Mathematics Colloquium, Fall 2007
Trjitzinsky Memorial Lectures
October 2-4, 2007

Michael Lacey will present the Trjitzinsky Memorial Lectures, October 2 - 4.
A reception will be held following the Tuesday, October 2, lecture in 314 Altgeld Hall.

• Lecture 1. Irregularities of Distribution and Related Questions, 4 p.m. Tuesday, Oct. 2, 314 Altgeld Hall
  A reception will be held following the Tuesday, October 2, lecture in 314 Altgeld Hall.

  Abstract: The subject of Irregularities of Distribution concerns identification of optimal rates of convergence to uniform distribution. This classical topic is mostly understood in average case analysis. Certain endpoint estimates remain stubbornly resistant, despite decades of research on the topic. These questions are in turn related to arising in Harmonic Analysis, Probability Theory, and Approximation Theory. Given N points P_N in the unit cube, define the Discrepancy Function by
  
  $$ D_N (x) = |P_N \cap [0,x)| - N |[0,x)| $$
  
  where x is the rectangle with antipodal vertices at the origin and at x in the unit cube. The importance of this function is highlighted by the classical Koksma-Hlawka inequality, a fact we recall in the lecture. We describe the average case analysis of D_N , namely a universal lower bound on the L² norm of D_N due to Klaus Roth. Its extensions, via square function inequalities, is presented. The endpoint lower bounds for D_N remain a mystery in dimensions three and higher.

• Lecture 2. Irregularities of Distribution and Related Questions, 4 p.m. Wednesday, Oct. 3, 245 Altgeld Hall
  Coffee and cookies will be served prior to the lecture in Room 321 Altgeld Hall.

  Abstract: The second lecture will highlight the connections between these questions and other areas of Analysis and Probability, which has recently come to the fore. We recall the very pretty analysis of the two dimensional versions of these questions due to Gabor Halasz. And then turn attention to the essential insights of Jozef Beck in the three dimensional questions, and the recent extensions of those ideas obtained by the speaker jointly with D. Bilyk, and A. Vagharshakyan.

• Lecture 3. Hankel Matrices, Commutators and Product BMO, 4 p.m. Thursday, Oct. 4, 245 Altgeld Hall
  Coffee and cookies will be served prior to the lecture in Room 321 Altgeld Hall.

  Abstract: Hankel matrices play a distinguished role in operator theory for the $\ell ^2(N)$ due to the fact that they intertwine in a natural way with the shift operator on that space. The characterization of bounded Hankel matrices is recalled, along with three classical proofs, all relating to function theory. These proofs display different advantages, throughly explored over the last fifty years. Hankel matrices with Hankelian entries are especially delicate, and don't fall directly under the three general approaches to Hankel matrices. We describe the emerging theory of such matrices. This is joint work with several collaborators, S. Ferguson, J. Pipher, S. Petermichl, and E. Terwilleger.