Experiments and expectations in computational algebra

3:00pm, Wednesday 24 June 2026

Abstract

Experimental results have long played a key role in number theory, suggesting conjectures, testing heuristics, and guiding the development of algorithms.  At the same time, the scale and complexity of modern computations raise new horizons for these experiments and new questions for how we use computational algebra software.  In this talk, I will discuss several recent examples.  The first is joint work with Blair Butler and Edgar Costa on a statistical approach to understanding ranks of elliptic curves.  The second is joint work with Andreas-Stephan Elsenhans on computing class groups of number fields.  I will conclude with some general questions about our expectations for computer algebra systems and how these expectations may evolve as formalization and AI/machine learning tools become increasingly integrated into mathematical practice.