ANALYSIS SEMINAR
Professor Pawel WOLFF
Department of Mathematics
Case Western Reserve University
"Hypercontractivity of Ornstein-Uhlenbeck semigroup and related weak-type estimates"
Date: Wednesday, September 10, 2008
Time: 4:30-5:30
Location: Department of Mathematics, Seminar Room, Yost 300
Abstract:
A Markov semigroup $(P_t)_{t \ge 0}$ with an invariant measure $\mu$ is hypercontractive if for some $1 < p < q < \infty$ there exists $t_0 > 0$ such that $P_{t_0}$ as an operator from $L^p(\mu)$ to $L^q(\mu)$ has norm $1$. I will present the most classical results from the theory of hypercontractive semigroups, namely hypercontractivity of Ornstein-Uhlenbeck semigroup (due to E. Nelson) and hypercontractivity of a convolution semigroup on the discrete cube (due to A. Bonami). Also, a brief overview of some of its applications will be given. Next, inspired by Talagrand's conjecture on weak type estimates for the convolution semigroup acting on $L^1$, I will discuss a related problem in the case of Ornstein-Uhlenbeck semigroup. Many ideas that will be presented here are due to K. Ball, F. Barthe, W. Bednorz and K. Oleszkiewicz.
ANALYSIS SEMINAR
Professor Pawel WOLFF
Department of Mathematics
Case Western Reserve University
"Hypercontractivity of Ornstein-Uhlenbeck semigroup and related weak-type estimates , Part 2
Date: Wednesday, September 17, 2008
Time: 4:30-5:30
Location: Department of Mathematics, Seminar Room, Yost 300
COLLOQUIUM
Professor Harsh MATHUR
Department of Physics
Case Western Reserve University
"Fractal Analysis and Drip Paintings"
Date: Friday, September 19, 2008
Time: 3:00-4:00
Location: Department of Mathematics, Seminar Room, Yost 300
Refreshments will precede the talk.
Abstract:
Fractal analysis has recently been used in a debate over the authenticity of a cache of newly discovered paintings that may be the work of Jackson Pollock. We have re-examined the basic ideas that underlie the fractal analysis: first, that the "definining visual character" of Pollock's drip paintings is their fractal nature; second, that the process by which he created fractals is chaotic motion over the canvas; and third, that Pollock's paintings have distinctive quantitative features that can be used to aid in the authentication of newly discovered works of uncertain provenance. The shortcomings of the fractal analysis and of the method as an aid in authentication will be discussed. Two new results in fractal analysis will also be presented. Their investigation was motivated by the study of drip paintings but the results are of much broader significance. First we have shown that the composite of two fractals is not scale invariant and that it has complex asymptotic scaling behaviour on fine scales. Second we have studied the statistics of the box counting staircase. These statistics are shown to provide a new way to characterize geometric objects and to provide a novel distinction between Fractals and Euclidean objects.
[1] K Jones-Smith and H Mathur, Nature 444, doi:10.1038/nature05398
[2] K Jones-Smith, H Mathur and L M Krauss, arxiv:0710.4917
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