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

多项式的（豪斯多夫）几何理论充满了美丽的结果，模式，并拥有丰富的知识。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）。森多夫和斯梅尔的理论在特定情况下得到了证明，但两者之间没有联系。这些方案包括：对于平面上给定的单连通区域，刻画了所有具有零点和临界点的多项式。最近，我们引入了复多项式的轨迹的概念。它是平面上的一个极小闭集，包含它的任何非极多项式的零点。我们建立了轨迹的一般性质，并指出它与其他经典定理：Laguerre定理，Rolle定理，Grace-Szego-沃尔什重合定理的联系。轨迹提供了这些定理的极值版本。我们使用的轨迹，以获得一个罗尔定理复杂的多项式，这是强于所有以前已知的结果。我的长期目标是研究轨迹的许多有趣的特性，分离轨迹的子类：最小区域，光滑边界或对称性;并开发轨迹的（近似）计算的有效算法。 轨迹和极导数之间的联系，引导我们到极凸的概念。极凸性推广了通常的凸性，非常适合描述多项式零点与临界点之间的关系。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。人们可以给出高斯-卢卡斯定理的一个新的改进，它补充了Dimitrov（1998）或Curgus & Mascioni（2004）最近的结果。在更高的维度上的极凸性也是令人感兴趣的。 我感兴趣的理论的频谱（SpF）和各向同性（IF）的功能。他们是显着的兴趣，并找到应用领域，如复杂的分析，优化，非光滑和矩阵分析，弹性和量子物理。最近，我们制定了一个直接的连接之间的SpF和IF通过引入一个家庭的运营商值的映射，称为k-各向同性功能。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的性质差异。当性质是可微性和算子单调性时，我们这样做。接下来的步骤是研究算子凸性和许多其他性质。 算子单调IF是近年来矩阵（非交换）凸性理论中的一个重要例子。实际上，它们都是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、非对易多项式、光谱面体中有应用。我的目标是将经典的凸分析扩展到这个非交换的设置。

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.