The 2015-16 D'Atri Memorial Lecturer is
Fernando Coda Marques

Professor of Mathematics at

Professor of Mathematics at

*Princeton University*
22nd D'Atri Memorial Lectures

Open to the Public – Use Lots 60A or 64 behind Hill Center

D'Atri Memorial Lecture Series

Min-max Theory for the Area Functional - A Panorama

Dates

Friday, February 19 | Tuesday, February 23

Abstract

In these lectures we will give a current panorama of the min-max theory for the area functional, initially devised by Almgren in the 1960s and improved by Pitts (1981). This is a deep high-dimensional version of the variational theory of closed geodesics. The setting is very general, being that of Geometric Measure Theory, and the main application until very recently was the construction of minimal varieties of any dimension in a compact Riemannian manifold. In the past few years we have discovered new applications of this old theory, including a proof of the Willmore conjecture, of the Freedman-He-Wang conjecture, and of Yau's conjecture (about the existence of infinitely many minimal hypersurfaces) in the positive Ricci curvature setting. We will give an overview of these results, describe our current efforts to develop the theory further (the first general Morse index estimates) and discuss future directions. Most of the material covered in these lectures is based on joint work with Andre Neves.

Lecture 1 :

"Min-max Theory for the Area Functional - A Panorama –

*Part I*"**Friday, February 19, 2016**

Time/Location : 2:00pm Hill Center 705

Lecture 2 :

"Min-max Theory for the Area Functional - A Panorama –

*Part II*"**Tuesday, February 23, 2016**

Time/Location : 1:40pm Hill Center 705

**Joseph D'Atri**was born in 1938, received an A.B. from Columbia College in 1959, and received a Ph.D. from Princeton University in 1964. He was a member of the Rutgers University Mathematics Department from 1963 until his death in 1993 and served as Chair of the Department from 1985 until 1990. Further information about Professor D'Atri and his work in geometry may be found in the book

*Topics in Geometry: in Memory of Joseph D'Atri*edited by Simon Gindikin and published in 1996 by Birkhauser as volume 20 in the series

*Progress in Nonlinear Differential Equations and Their Applications*.

Previous D'Atri Memorial Lectures

Lecturer | Date | Title(s) |
---|---|---|

Frank Merle | Jan 2015 |
A Road Map for the Soliton Resolution Conjecture Universality Questions for Blow-up Behavior for the Mass Critical Korteweg-de-Vries Equations |

Nicola Fusco | Jan 2014 |
Stability and Minimality of a Non-local Variational Problem The Quantitative Isoperimetric Inequality |

Vladimir Sverak | Dec 2012 | Mathematical Aspects of the Navier-Stokes and Euler's Equations |

Sun-Yung Alice Chang | Feb 2012 | Conformal Invariants: Perspectives From Geometric PDE Higher Order Isoperimetric Inequalities – An Approach via Method of Optimal Transport |

Gang Tian | Dec 2010 | Geometric Equations in Dimension 4 B-field Renormalization Flow on Complex Surfaces |

Carlos Kenig | Feb 2010 | Some Recent Applications of Unique Continuation to Mathematical Physics, PDE, and Fourier Analysis
The Global Behavior of Solutions to Critical Non-Linear Dispersive and Wave Equations |

Yakov Eliashberg | Oct 2008 | Symplectic Topology of Stein Manifolds On the Border between Flexible and Rigid Mathematics |

Henri Berestycki | Nov 2006 | Reaction-Diffusion Equations in Nonhomogeneous Media |

Neil Trudinger | Mar 2006 | Nonlinear Elliptic PDE and Optimal Transportation Nonlinear Elliptic PDE and Geometric Invariance |

Helmut Hofer | Nov 2004 | Somewhere Between Hamiltonian Dynamics and Symplectic Geometry |

Paul Rabinowitz | 2004 | An Aubry-Mather Theory for Partial Differential Equations |

Richard Hamilton | 2003 | On the Ricci Flow |

Luis A. Caffarelli | 2002 | Constraint Methods for Nonlinear Homogenization in Periodic and Random Media, I, II |

Brian White | 2001 | Total Curvature, Soap Films and the Unreasonable Effectiveness of Mathematics |

Stefan Hildebrandt | 2000 | On Two-dimensional Parametric Variational Problems Uniqueness Results for Minimal Surfaces with Free Boundaries |

Richard Schoen | 1999 | The Plateau Problem in Complex Geometry |

Nigel Hitchin | 1998 | Hyperkahler Geometry |

Misha Gromov | 1997 | Metric Geometry of Algebraic Manifolds |

Eugenio Calabi | 1995 | On Singular Sympletic Structures |

John Milnor | 1994 | Geometry and Dynamics |

Jacques Faraut | 1993 | Ordered Symmetric Spaces |