学术报告—曹军教授
邀请专家：曹军教授 (浙江工业大学)
时间：2025年3月21日 15:30-16:30
地点：新葡京娱乐城门户网站 4108
题目：薛定谔算子及其对应的调和分析
摘要： 狄氏型是在度量测度上建立光滑性分析的重要工具。在狄氏型理论中，正则性通常是许多丰富理论的重要基础前提。在这个报告中，我们将介绍狄氏型的正则性与一些Sobolev型不等式的密切关系。
专家简介： 曹军，浙江工业大学数学科学学院教授，主要从事调和分析及其应用方向研究，相关结果发表在《J. Math. Pures Appl.》、《J. Lond. Math. Soc.》、《Trans. Amer. Math. Soc.》等国内外期刊，主持国家自然科学基金面上项目和浙江省杰出青年基金项目。
学术报告—Naotaka Kajino教授
邀请专家：Naotaka Kajino教授 (日本京都大学数理解析研究所)
时间：2025年3月21日 14:00-15:00
地点：新葡京娱乐城门户网站 4108
题目：Impossibility of quasisymmetric Gaussian uniformization, via decay rates of harmonic functions, for Brownian motion on some planar Sierpiński carpets
摘要： It is an established result in the field of analysis of heat equations andassociated diffusions (Markov processes with continuous sample paths) on fractals, that the heat kernel (the transition density of the diffusion) typically satisfies analogs of Gaussian bounds which involvea space-time scaling exponent β greater than two and therebyare called SUB-Gaussian bounds. The exponent β ，called the walk dimension of the diffusion, could be considered as representing “how close the geometry of the fractal is to being smooth”. It has been observed by Kigami in [Math. Ann. 340 (2008) 781-804] that, in the case of the standard two-dimensional Sierpiński gasket, one can decrease this exponent to two (so that Gaussian bounds hold) by suitable changes of the metric and the measure while keeping the associated Dirichlet form (the quadratic energy functional) the same. Then it is natural to ask how general this phenomenon is for diffusions on fractals. In fact, it turns out that the above phenomenon, that one can decrease the exponent β to two so that Gaussian bounds hold, seems to happen only for a very limited class of self-similar fractals. This talk is aimed at presenting the result that this phenomenon indeed does NOT happen for the Brownian motion on a class of two-dimensional Sierpiński carpets, as well as for the Brownian motion on the standard three- and higher-dimensional Sierpiński gaskets. The key to the proof is some knowledge about decay rates of harmonic functions, which for Sierpiński carpets seems new and is of independent interest. This talk is based on joint works with Mathav Murugan (University of British Columbia). The results for planar Sierpiński carpets is in progress, and that for the standard higher-dimensional Sierpiński gaskets is given in[Invent. math. 231 (2023), 263-405].
专家简介： Professor Kajino is working on probability theory, with the principal research interest in analysis of Laplacians and their associated diffusion processes on fractals, especially how analytic properties of Laplacians reflect the geometry of the fractals. The classes of fractals include the classical Euclidean self-similar fractals such as the well-known Sierpinski gasket and Sierpinski carpet, and self-conformal fractals appearing naturally in complex function theory such as the limit sets of Kleinian groups and the Julia sets of complex dynamical systems. Recently he has been studying also L^p-type nonlinear energy functionals on fractals and their associated p-harmonic functions and p-energy measures for p > 1.
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2025.03.20