Lecturer

Professor Tristan Rivière

Coordinator

Alexis Michelat

Official website : Link Fundamental notions of Geometric Measure Theory and Differential Geometry of Minimal Submanifolds from respectively :

• “Lectures on Geometric Measure Theory" Australian National University (1983), by Leon Simon.
• "Introduction to Geometric Measure Theory" (2014) Lecture notes (2014), by Leon Simon (this is a more up to date version of the previous notes by the same author).
• “A course in Minimal Surfaces" AMS (2011), by Tobias H. Colding and William P. Minicozzi II.

The study of minimal surfaces takes its origins in the works of Euler and Bernouilli from the eigtheen century. Since then, minimal surfaces have become central objects in mathematics much beyond the field of geometry with applications in analysis, applied mathematics, theoretical physics and natural sciences. There have been tremendous developments in the field in the last few years, with a particular emphasis on the variational methods, which permitted to solve several old problems. In the seminar we shall concentrate first on the Almgren-Pitts min-max theory for the construction of codimension one minimal surfaces in arbitrary closed manifolds as well as on the Gromov-Guth min-max widths. We will then carefully study a series of recent works by Marques and Neves about the realization of these widths by minimal hypersurfaces. The paramount of the seminar will be the presentation of their very recent proof of a conjecture by Yau asserting the existence of infinitely many embedded minimal hypersurfaces on any closed manifold of dimension strictly less than 8 for generic metrics.

Date	Lectures	Speaker
28 February 2018	Introduction to Geometric Measure Theory I	Alexis Michelat
7 March 2018	Introduction to Geometric Measure Theory II	Alexis Michelat
14 March 2018	Gromov and Guth p-Sweep-Outs for Hypersurfaces	Alexis Michelat
21 March 2018	Existence of infinitely many minimal hypersurfaces in positive Ricci curvature I	Alessandro Pigati
28 March 2018	Existence of infinitely many minimal hypersurfaces in positive Ricci curvature II	Alessandro Pigati
11 April 2018	Geometric aspects of semilinear elliptic PDEs and minimal hypersurfaces on closed manifolds II	Dr. Marco Guaraco
18 April 2018	No Lecture
25 April 2018	No Lecture
2 May 2018	Bumpy Metric Theorem and Morse Index of Minimal Hypersurfaces	Yang Liu and Jaume De Dios
9 May 2018	Weyl Law for the Volume Spectrum I	Alexis Michelat
16 May 2018	Weyl Law for the Volume Spectrum II	Alessandro Pigati
23 May 2018	Denseness and equi-distribution of minimal hypersurfaces	Prof. Dr. Alessandro Carlotto

• “A course in Minimal Surfaces" AMS (2011), by Tobias H. Colding and William P. Minicozzi II.
• "Geometric Measure Theory" Springer-Verlag (1969), by Herbert Federer.
• "Seminar on geometric measure theory" DMV Seminar, 7. Birkhäuser Verlag, Basel, 1986. 117 pp. ISBN: 3-7643-1815-5, by Robert Hardt and Leon Simon.
• ”Existence and regularity of minimal surfaces on Riemannian manifolds” Princeton University Press (1981), by Jon T. Pitts.
• “Lectures on Geometric Measure Theory" Australian National University (1983), by Leon Simon.
• "Introduction to Geometric Measure Theory"s Lecture notes (2014), by Leon Simon.