Opportunities and Challenges in 21st Century Mathematical Computation: ICERM Workshop Report

“Experimental mathematics ” has emerged in the past 25 years or so to be-come a competing paradigm for research in the mathematical sciences. So what exactly is “experimental mathematics”? While several definitions have been offered (e.g., [6]), perhaps the most succinct definition is given in the book The Computer as Crucible: Experimental mathematics is the use of a computer to run computa-tions — sometimes no more than trial-and-error tests — to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search. [7, pg. 1] Here we should distinguish “experimental mathematics ” from “computational mathematics ” and “numerical mathematics. ” While there is no clear delin-eation, the latter two terms generally encompass computational methods for concrete applied mathematics and engineering applications, whereas “experi-mental mathematics ” usually applies more specifically to computations that advance the state of the art in mathematical research. While the overall approach and philosophy of experimental mathematics has not changed greatly in the past 25 years, its techniques, scale and sociology hav