# 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.

**Spring 2021 Math Colloquium**

**Tuesdays @ 4pm California Time**

**"tea" (meet-and-greet) @ ~15m before the talk**

**Schedule ***(click dates for title and abstract)*

4/20 | Burak Hatinoglu | UCSC |

5/4 | Suresh Eswatharan | University of Dalhousie |

5/25 | William Goldman | University of Maryland |

6/1 |

**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

**Suresh Eswatharan, University of Dalhousie**

*Entropy of logarithmic modes*

Given a map T acting on a space M and a T-invariant measure μ, the Kolmogorov-Sinai entropy of μ is a non-negative number that describes, in some sense, the complexity of a μ-typical orbit of T. In this talk, we consider hyperbolic surfaces M and the Kolmogorov-Sinai entropy of special flow-invariant measures, namely semiclassical measures, that arise from linear combinations of Laplace-Beltrami eigenmodes on M, namely ε-logarithmic modes. Here, ε quantifies the spectral width of these modes. We give a lower bound for the Kolmogorov-Sinai entropy of these semiclassical measures and show it carries a continuous dependence on ε. This generalizes some earlier work of Anantharaman-Koch-Nonnenmacher for eigenmodes.

**William Goldman, University of Maryland**

*Dynamics on character varieties and geometric structures*

The classification of gometries on low-dimensional manifolds leads to interesting dynamical systems on moduli spaces associated with surfaces. In my talk I will survey some examples and describe some future directions.