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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614202
• 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”指的是该主题涉及微积分的扩展和概括（由牛顿和莱布尼茨发明，用于研究行星的运动，通常被称为科学的伟大智力成就之一）。分析使人们能够以精确的方式确定量如何依赖于其他变量，例如，地球卫星的速度如何随时间演变。一般来说，这种关系称为数学函数。当然，可能有“良好的”函数，如卫星的速度，但也可能有非常不受控制和剧烈波动的关系，我的建议是致力于新的代数和几何方法，以解决一些长期存在的问题，如尚未解决的著名的电晕问题，某些家庭的功能（该定义起源于太阳的日冕，围绕太阳的等离子体光环）和与H.庞加莱猜想（也因庞加莱猜想而闻名，庞加莱猜想是最近由G。Perelman）。使用我早期工作中开发的一些分析和几何方法，我希望在这些问题中的每一个问题上都取得重要进展，这将保证加拿大数学家在这些领域的领导地位。