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

多项式的(Hausdorff)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)，斯梅尔均值猜想(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。有这样的程序：对于平面上给定的单连通区域，刻画所有在其上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，它包含它的任一非极多项式的零。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-萨戈-沃尔什重合定理。这些轨迹提供了这些定理中每一个的极端版本。我们利用这些轨迹得到了复多项式的一个罗尔定理，它比所有已知的此类结果都要强。我的长期目标是研究轨迹的众多有趣的性质，分离出轨迹的子类：最小的面积、光滑的边界或具有对称性；并开发高效的算法来(近似)计算轨迹。轨迹和极导数之间的联系，将我们引向极凸性的概念。极凸性，推广了通常的凸性，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。人们可以给出Gauss-Lucas定理的一种新的改进，补充了Dimitrov(1998)或Curgus&Mascioni(2004)最近的改进。更高维度的极凸性也很有趣。我对谱(SPF)和各向同性(IF)函数理论感兴趣。它们在复杂分析、优化、非光滑和矩阵分析、弹性和量子物理等领域具有重要的意义和应用。最近，我们通过引入一族算子值映射，称为k-各向同性函数，建立了SPF和IF之间的直接联系。K=1的情况减少到SPF，而k=n的情况减少到IF。K-各向同性函数解释了SPF和IF在性质上的差异。当性质是可微性和算子单调性时，我们就这样做了。下一步是研究操作符的凸性和许多其他性质。算子单调IF是最近的矩阵(非交换)凸性理论中的一个重要例子。事实上，它们就是R上的单值矩阵凸函数。Effros&Winkler(1995)推广了Wittstock(1984)的定义。它在量子力学、信息论、非对易多项式、多面体等领域有着广泛的应用。我的目标是将经典的凸分析推广到这种非对易的环境中。

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.