Geometry of Polynomials, Operator-Valued Maps, Polar and Non-Commutative Convex Analysis

多项式几何、算子值映射、极坐标和非交换凸分析

• 批准号: RGPIN-2020-06425
• 负责人: Sendov, Hristo
• 金额: $ 1.97万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741204
• 关键词: Geometry; Polynomials; Operator; Valued; Maps

The theory of (Hausdorff) Geometry of Polynomials is full of beautiful results, patterns, and possesses a wealth of conjectures. Notable are Sendov conjecture (1962), Smale mean-value conjecture (1981), Borcea variance conjectures (1998), Rerh conjecture (2012). The Sendov and Smale conjectures are proven in particular cases, but no connection between the two is known. There are programs such as: for a given simply connected domain in the plane, characterize all polynomials having zeros and critical points in it. Recently, we introduced the notion of a locus of a complex polynomial. It is a minimal closed set in the plane that contains a zero of any of its apolar polynomials. We established general properties of the loci and showed its connections with other classical theorems: Laguerre's, Rolle's, Grace-Szego-Walsh' Coincidence theorems. The loci provide extremal versions of each of these theorems. We used the loci to obtain a Rolle's theorem for complex polynomials, that is stronger than all previously known such results. My long-term goal is to investigate the numerous intriguing properties of loci, isolate subclasses of loci: smallest area, smooth boundary, or with symmetries; and develop efficient algorithms for the (approximate) computation of a locus. Connections between loci and polar derivatives, lead us to the notion of polar convexity. Polar convexity, generalizes the usual convexity and is well-suited for describing relationships between zeros and critical points of polynomials. We used it to strengthen Laguerre's theorem. My goal is to obtain stronger versions of classical results. One can give a new refinement of the Gauss-Lucas theorem, one complementing recent ones by Dimitrov (1998) or Curgus & Mascioni (2004). Polar convexity in higher dimensions is of interest too. I am interested in the theory of spectral (SpF) and isotropic (IF) functions. They are of significant interest and find applications in areas such as complex analysis, optimization, non-smooth and matrix analysis, elasticity, and quantum physics. Recently, we formulated a direct connection between SpF and IF by introducing a family of operator-valued maps, called k-isotropic functions. The case k=1 reduces to the SpF and the case k=n to the IF. The k-isotropic functions explain the differences in properties of the SpF and the IF. We did so when the properties are differentiability and operator monotonicity. The next steps are to look at operator convexity and numerous other properties. The operator monotone IF are an important example in the recent theory of matrix (non-commutative) convexity. In fact, they are exactly the single-valued matrix convex functions on R. The foundations were laid down by Effros & Winkler (1995) extending a definition of Wittstock (1984). It finds applications in quantum mechanics, information theory, non-commutative polynomials, spectrahedra. My goal is to extend the classical convex analysis to this non-commutative setting.

（豪斯多夫）多项式几何理论充满了美丽的结果、模式，并拥有丰富的猜想。值得注意的是 Sendov 猜想 (1962)、Smale 均值猜想 (1981)、Borcea 方差猜想 (1998)、Rerh 猜想 (2012)。森多夫和斯梅尔猜想在特定情况下得到了证明，但两者之间的联系尚不清楚。有一些程序，例如：对于平面中给定的简单连通域，表征其中具有零点和临界点的所有多项式。 最近，我们引入了复多项式轨迹的概念。它是平面中的最小闭集，包含其任何非极性多项式的零点。我们建立了轨迹的一般性质，并展示了它与其他经典定理的联系：拉盖尔、罗尔、格雷斯-塞戈-沃尔什的符合定理。轨迹提供了每个定理的极值版本。我们使用这些轨迹获得了复杂多项式的罗尔定理，该定理比以前已知的所有此类结果都更强。我的长期目标是研究基因座的众多有趣特性，分离基因座的子类：最小面积、平滑边界或具有对称性；并开发用于轨迹（近似）计算的有效算法。 轨迹和极导数之间的联系使我们得出极凸性的概念。极凸性概括了通常的凸性，非常适合描述多项式的零点和临界点之间的关系。我们用它来加强拉盖尔定理。我的目标是获得经典结果的更强版本。人们可以对高斯-卢卡斯定理进行新的改进，补充 Dimitrov (1998) 或 Curgus & Mascioni (2004) 最近提出的定理。高维中的极凸性也很有趣。 我对光谱 (SpF) 和各向同性 (IF) 函数的理论感兴趣。它们引起了人们的极大兴趣，并在复分析、优化、非光滑和矩阵分析、弹性和量子物理等领域得到了应用。最近，我们通过引入一系列算子值映射（称为 k 各向同性函数），在 SpF 和 IF 之间建立了直接联系。 k=1 的情况简化为 SpF，k=n 的情况简化为 IF。 k 各向同性函数解释了 SpF 和 IF 特性的差异。当属性是可微性和运算符单调性时，我们这样做了。下一步是研究算子凸性和许多其他属性。 算子单调 IF 是最近矩阵（非交换）凸性理论中的一个重要例子。事实上，它们正是 R 上的单值矩阵凸函数。Effros & Winkler (1995) 扩展了 Wittstock (1984) 的定义，奠定了基础。它在量子力学、信息论、非交换多项式、谱面体中都有应用。我的目标是将经典的凸分析扩展到这种非交换设置。