3:00pm - 5:00pmProbability and randomness in commutative algebra and algebraic geometry
Chair(s): Dane Wilburne (Brown University, United States of America), Christopher O'Neill (San Diego State University)
Randomness has long been used to study polynomials. Several classical instances include Lit- tlewood and Offord’s examination of the expected number of real roots of an algebraic equation defined by random coefficients, as well as work of Kac and Kouchnirenko on varieties defined by random coefficients on a fixed Newton polytope support. Additionally, the use of smooth analysis, which measures the expected performance of an algorithm under slight random perturbations of worst-case inputs, has been used in the context of algebraic geometry. The aim of this minisymposium is to highlight a recent surge of interactions between the fields of probability and commutative algebra/algebraic geometry, in which questions of expected (average, typical) or unlikely (rare, non-generic) behavior of ideals and varieties are studied formally using probability distributions. Recent work has seen the successful application of techniques from statistics and probabilistic combinatorics in this setting. Our goal is to bring researchers working in this intersection together to share their work and form potential new collaborations.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
What can be predicted in algebraic geometry?
Lily Silverstein
UC Davis
This talk explores how supervised machine learning can be used to predict the algebraic and combinatorial properties of polynomial ideals prior to doing a computation. This is practical for fast approximations of algebraic invariants, and also for the problem of "algorithm selection." When several algorithms exist for performing an exact computation, we train a neural network to automatically select the fastest algorithm, on a case-by-case basis, by learning features of the input that are predictive of algorithm performance.
Joint work with Jesus De Loera, Robert Krone, and Zekai Zhao.
Degree of Random Monomial Ideals
Jay Yang
University of Minnesota
In joint work with Lily Silverstein and Dane Wilburne, we investigate the behavior of the standard pairs of a random monomial ideal. We then use this to explore the degree and arithmetic degree of random monomial ideals.
Stochastic Exploration of Real Varieties
David Kahle
Baylor University
Nonlinear systems of polynomial equations arise naturally in many applied settings. The solution sets to these systems over the reals are often positive dimensional spaces that in general may be very complicated yet have very nice local behavior almost everywhere. Standard methods in real algebraic geometry for describing positive dimensional real solution sets include cylindrical algebraic decomposition and numerical cell decomposition, both of which can be costly to compute in many practical applications. In this talk, we communicate recent progress towards a Monte Carlo framework that provides a probabilistic method for exploring such real solution sets. After describing how to construct probability distributions whose mass focuses on a variety of interest, we show how state-of-the-art Hamiltonian Monte Carlo methods can be used to sample points near the variety that may then be magnetized to the variety using endgames. We conclude by showcasing trial experiments using practical implementations of the method in the probabilistic programming language Stan.
Random numerical semigroups
Christopher O'Neill
San Diego State University
A numerical semigroup is a subset of the natural numbers that is closed under addition. Consider a numerical semigroup S selected via the following random process: fix a probability p and a positive integer M, and select a generating set for S from the integers 1,...,M where each potential generator has probability p of being selected. What properties can we expect the numerical semigroup S to have? For instance, how many minimal generators do we expect S to have? In this talk, we answer several such questions, and describe some surprisingly deep geometric and combinatorial structures that arise naturally in the process.