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This week's seminar has been replaced by a colloquium.

Title: Chebyshev constants and the inheritance problem Speaker: Vilmos Totik Time: 4:00pm–5:00pm Place: PHY 108

Recently a lower bound was given by K. Schiefemayr for the Chebyshev constants corresponding to a set of finite intervals. In this talk we give a matching upper bound with the aid of the statement of a so-called inheritance problem: To divide an inheritance \(m\) brothers turn to a judge. Secretly, however, each of them bribes the judge. What a given brother inherits depends continuously and monotonically in the bribes: it is monotone increasing in his own bribe and it is monotone decreasing in everyone else's bribe. Show that if the eldest brother does not give too much to the judge, then the others can choose their bribes so that the decision will be fair, i.e., each of them gets the same share as without bribes.

Title: Variational identities and their applications to soliton equations Speaker: Wen-Xiu Ma Time: 4:00pm–5:00pm Place: PHY 108

We will show that associated with matrix spectral problems, there exist variational identities involving solutions to stationary zero curvature equations. The resulting variational identities can be used to generate Hamiltonian structures for the corresponding soliton equations.

Title: Dirichlet's problem and lightning bolts Speaker: Erik Lundberg Time: 4:00pm–5:00pm Place: PHY 108

If Dirichlet's problem is posed with any entire data on an ellipse, then the solution is known to be entire. We consider other algebraic curves and present a method for locating singularities developed by analytically continuing a solution with entire data. The technique uses annihilating measures supported on finite sets called “lightning bolts” which go back to Kolmogorov and Arnold's solution of Hilbert's 13th problem.

Title: Approximation on Hausdorff compacts and applications Speaker: Arthur Danielyan Time: 4:00pm–5:00pm Place: PHY 108

We consider a function defined on a compact Hausdorff space and discuss the problem of pointwise approximation of such function by elements of a closed subspace of continuous (on the compact) complex valued functions. The approximation method is universal in the sense that it gives also uniform approximation on certain subsets whenever such approximation is possible at all.

Title: Unbounded Subnormal Operators, Part III Speaker: Sherwin Kouchekian Time: 4:00pm–5:00pm Place: PHY 108

Title: Unbounded Subnormal Operators, Part II Speaker: Sherwin Kouchekian Time: 4:00pm–5:00pm Place: PHY 108

Title: Unbounded Subnormal Operators, Part I Speaker: Sherwin Kouchekian Time: 4:00pm–5:00pm Place: PHY 108

We start with a short review of some aspects of the theory of subnormal operators. Then we move on to the concept of subnormality in the unbounded case. It will be shown that the defined notion of subnormality is not only an exercise for the sake of generalization, as there are important concrete examples of unbounded subnormal operators. For instance, the famous Creation Operator is such an example. Finally, we will focus our attention to the class of unbounded multiplication operators and, if time permits, present some of our obtained results in this regard.