Topology of Surfaces with Finite Willmore Energy
Dienstag, 21.1.20, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we care about the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type \(C^{\balpha}\) regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in \(\bmathbb{R}^n\) with finite Willmore energy. Especially, we prove a removability of singularity of multiplicity one surface with finite Willmore energy and a uniqueness theorem of the catenoid under no a priori topological finiteness assumption.\n
On the mean field limit of the Doi-Onsager model for liquid crystals
Dienstag, 4.2.20, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
The microscopic Doi-Onsager model is one of the most fundamental theories for liquid crystals. Formally, one can derive the macroscopic liquid crystals theories such as the Oseen-Frank/Ericksen-Leslie theory from it. We will discuss this issue in a rigorous framework. In particular, we show that when the typical molecular interaction distance tends to zero (called the mean field limit), minimizers or critical points of Onsager's energy functional converge to minimizing harmonic maps or weak harmonic maps respectively. If time permits, we will also show that, under the same limit, solutions of the dynamical Doi-Onsager equation without hydrodynamics will converge to weak solutions of the harmonic map heat flow, which can be viewed as a special Oseen-Frank gradient flow. These are based on joint works with Yuning Liu (NYU Shanghai).