Laura Desideri Mathematisches Institut Auf der Morgenstelle 10 D - 72 076 Tübingen Office: 5 P 25 Telephone: + 49 (0) 7071 29 78 587 E-mail: desideri(at)mathematik.uni-tuebingen.de

Since September 2010, I am a post-doc in Prof. Dr. Franz Pedit's team in the Geometrie Werkstatt at the Mathematisches Institut of Tübingen University.

#### Teaching

In the Summer Semester 2010-2011, I give a lecture on Surface Theory.

I can find on my "old" web page in Jussieu a summary of my previous teaching (in French).

#### Research

The fields I am working in are differential geometry, more precisely minimal surfaces, and the theory of isomonodromic deformations of Fuchsian systems.

I realized my PhD thesis Problème de Plateau, équations fuchsiennes et problème de Riemann-Hilbert at the Institut de Mathématiques de Jussieu (Paris VII University), under the supervision of Frédéric Hélein. I defended it on December 2009 in Paris. You may download the dissertation in pdf or in ps. The dissertation is also available at thesis online (tel).

During my PhD, I studied an almost forgotten technic to solve the Plateau problem, which has been developed by René Garnier (in 1928) in the case of polygonal boundary curves in Euclidean 3-space. Garnier's method is very different than the variational one, and provides minimal disks without any branch point. However, his proof is sometimes really complicated, and even obscure or incomplete. Following Garnier's initial ideas, I wrote a new proof of this now classical result, and I generalized it to the case of maximal surfaces in Minkowski 3-space.

Garnier's method relies on the fact that we can associate a second-order Fuchsian equation with any minimal surface with a polygonal boundary curve. The monodromy of the equation is determined by the oriented directions of the edges of the polygon. To solve the Plateau problem, we are thus led to solve a Riemann-Hilbert problem, and to construct isomonodromic deformations of Fuchsian equations. When the polygonal boundary curve is quadrilateral, these deformations are given by the sixth Painlevé equation.

You will find here an eight-page English summary of my PhD dissertation.

Garnier's paper Le problème de Plateau is available on Numdam.

Curriculum Vitæ: in English, en français.

#### Preprints

• with R. Jakob, Immersed solutions of Plateau's problem for piecewise smooth boundary curves with small total curvature (submitted), arXiv:1103.1340

• The Plateau problem for polygonal boundary curves in Minkowski 3-space (submitted), arXiv:1012.3582

#### Publications

• Problème de Plateau, équations fuchsiennes et problème de Riemann-Hilbert (French, 122 pages), arXiv:1003.0978. To appear in Mémoires de la Soc. Math. Fr.