Abstract:
Plateau’s problem is a main interest in geometric measure theory. It aims at understanding the behavior of physical objects that admit certain minimizing property, such as soap films. Physical soap films are probably more accurately modeled by the (M, 0, delta)-minimal sets of Frederick Almgren, but the lack of algebraic coherence makes it difficult to prove minimality.
In the 90’s, K. Brakke, G. Lawlor & F. Morgan introduced the method of paired calibration to prove various minimality of sets satisfying a given separation condition. It is very often used in the classification of singularities for codimension 1 minimal sets in Plateau’s problem. Compared to ordinary algebraic calibration methods, a major advantage of paired calibrations is that it ignores algebraic multiplicities, which corresponds to the spirit of Plateau’s problem. However, in general we do not know a generalisation to codimension larger than one, and, at first glance, products of paired calibrations can fail to be paired calibrations.
In this talk, we will first introduce the background and definitions for Almgren minimal sets, classification of singularities for Almgren’s minimal sets, and give examples where minimality is proved by paired calibrations. Then we will give an example where we use product of paired calibrations to prove minimality of the set Y × Y.
Biography:
Xiangyu Liang is a Professor at the School of Mathematics and System Sciences, Beihang University. She received B.S in Mathematics from Tsinghua University in 2006, and Ph.D. in 2011 from University of Paris-Sud. From 2012 to 2014 she was a Postdoc at the University of Warwick. She next became permanent faculty at the University of Lyon 1, France. In 2016 she joined Beihang University through the 12th “Recruitment Program of Global Experts” for Young professionals. Xiangyu Liang’s main area of interest is geometric measure theory. Her research is mainly done on Plateau’s problem, classification of singularities, tangent structure and related problems.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai