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Title: Serre Duality on Compact Riemann Surfaces, Part III Speaker: Bryce Virgin Time: 4:00pm–5:00pm Place: CMC 130

Due to the Thanksgiving holiday, there will be no seminar this week.

This week's seminar is cancelled.

Due to the Veteran's Day holiday, there will be no seminar this week.

Title: Serre Duality on Compact Riemann Surfaces, Part II Speaker: Bryce Virgin Time: 4:00pm–5:00pm Place: CMC 130

Title: Serre Duality on Compact Riemann Surfaces Speaker: Bryce Virgin Time: 4:00pm–5:00pm Place: CMC 130

Riemann surfaces are connected complex manifolds of complex dimension one. First defined in Bernhard Riemann's Ph.D. dissertation in 1851, they have since been a central object of study in the field of complex analysis. Accordingly, a theory which allows one to answer important questions about Riemann surfaces is therefore highly desirable. Sheaves and their cohomology provide an option for recovering nontrivial global information about Riemann surfaces, using the structure of algebras of functions on the surface defined by the local condition of holomorphicity. In this talk, we will briefly discuss the application of cohomology to the investigation of Riemann surfaces as a motivation for its study, and then outline a proof of the ancillary version of Serre Duality used in these applications.

Title: Flow of zeros of polynomials generated by repeated differentiation, Part III Speaker: E. A. Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

Title: Flow of zeros of polynomials generated by repeated differentiation, Part II Speaker: E. A. Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

Title: Flow of zeros of polynomials generated by repeated differentiation Speaker: E. A. Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

I plan to discuss the following problem. Let \(P_n(z)\) be a sequence of polynomials; \(\deg P_n=n\) and their unit normalized counting measure of zeros iare weakly convergent to a known measure. Let \(0 < t <1\) and \(Q_n\) be the derivative of \(P_n\) of order \(tn\). What can we say about the convergence of unit normalized counting measure of zeros of \(Q_n\)?

A large number of important particular results are known (history goes back to 18th century). A more or less general solution has been obtained recently by A. Martínez-Finkelshtein and me. There are connections of the problem and its solution to many other problems in analysis (special functions, approximation theory, potential theory, Riemann surfaces to mention a few). I will try to outline the situation in comparatively elementary terms.

Title: Extensions to the Douady–Earle extension, Part III Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130

Title: Extensions to the Douady–Earle extension, Part II Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130

Title: Extensions to the Douady–Earle extension Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130

The Douady–Earle extension associates a unit disk diffeomorphism (conformal map) to a unit circle homeomorphism, by relating both to a positive measure supported on the circle. A classical result of Milnor and Thurston classifies the limit of iterative compositions of these maps, leading to a hyperbolic fixed point related to the measure's barycenter. We apply this method to characterize the long-time dynamics of a discrete (atomic) measure on the circle and generalize the result to include stochastic perturbations to the complex dynamics.

Title: On the conjugate functions Speaker: Arthur Danielyan Time: 4:00pm–5:00pm Place: CMC 130

The conjugate harmonic functions are well known. However, in the theory of Fourier series, a conjugate function is defined also for the functions given just on the real line. By a classical theorem, for any periodic and (Lebesgue) integrable function the conjugate function exists almost everywhere. In this talk, we discuss a theorem on the continuity of the conjugate functions. As an application, we present the complete solution of the following open problem (from a well-known problem list of 1980): Let \(u\) be a continuous and real function on the unit circle. Give a necessary and sufficient condition on \(u\) such that \(u\) is the real part of a function in the disc algebra.