## Brussels-London

geometry seminar

Day-long geometry seminars with three talks on a common theme

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The seminar involves geometers from the Université libre de Bruxelles and the following London colleges:

University College, King's College, Queen Mary's College, and Imperial College.

#### Brussels-London XVI

# Geometric analysis on metric spaces

#### Université Libre de Bruxelles, 26th November 2018

The talks will take place at the Université Libre de Bruxelles. The precise location of the lectures will be confirmed nearer the time.

Registration is free, but necessary so that we know how many people to expect. To register please send an email to Edwine Lukamba. Lunch will be provided for all registered attendees.

#### 13:30 Alexander Lytchak

#### 15:00 Stefan Wenger

#### 16:30 Ilaria Mondello

**Alexander Lytchak.** Classical Plateau problem in non-smooth spaces.

In the talk I will discuss a solution of the most classical formulation of the Plateau problem in general metric spaces and applications of arising minimal discs. The solution provides a simplification even in the classical Euclidean setting. The talk will be based on a joint work with Stefan Wenger.

**Stefan Wenger.** Finding good parametrizations for metric surfaces

By the classical uniformization theorem, every smooth Riemann surface is conformally diffeomorphic to a surface of constant curvature. What happens if the smooth Riemannian metric is replaced by a non-smooth distance? Does the so obtained metric surface still admit parametrizations with good geometric and analytic properties? Such questions have been widely studied in the field of Analysis on metric spaces and are important for example in view of applications to Geometric Group Theory. I will show how one can use recently established existence and regularity results for area and energy minimizing discs in metric spaces to obtain canonical parametrizations of metric surfaces. In particular, we obtain a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parametrizations of linearly locally connected, Ahlfors 2-regular metric 2-spheres. Joint work with Alexander Lytchak.

**Ilaria Mondello.** Synthetic Ricci lower bounds: new geometric examples.

Singular manifolds appear naturally in geometry when considering quotients of smooth manifolds, their Gromov-Hausdorff limits or geometric flows. An important question in the study of such singular manifolds is to define a relevant notion of curvature, or curvature bounds. The work of Lott-Sturm-Villani and Ambrosio-Gigli-Savaré showed that it is possible to define a curvature-dimension condition on metric measure spaces, that corresponds to a Ricci lower bound in the case of smooth Riemannian manifolds. If some constructions on manifolds (quotients, cones, spherical suspension...) give examples of metric spaces satisfying the curvature-dimension condition, there is not any easy criterion to establish whether the RCD(K,N) condition holds on a manifold with simple singularities. In this talk, we present a geometric criterion for a compact stratified space to satisfy the RCD(K,N) condition: this gives a new large class of examples, including among others manifolds with conical singularities, isolated or not. The talk is based on a joint work with J. Bertrand, C. Ketterer and T. Richard.

#### Getting to the ULB

The entire event will take place in the mathematics department of the Université libre de Bruxelles. The department is in building NO, marked on this map which also shows the public transport which serves the campus: metro station "Delta" (line 5) and buses 71, 72 and 95. Note only roads, and not footpaths are shown on this map, the footpaths appear on this alternative plan of the campus. The talks will take place in the Salle de Profs on the 9th floor of NO.

Most international visitors will arrive at Gare du Midi (where the Eurostar and Thalys terminals are). From here the easiest route to the ULB is by metro. Take line 2 or 6 in the direction "Elisabeth", and change at "Arts-Loi" for line 5, direction "Hermann Debroux", which will stop at "Delta". The journey should take approximately 30 minutes.

#### Registration

Registration is free, but necessary in order that we know how many people to cater for. In order to register for the seminar please send an email to Edwine Lukamba. Lunch will be provided for all registered attendees.

Register

#### Past seminars

Click on the seminar names to see the speakers, titles and abstracts.

#### Scientific committee

The organisers of the Brussels-London geometry seminar are Joel Fine (ULB), Jason Lotay (UCL) and Felix Schülze (UCL).

In addition to the organisers, the scientific committee includes:

- Mélanie Bertelson (Université libre de Bruxelles)
- Simone Gutt (Université libre de Bruxelles)
- Reto Müller (Queen Mary's College, London)
- Dmitri Panov (King's College, London)
- Bruno Premoselli (Université libre de Bruxelles)
- Michael Singer (University College, London)
- Guiseppe Tinaglia (King's College, London)

#### Funding

The seminar is supported by the London Mathematical Society, the ERC and the FNRS. Thanks to this support, there is funding to help graduate students and postdocs attend this seminar. If you are based in London, please contact Jason Lotay or Joel Fine.

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