Event Title

The Topology of Random Manifolds

Description

We will investigate zero sets of random functions, starting in the univariate case with a simple question motivated by the fundamental theorem of algebra, "How many zeros of a random polynomial are real?". After discussing the answer provided by M. Kac, we then consider the case of a random manifold determined by the zero set of a random function in several variables. We will focus on a question motivated by seeking a broad point of view on Hilbert's sixteenth problem: "What is the topology of a random real algebraic manifold?". Recent progress addresses the average number of connected components of a random manifold, the embedding of components in the ambient space, and higher-dimensional Betti numbers. We survey these results and give an overview of related problems representing several new avenues of research.

Presenter Profile Page(s)

http://brain2.math.fau.edu/~elundber/

Date of Event

February 7 from 12:00 - 1:00 PM

Location

Mailman-Hollywood Auditorium 2nd Floor

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Feb 7th, 12:00 PM Feb 7th, 1:00 PM

The Topology of Random Manifolds

Mailman-Hollywood Auditorium 2nd Floor

We will investigate zero sets of random functions, starting in the univariate case with a simple question motivated by the fundamental theorem of algebra, "How many zeros of a random polynomial are real?". After discussing the answer provided by M. Kac, we then consider the case of a random manifold determined by the zero set of a random function in several variables. We will focus on a question motivated by seeking a broad point of view on Hilbert's sixteenth problem: "What is the topology of a random real algebraic manifold?". Recent progress addresses the average number of connected components of a random manifold, the embedding of components in the ambient space, and higher-dimensional Betti numbers. We survey these results and give an overview of related problems representing several new avenues of research.