The Brownian intersection exponents are computed by Lawler, Schramm and Werner (Acta Math, 2001a, 2001b, 2002) [conjectured by Duplantier and Kwon (Phys Rev Lett, 1988)]. These exponents are useful in determining the Hausdorff dimension of some exceptional sets of planar Brownian motion. In this talk, we discuss our recent development on such exponents. First, we show that a family of generalized exponents can be related to the fractal geometry of the Brownian loop soup (a Poissonian ensemble of Brownian loops in 2D). Second, we prove that the corresponding 3D exponents are analytic, which remains open to this day.

报告人简介：高一帆，现在是香港城市大学博士后，主要研究方向是统计物理和随机几何（渗流、高斯自由场、SLE等），于2022年在北京大学获得博士学位，导师为李欣意和章复熹老师。 取得了诸多重要成果，被Ann. Probab.和Comm. Math.Phys.等杂志接收或发表。