Dr. Alexei Miasnikov and Dr. Robert Gilman of the Department of Mathematical Sciences at Stevens Institute of Technology recently received a two-year research grant from the National Security Agency's (NSA) Mathematical Sciences Program to test their novel approach to the Andrews-Curtis conjecture, a famous unsolved problem from topology and combinatorial group theory. Their innovative computation-based approach will pave the way for future exploration of this and other conjectures.
The Andrews-Curtis conjecture demonstrates that even with today’s technology, many areas of mathematics remains undefined. Dr. Gilman and Dr. Miasnikov’s investigation will illuminate our understanding with a new method based on computation with finite - but huge - approximations to the infinite Andrews-Curtis graphs. “We have a novel approach in using computers to solve these theoretical problems,” Dr. Miasnikov explains. “Our research could change the way people think about the conjecture. Beyond that, our computations have potential applications to theoretical math as well as physics.”
Proposed in 1965, the Andrews-Curtis conjecture asserts that a certain family of infinite graphs derived from finite presentations of the trivial group is connected. Although the groups involved are trivial, the conjecture is not.
The conjecture is thought by many to be false, and a list of possible counterexamples to the conjecture was proposed in 1985, but even these needed to be verified. Mathematicians have since attempted to prove or disprove the counterexamples as a way to prove or disprove the conjecture. In 1999 Dr. Miasnikov and his nephew Alex Myasnikov disproved one of the proposed counterexamples through the use of genetic algorithms. Though some other counterexamples have been disproven, many may remain beyond the reach of present day computers.
“From a group theory standpoint, the Andrew-Curtis conjecture seemed too good to be true,” Dr. Gilman says, “but so far no one has been able to prove or disprove it.” His work along with Dr. Miasnikov is a step toward better understanding unsolved problems in mathematics.
“The grant serves as an acknowledgement of Dr. Misasnikov and Dr. Gilman’s success in mathematics, and speaks to their outstanding contributions to the field,” says Dr. Michael Bruno, Dean of the Charles V. Schaefer, Jr. School of Engineering and Science. “Their techniques have the potential to unravel some mysteries of mathematics and open new avenues of exploration.”
About the Professors
Dr. Robert Gilman is a Professor of Mathematical Sciences in the Charles V. Schaefer, Jr. School of Engineering and Science. He is Director of the Algebraic Cryptography Center, which focuses on research in specific thrust areas, including Cryptography and Cryptanalysis, Generic Complexity, Statistical Algebra, and Asymptotic Group Theory.
Department Director of Mathematical Sciences in the Charles V. Schaefer, Jr. School of Engineering and Science, Dr. Miasnikov is a also member of the Algebraic Cryptography Center at Stevens. Dr. Miasnikov is also a recent recipient of the prestigious and competitive Marsden Fund Award as a Principal Investigator in a project that studies mathematical structures described by finite state machines such as automata and beyond.
About the NSA Mathematical Sciences Program
The NSA Mathematical Sciences Program supports self-directed, unclassified research in the areas of Algebra, Number Theory, Discrete Mathematics, Probability, and Statistics.