苏州大学数学科学学院邀请美国加州大学戴维斯分校夏青岚教授作了题为“Ramified optimal transportation and its applications(网状最佳运输及其应用)”的讲座。苏州大学数学科学学院现有数学一级学科博、硕士学位授予点(下设基础数学、应用数学、计算数学、概率论与数理统计、运筹学与控制论、数学教育六个二级学科博、硕士点),统计学一级学科博、硕士学位授予点(下设数理统计、应用概率、金融风险管理、生物统计、经济统计五个二级学科博、硕士点);应用统计、金融工程、学科教育(数学)三个专业硕士学位点。讲座的主要内容是:
最佳的运输问题的目标是寻求从源目标的具有成本效益的运输。在数学中,有至少两个非常重要的类型优化交通:蒙赫 - 坎托罗维奇问题,分枝最佳交通工具。在这次报告中,我会给出一个简要介绍了理论的网状最佳运输通道.一个动机的理论来源于自然界中发现的分支结构的研究。许多生命系统,如树木,在叶中脉,以及动物心血管/循环系统展览分支结构,为主要非生命系统,如河道网络,铁路,航空网络,电力供应和通信网络.为什么做自然和工程师都喜欢这些枝状结构?什么是对非分支结构,这些分支结构的优势是什么?这些问题部分激励我们去探索其背后的数学。在这次演讲中,我将讨论如何建立一个数学理论的这一普遍现象的最佳传送路径条款。两个概率的措施之间的最优传送路径可以被看作是在概率测度的空间的短程线。在这次演讲中,我也将调查多学科领域的理论,如数学生物学一些应用(如动力形成树的叶子),度量几何结构(例如,在quasimetric空间测地问题),分形几何(如修改后的弥散限制聚集),几何分析(的措施,如运输尺寸)和数理经济学(如网状最优分配问题)。
The optimal transportation problem aims at finding a costefficient transport from sources to targets. In mathematics, there areat least two very important types of optimal transportation:Monge-Kantorovich problem and ramified optimal transportation. In thistalk, I will give a brief introduction to the theory of ramifiedoptimal transportation.One motivation of the theory comes from the study of the branching structures found in nature. Many living systems such as trees, the veins on a leaf, as well as animal cardiovascular/circulatory systems exhibit branching structures, as domany non-living systems such as river channel networks, railways,airline networks, electric power supply and communication networks.Why do nature and engineers both prefer these ramifying structures? What are the advantages of these branching structures over non-branching structures? These questions partially motivates us to explore the mathematics behind them. In this talk, I will talk about how to set up a mathematical theory for this general phenomenon in terms of optimal transport paths. An optimal transport path between two probability measures can be viewed as a geodesic in the space of probability measures. In this talk, I will also survey some applications of the theory in multidisciplinary areas such as mathematical biology (e.g. the dynamical formation of tree leaves),metric geometry (e.g. the geodesic problems in quasimetric spaces),fractal geometry (e.g. the modified diffusion-limited aggregation),geometric analysis (e.g. transport dimension of measures) and mathematical economics (e.g. ramified optimal allocation problem).