Research Interests:

I study random functions of one or more ( real or complex) variable, and largely concerned with when the value 0 is obtained. This research includes studying random polynomials, and random analytic functions. These functions are chosen to respect the underlying space (or manifold), e.g. to have a translationaly invariant zero set (as a stochastic process). This type of construct is very general, and turns out to be surprisingly general (including such general subjects as the distribution of zeros of random sections of the Nth tensor power of a positive line bundle on a compact Kahler Manifolds). To study this subject techniques from complex geometry, probability theory and real geometry are used to attack problems.

Simply speaking my particular research projects center around events where the zeros are either over or under crowded. As a simple example consider the where where there are no zeros in a region where many are expected. This event is often called a hole, and is an example of an event which is a large deviation from the mean. To compute the hole probability a several techniques are used. In particular, various techniques from the following areas have been used to compute large deviation results:

Research Papers and Technical reports:

1. “Estimates on the Probability of Outliers for Real Random Bargamann-Fock Functions,” 2008, ArXiv:math.cv/0807.0604.

2. joint with B. Shiffman, S. Zelditch, “Overcrowding and Hole Probabilities for Random Zeros on Complex Manifolds,” 2008, ArXiv:math.cv/0805.2598.

3. “The Order of Decay for the Hole Probability for Random SU(m+1) Polynomials,” ArXiv:math.cv/0610686 (2007).

4. “The Hole Probability for Random SU(2) Polynomials,” Arxiv:math/0610686 (2006).

5. The zeros of Flat Gaussian Random Functions on Cn and Hole Probability,” MMJ 55, (2007), no. 2, 269-284.

Talks given:

Coming Soon

Bibliography:

Coming Soon

Background:

CV_Zrebiec