Random Algebraic Geometry: Together with Antonio Lerario and Saugata Basu I am studying topology of the zero sets of random polynomials. We have a small seminar this semester learning this new area and also related topics in random matrix theory.

Our main focus is on the expected Betti numbers of random polynomials with various distributions on the coefficients. For metric properties such as number of zeros and volume of zero sets of random algebraic polynomials and systems of equations, with an appropriate distribution on coefficients, the expectation can be calculated exactly using the beautiful integral geometry methods described in the paper by A. Edelman and E. Kostlan on on real zeros of random polynomials. Similar methods were also used successfully in a series of papers by Shub and Smale.

On the other hand, for Betti numbers, there are not yet any unified methods, but some promising recent progress has been made in separate settings. Welschinger and Gayet studied the Kostlan (complex Fubini-Study) distribution, and obtained upper bounds for Betti numbers. Nazarov and Sodin used a "barrier method" to obtain lower bounds for the expected number of nodal domains of a random spherical harmonic, showing that the order of expectation is maximal. Antonio Lerario and I recently studied the number of connected components of a random homogeneous polynomial in the real Fubini-Study ensemble and showed that (preprint) the order of expectation is maximal. Viewing spherical harmonics as eigenfunctions on the sphere, our result extends Nazarov and Sodin's by allowing arbitrarily many variables, and a full spectrum of frequencies up to some eigenvalue (rather than just a single eigenspace). We used Nazarov and Sodin's barrier method, which had seemed ad hoc since it utilized some specific aspects of harmonic function theory, but it was possible to extend via a detailed study of the Gegenbauer polynomials. My preprint with Lerario also discusses interpretation of these results and ensuing questions related to a random version of Hilbert's sixteenth problem. That a whole spectrum of eigenfunctions would exhibit this behavior had been suggested in an inspiring letter from P. Sarnak to B. Gross and J. Harris (letters) where he described the random curve as "4% Harnack" based on numerical computations from the thesis of his undergraduate student, M. M. Nastasescu, which indicated that the number of ovals is approximately 4% of that of a Harnack curve.

The random study of the topology of real algebraic sets is a very new field and there are many exciting open questions:

For instance, what about intersections of hypersurfaces? What happens as the number of variables goes to infinity (as opposed to the large degree limit)? What is the variance of the number of connected components? On the more analytic side, what are the scaling limits of these random curves, and are they related to SLE? Somewhere between analysis and topology, the work of Liviu Nicolaescu studies the critical points of random functions which suggest that the variance is proportional to the expectation which would indicate that the number of connected components (if highly correlated with the number of critical points) is highly concentrated about its mean.

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