# Difference between revisions of "PDE Geometric Analysis seminar"

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|Haozhao Li (University of Science and Technology of China) | |Haozhao Li (University of Science and Technology of China) | ||

|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]] | |[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]] | ||

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Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen. | Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen. | ||

+ | |||

+ | |||

+ | ===Haozhao Li (University of Science and Technology of China)=== | ||

+ | Regularity scales and convergence of the Calabi flow | ||

+ | |||

+ | We define regularity scales to study the behavior of the Calabi flow. | ||

+ | Based on estimates of the regularity scales, we obtain convergence theorems | ||

+ | of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence | ||

+ | of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for | ||

+ | the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra | ||

+ | assumption that the scalar curvature is uniformly bounded. |

## Revision as of 17:46, 21 January 2015

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

## Contents

### Previous PDE/GA seminars

### Tentative schedule for Fall 2015

# Seminar Schedule Spring 2015

date | speaker | title | host(s) |
---|---|---|---|

January 21 (Departmental Colloquium: 4PM, B239) | Jun Kitagawa (Toronto) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics | Feldman |

January 26 | |||

February 2 | Jessica Lin (Madison) | TBA | Kim |

February 9 | |||

February 16 | |||

February 17 (joint with Analysis Seminar: 4PM, B139) | Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier | Seeger |

February 23 | Jennifer Beichman (Madison) | TBA | Kim |

March 2 | Benoit Pausader (Princeton) | TBA | Kim |

March 9 | Haozhao Li (University of Science and Technology of China) | Regularity scales and convergence of the Calabi flow | Wang |

March 16 | |||

March 23 | Ben Fehrman (University of Chicago) | TBA | Lin |

March 30 | Spring recess Mar 28-Apr 5 (S-N) | ||

April 6 | |||

April 13 | |||

April 20 | Yuan Lou (Ohio State) | TBA | Zlatos |

April 27 | |||

May 4 |

## Abstracts

### Jun Kitagawa (Toronto)

Regularity theory for generated Jacobian equations: from optimal transport to geometric optics

Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.

### Haozhao Li (University of Science and Technology of China)

Regularity scales and convergence of the Calabi flow

We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.