ANALYSIS  SEMINAR (UTK)

Abstract: In Complex Dynamics, we study the iteration of holomorphic or meromorphic functions on the complex plane or the Riemann sphere. Of particular interest is the behavior of the critical orbits of function families with one or more parameters. The simplest of such families, z^2 +c, is well-known to define the famous Mandelbrot set fractal as the set of c-values for which the unique critical orbit is bounded. In this talk we will examine the function family R(z) = z^n +b +a/(z^d), and we will explore old and new results establishing the location of baby Mandelbrot sets in parameter space for increasingly general versions of this family. In the most general case, which we call maximally generalized McMullen maps, this family has multiple independent critical orbits, and the dynamics in this case are not yet well understood.

Abstract: While the boundedness properties of Calderón-Zygmund singular integral operators are classical in harmonic analysis, a theory for compact CZ operators has more recently been established. We present new developments in the theory of compact CZ operators. In particular, we give a new formulation of the T1 theorem for compactness of CZ operators, which, compared to existing compactness criteria, more closely resembles David and Journé’s original T1 theorem for boundedness and follows from a simpler argument. Additionally, we discuss the extension of compact CZ theory to weighted Lebesgue spaces via sparse domination methods. This talk is based on joint works with Mishko Mitkovski, Paco Villarroya, and Brett Wick. 

Abstract: A long-standing problem in metric geometry is to recognize those metric spaces that admit a “good” parameterization by the Euclidean space or the unit sphere. In this talk, we will discuss the quasisymmetric uniformization problem and some recent advancements in terms of the Menger curvature, a curvature suited for metric spaces with no a priori smooth structure. The talk is based on a joint work with G. C. David (Ball State University).

Abstract: A Dirichlet-type space is a generalization of the Hardy and Drury–Arveson spaces, both of which play a central role in functional analysis and operator theory. Within this class of spaces, our focus is on those that are not complete Pick spaces. A key aspect of our study concerns cyclic functions, a concept of significant interest in functional analysis. To gain further insight into the structure and behavior of cyclic functions in these spaces, we introduce the notion of sequentially weak* cyclicity and explore its implications.

Abstract:  In this talk, we will discuss a structure result for cubic Blaschke products that says there is a unique hyperbolic inflection point in the unit disk and, moreover, this point lies at the hyperbolic midpoint of the two critical points. Using this structure result for cubic Blaschke products, we give a new classification for their dynamics based on an explicit expression in terms of the parameters. This is based on joint work with Alexandra Hill.

Abstract: Conformal welding homeomorphisms are circle homeomorphisms that arise naturally in Teichmuller theory, mathematical physics and dynamics. It is well known that not every circle homeomorphism is a conformal welding, and that non-conformally equivalent curves can give rise to the same welding. However, in this talk we will see that every circle homeomorphism is the composition of two conformal weldings. Our approach uses the log-singular circle homeomorphisms introduced by Bishop, as well as flexible curves, which are the curves that correspond to them. We will also see how a large class of non-conformally equivalent flexible curves can correspond to the same log-singular circle homeomorphism.