Mathematics Colloquium

The Mathematics Department Colloquium is a quarterly series of invited speakers from all mathematical fields and geared toward a broad audience. Talks are generally every Tuesday from 4-5 PM, with an informal tea beforehand. For the duration of COVID-19, it will be held via Zoom, with a meet-and-greet ~15m before the talk-- bring your own beverage.

[Archives]

Spring 2021 Math Colloquium
Tuesdays @ 4pm California Time
"tea" (meet-and-greet) @ ~15m before the talk
Meeting ID: 276-046-2984, Passcode: 962595. [Direct link]
For further information please contact
Professor Francois Monard or call 831-459-1525

Schedule (click dates for title and abstract) 

4/6
4/13
4/20 Burak Hatinoglu UCSC
4/27
5/4 Suresh Eswatharan University of Dalhousie
5/11
5/18
5/25 William Goldman University of Maryland
6/1

Tuesday, April 20, at 4pm

Burak Hatinoglu, UCSC

Norm estimates and asymptotics of Chebyshev polynomials

The Chebyshev polynomial Tn,K of degree n associated to an infinite compact subset K of the complex plane is the unique, degree n monic polynomial, which minimizes the sup-norm on K among all degree n monic polynomials. Classical theorem of Gabor Szeg ̈o says that

limn→∞||Tn,K||1/n
K = Cap(K),

where Cap(K) denotes the logarithmic capacity of the set K. This result is one of the oldest results connecting the approximation theory and the potential theory. In his seminal 1969 paper, “Extremal Polynomials Associated with a System of Curves in the Complex Plane”, Harold Widom systematically considered the ratios ||Tn,K||K/Capn(K) for finite unions of smooth Jordan curves and arcs.

Even though it is a classical topic, the research on Chebyshev polynomials is still active and many results were obtained in the last 10 years.

In this talk, after talking about some basic properties and necessary background, I will discuss some recent results on norm estimates and asymptotics of Chebyshev polynomials. I will mostly focus on results related with the ratios ||Tn,K||K/Capn(K), which are also called Widom factors after Harold Widom’s above mentioned fundamental paper. As time permits, I will also talk about related results for orthogonal polynomials with respect to a probability measure supported on the set K.

I will try to make the talk accessible to graduate students


Tuesday, May 4, at 4pm

Suresh Eswatharan, University of Dalhousie

tba


Tuesday, May 25, at 4pm

William Goldman, University of Maryland

tba