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

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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.