报告题目：A New Theoretical Framework for Nonlinear Analysis via the MCCF System and Sharpness Operators-The Resolution to Schauder Conjecture

报 告 人：袁先智 教授（华东理工大学）

报告时间：2026年8月3日（星期一）上午11：20-12：00

报告地点：数学科学学院115（大报告厅）  

校内联系人： 朱传喜 教授   联系方式：84708351-8420

报告摘要：Dating back to the foundational works of Schauder (1930), Leray-Schauder (1934), and Tychonoff (1935), a major open quest in non-linear functional analysis has been to determine the ultimate geometric boundaries of topological fixed-point principles. For nearly a century, Schauder's Problem 54 in \textit{The Scottish Book} and Eilenberg's Problem 138 have remained two of the most stubborn open challenges in infinite-dimensional topology due to the catastrophic collapse of linear duality ($E^* = \{0\}$) in non-locally convex spaces. To overcome this fundamental structural barrier, this paper establishes a definitive, constructive resolution to both problems within general Hausdorff topological vector spaces (TVSs). By introducing a novel combinatorial-topological framework termed the \textit{Monotone Continuous Covering Family (MCCF) system} and deploying \textit{Yuan's Sharpness Operator}  (Yuan 2025-2026) and related references wherein, we substitute traditional linear separation principles with a powerful non-linear $``$topological concentration$"$ mechanism, utilizing the Yuan Sharpness Operator as a universal finite-rank approximation tool in general TVSs. Under this infrastructure, we prove that: (1) every continuous mapping defined on a non-empty compact subset of a Hausdorff TVS possesses the finite-rank-approximation property, thereby universally satisfying Klee's (1960) admissibility criterion; and (2) every compact $p$-convex set for $p \in (0,1]$ is a definitive Absolute Retract (AR) for the class of normal spaces in the sense of van Mill (1989).

报告人简介 ：袁先智教授先后在四川大学(1982-1989)，加拿大多伦多大学(97-98)，戴尔豪斯大学(90-94) 获得数学学士，数学硕士，金融工程硕士，统计学硕士， 数学博士学位。袁博士目前是华东理工大学等高校的特聘或访问教授；国际金融工程期刊（Int. Journal of Financial Engineering, IJFE）的主编；也是原国际非线性分析期刊（Nonlinear Analsysi, TMA）等学术期刊的编委成员；是2013年和2017年上海和四川分别引进的省部级人才，袁博士目前也是某科技公司的首席科学家.

袁博士在国内外SCI和SSCI学术刊物发表了180多篇专业论文，出版6本专著。在非线性分析、数理经济、金融工程、博弈论、金融科技等研究方面取得了一系列处于国际领先的系统性结果。