In the first lecture I will focus on the aspects of the theory for elliptic (time independent) PDE, while in the second part I will consider parabolic and dispersive evolution equations.

G. Uhlmann -* **"Harry Potter's Cloak via Transformation Optics"*

Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc. including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last fifteen years or so there have been several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion one of them, the so-called "transformation optics" that has received the most attention in the scientific literature.

G. Uhlmann -* **"Journey to the Center of the Earth"*

We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among other.

The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.

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Anyone is welcome to attend!

There is no formal **registration**, but please send an email to Frau Minzlaff (corina.minzlaff@rub.de) so that we can estimate the number of participants. Everyone is welcome to join us for a **dinner** on the Thursday evening.

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**Hotels**

Here are some of the hotels in Bochum, please contact Frau Minzlaff if you would like assistance with booking a room.

**Park Inn** - conveniently located next to central station, large rooms, but not cheap (ca. 92 Euro incl. breakfast)).

**Ibis City** und **Ibis Zentrum** - next to central station, small rooms, good price but nothing special - Ibis Zentrum is probably a bit more quiet (ca. 67 incl. breakfast).

**Art Hotel Tucholsky **in the "Bermuda triangle" (party mile), that is, can be noisy at night, but if you reserve a room to the back yard it's very fine, art hotel with funny accessoires - great breakfast, probably the most interesting hotel in the list (ca. 75 Euros incl. breakfast)

**Jugendherberge (youth hostel) Bochum **again in the "Bermuda triangle", it is supposed to be very good and has differnet kinds of rooms (ca. 62 Euros for single rooms incl. breakfast)

General directions including a map of the campus can be found here.

For assistance or questions please contact Corina Minzlaff.

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*We hope to see you there!*