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.
This area is a recently active area, with an old history. More COMING SOON
Simply speaking my particular research projects center around events such as the one 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:
Value distribution theory,
Probability theory, e.g. Slepian's lemma
Several Complex Variables, e.g. regularity of subharmonic functions
Complex Geometry, e.g. the asymptotics of Szego kernels
Paper's:
The zeros of Flat Gaussian Random Functions on C^n and Hole Probability. link
In this paper, based on my thesis work, an argument made by Sodin and Tsirelison to estimate the order of the decay of the hole probability in one variable was adapted in order to solve the analogous problem in several variables. This paper uses standard techniques from several complex variables, including Jensen's formula, and regularity properties of subharmonic functions.
Hole probability of random SU(m+1) polynomials: link
Talks given:
Coming Soon
Bibliography:
Coming Soon
Background: