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Title: Quadratic differentials and extremal problems revealed: the shocking details Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130

Following the comprehensive presentations given by professors Khavinson, Rakhmanov, and Solynin, on the topics mentioned in this title, we'll consider the problem of dynamics of critical trajectories and arrive at what Michael Berry once called “a Victorian discontinuity”.

Title: On extremal problems solved by Quadratic Differentials (QD), Part IV Speaker: E. A. Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

Title: On extremal problems solved by Quadratic Differentials (QD), Part III Speaker: E. A. Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

No seminar this week.

Title: On extremal problems solved by Quadratic Differentials (QD), Part II Speaker: E. A. Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

Title: On extremal problems solved by Quadratic Differentials (QD), Part I Speaker: E. A. Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130

There is a large class of extremal problems in geometric function theory whose solutions may be presented as a union of critical trajectories of a QD. Such are, say, Minimal Capacity type of problems and (more general) Moduli-type problems. The plan is to make an elementary introduction to the field.

Title: Extremal domains for the analytic content, Part II Speaker: Dmitry Khavinson Time: 4:00pm–5:00pm Place: CMC 130

Title: Extremal domains for the analytic content, Part I Speaker: Dmitry Khavinson Time: 4:00pm–5:00pm Place: CMC 130

The analytic content \(\lambda(K)\) of a set \(K\) is the distance in the uniform norm from \(\bar{z}\) to functions analytic in a neighborhood of \(K\). It was shown more than 30 years ago (H. Alexander & D. Khavinson) that $$ \frac{2\mathrm{Area}(K)} {\mathrm{Perimeter}(K)} \le\lambda(K) \le\sqrt{\frac{\mathrm{Area}(K)}\pi}. $$ The upper bound is attained only for disks (H. Alexander & D. Khavinson). The lower bound was conjectured to hold only for disks and annuli (Khavinson's thesis, 1982). In this talk I shall outline the recent proof of this conjecture obtained in a joint work with A. Abanov (Texas A&M), C. Bénéteau and R. Teodorescu (USF).