Geometric flows in symplectic geometry
数学专题报告
报告题目(Title)：
Geometric flows in symplectic geometry
报告人(Speaker)：
Xiangwen Zhang (University of California, Irvine)
地点(Place)：
ZOOM ID: 913 6112 7888 Passcode: 877592
时间(Time)：
21 Oct 2022, 9am Beijing time.
邀请人(Inviter)：
熊金钢
报告摘要
Geometric flows have been proven to be powerful tools in the study of many important problems arising from both geometry and theoretical physics. Aiming to study the equations from the flux compactifications of Type IIA superstrings, we introduce the so-called Type IIA flow, which is a flow of closed and primitive 3-forms on a symplectic Calabi-Yau 6-manifold. Remarkably, the Type IIA flow can also be viewed as a flow as a coupling of the Ricci flow with a scalar field. In this talk, we will discuss the recent progress on this flow. This is based on a joint project with Fei, Phong and Picard.
*PAST TALKS:
1.Stability in Gagliardo-Nirenberg-Sobolev inequalities: nonlinear flows, regularity and the entropy method, Matteo Bonforte (Universidad Autónoma de Madrid), 16 Sept 2022.
Abstract: We discuss stability results in Gagliardo-Nirenberg-Sobolev inequalities, a joint project with J. Dolbeault, B. Nazaret and N. Simonov. We have developped a new quantitative and costructive "flow method", based on entropy methods and sharp regularity estimates for solutions to the fast diffusion equation (FDE). This allows to study refined versions of the Gagliardo-Nirenberg-Sobolev inequalities that are nothing but explicit stability estimates. Using the quantitative regularity estimates, we go beyond the variational results and provide fully constructive estimates, to the price of a small restriction of the functional space which is inherent to the method.
2.Reverse conformally invariant Sobolev inequalities on the sphere, Tobias Konig (Goethe University Frankfurt), 23 Sept 2022.
Abstract: In this talk I will present a result about the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the n-dimensional sphere S^n involving an operator of order 2s > n. In this case the Sobolev exponent is negative. Our results extend existing ones to noninteger values of s. In particular, we obtain the sharp threshold value of s for the validity of a corresponding Sobolev inequality in all dimensions n >= 2. This is joint work with Rupert L. Frank (LMU Munich and Caltech) and Hanli Tang (Beijing Normal University).
3.Infinite time blow-up for the Keller-Segel system in the plane, Juan Dávila (University of Bath, UK), 30 Sept 2022.
Abstract: We study the Keller-Segel system in the plane with an initial condition with sufficient decay and critical mass 8 pi. We find a function u0 with mass 8 pi such that for any initial condition sufficiently close to u0 and mass 8 pi, the solution is globally defined and blows up in infinite time. This proves the non-radial stability of the infinite-time blow up for some initial conditions, answering a question by Ghoul and Masmoudi (2018). This is joint work with Manuel del Pino (U. of Bath), Jean Dolbeault (U. Paris Dauphine), Monica Musso (U. of Bath) and Juncheng Wei (UBC).
4.The singular set in the Stefan problem, Xavier Ros-Oton (Universitat de Barcelona), 7 Oct 2022.
Abstract: The Stefan problem, dating back to the XIXth century, is probably the most classical and important free boundary problem. The regularity of free boundaries in the Stefan problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is $C^\infty$ in space and time, outside a certain set of singular points. The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs and Geometric Analysis. In particular, a major question in such a context is to establish estimates for the size of the singular set. The goal of this talk is to present some new results in this direction for the Stefan problem. This is a joint work with A. Figalli and J. Serra.
5.Higher order asymptotics for fast diffusions on bounded domains, Beomjun Choi (POSTECH, South Korea), 14 Oct 2022.
Abstract: Sobolev-subcritical fast diffusion with vanishing boundary condition leads to finite-time extinction, with a vanishing profile selected by the initial datum. In a joint work with R. McCann and C. Seis, we quantify the rate of convergence to this profile for general smooth bounded domains. In rescaled time variable, the solution either converges exponentially fast or algebraically slow. In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes, giving a higher order asymptotics. We also improve on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the presence of zero modes.
* This seminar is jointly organized by Tianling Jin at The Hong Kong University of Science and Technology.
相关系列报告：
https://www.math.hkust.edu.hk/~tianlingjin/PDEseminar.html
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