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Mathematics & Statistics

Analysis
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Friday, November 30, 2012

Title
Speaker
Time
Place

Quadratic differentials and extremal problems revealed: the shocking details
Razvan Teodorescu
4:00pm-5:00pm
CMC 130

Abstract

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”.

Friday, November 9, 2012

Title
Speaker
Time
Place

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

Friday, November 2, 2012

Title
Speaker
Time
Place

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

Friday, October 26, 2012

No seminar this week.

Friday, October 19, 2012

Title
Speaker
Time
Place

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

Friday, October 12, 2012

Title
Speaker
Time
Place

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

Abstract

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.

Friday, October 5, 2012

Title
Speaker
Time
Place

Extremal domains for the analytic content, Part II
Dmitry Khavinson
4:00pm-5:00pm
CMC 130

Friday, September 28, 2012

Title
Speaker
Time
Place

Extremal domains for the analytic content
Dmitry Khavinson
4:00pm-5:00pm
CMC 130

Abstract

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).