## Alessandro Carlotto: Catalogue data in Spring Semester 2022 |

Name | Dr. Alessandro Carlotto |

Field | Mathematics |

Department | Mathematics |

Relationship | Assistant Professor |

Number | Title | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|

401-3830-22L | Seminar on Minimal Surfaces (an Invitation to Geometric Analysis) The total number of students who may take this course for credit is limited to twenty; however further students are welcome to attend. | 4 credits | 2S | A. Carlotto | |

Abstract | This course is meant as an invitation to some key ideas and techniques in Geometric Analysis, with special emphasis on the theory of minimal surfaces. It is primarily conceived for advanced Bachelor or beginning Master students. | ||||

Learning objective | The goal of this course is to get a first introduction to minimal surfaces both in the Euclidean space and in Riemannian manifolds, and to see some analytic tools in action to solve natural geometric problems. Students are guided through different types of references (standard monographs, surveys, research articles), encouraged to compare them and to critically prepare some expository work on a chosen topic. This course takes the form of a working group, where interactions among students, and between students and instructor are especially encouraged. | ||||

Content | The minimal surface equation, examples and basic questions. Parametrized surfaces, first variation of the area functional, different characterizations of minimality. The Gauss map, basic properties. The Douglas-Rado approach, basic existence results for the Plateau problem. Monotonicity formulae and applications, including the Farey-Milnor theorem on knotted curves. The second variation formula, stability and Morse index. The Bernstein problem and its solution in the two-dimensional case. Total curvature, curvature estimates and compactness theorems. Classification results for minimal surfaces of low Morse index. | ||||

Literature | The three basic references that we will mostly refer to are the following ones: [Whi16] B. White, Introduction to minimal surface theory. Geometric analysis, 387-438, IAS/Park City Math. Ser., 22. American Mathematical Society, Providence, RI, 2016. [CM11] T. Colding, W. Minicozzi, A course in minimal surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp. [Oss86] R. Osserman, A survey of minimal surfaces. Second edition. Dover Publications, Inc., New York, 1986. vi+207 pp. Further, more specific references will be listed during the first introductory lectures. | ||||

Prerequisites / Notice | In addition to the first four semesters of the Bachelor program in Mathematics (in particular all courses in Real and Complex Analysis, Measure Theory, Topology), some background in Differential and Riemannian Geometry is certainly a must. At the very least, students are expected to have taken Differential Geometry 1, and possibly be enrolled in the follow-up course Differential Geometry 2. In addition, some prior exposure to partial differential equations (primarily of elliptic type, and especially on basic topics like Schauder estimates and the maximum principle), although not strictly necessary, may certainly help. | ||||

401-5350-00L | Analysis Seminar | 0 credits | 1K | A. Carlotto, A. Figalli, N. Hungerbühler, M. Iacobelli, L. Kobel-Keller, T. Rivière, J. Serra, University lecturers | |

Abstract | Research colloquium | ||||

Learning objective | |||||

Content | Research seminar in Analysis |