Algebraic and Geometric Approaches to Some Long-standing Problems of Analysis

一些长期存在的分析问题的代数和几何方法

• 批准号: RGPIN-2015-06535
• 负责人: Brudnyi, Alexander
• 金额: $ 1.82万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=646415
• 关键词: Algebraic; Geometric; Approaches; Some; Long

My research interests are in the general areas of Analysis. The ''Analysis'' refers to the fact the subject deals with extensions and generalizations of the Calculus (which was invented by Newton and Leibnitz to study the motion of the planets and is often referred to as one of the great intellectual achievements of Science). Analysis allows one to determine, in a precise manner, exactly how quantities depend on other variables, for example, how the velocity of an earth satellite evolves over time. In general, such relationships are called mathematical functions. Of course, there may be ''well-behaved'' functions such as the velocity of a satellite, but there may also be very uncontrolled and wildly fluctuating relationships, such as the price of a share on the stock market over time. My proposal is devoted to new algebraic and geometric approaches to some long-standing problems of Analysis such as the still unsolved famous Corona problem for certain families of functions (the definition is originated from Sun's corona, an aura of plasma that surrounds the Sun) and the Center problem for differential equations related to the still unsolved classical Center-Focus problem of H. Poincare (known also for the fame Poincare conjecture, one of the seven Millennium Prize Problems recently solved by G. Perelman). Using some analytic and geometric methods developed in my earlier work, I expect to make an essential progress in each of these problems which would guarantee the leadership of Canadian mathematicians in these areas. ******

我的研究兴趣是分析的一般领域。“分析”指的是微积分(由牛顿和莱布尼茨发明，用于研究行星运动，通常被称为科学的伟大智力成就之一)的推广和推广。通过分析，人们可以准确地确定量是如何依赖于其他变量的，例如，地球卫星的速度如何随着时间的推移而演变。一般而言，这种关系称为数学函数。当然，可能有像卫星速度这样的“行为良好的”函数，但也可能有非常失控和剧烈波动的关系，比如股票市场上的股票价格随着时间的推移。我的建议致力于用新的代数和几何方法来解决一些长期存在的分析问题，例如某些函数族的仍未解决的著名日冕问题(该定义源于太阳的日冕，围绕太阳的等离子体光环)，以及与H·庞加莱的经典中心焦点问题有关的微分方程中心问题(也因著名的庞加莱猜想而闻名，佩雷尔曼最近解决的七个千禧年奖问题之一)。利用我早期工作中开发的一些解析和几何方法，我希望在每一个问题上都取得实质性进展，这将保证加拿大数学家在这些领域的领导地位。