Innovative new framework opens door to solving unapproachable mathematical questions

Dr. Alexei Miasnikov, Director of the Department of Mathematical Sciences at Stevens Institute of Technology, in collaboration with Professor Olga Kharlampovich of Hunter College, has been awarded a grant by the National Science Foundation to continue development of a framework using first-order logic that allows mathematicians to approach previously unsolvable problems. The grant is titled "Collaborative research: model theory and algebraic geometry in groups and algebras, non-standard actions, algorithmic problems."  Supported by two separate NSF panels, it will potentially provide insights into several fundamental unsolved problems in group theory and algebra. “The frontier has been widened immensely,” says Dr. Miasnikov. “Many answers are now within our reach.”

Dr. Alexei Miasnikov
Alexei Miasnikov

“This prestigious grant is indicative of the strength of theoretical research at Stevens,”says Dr. Michael Bruno, Dean of the Charles V. Schaefer, Jr. School of Engineering and Science.“Dr. Miasnikov’s groundbreaking work has already created new threads of research, and this grant ensures that it will go even further to transform our understanding of essential mathematical questions.”

Most of the fundamental properties of objects that occur in everyday mathematical practice can be described in a very particular, stripped down, but universal language called first-order logic. This powerful language is used in philosophy, linguistics, and computer science in addition to mathematics.  In their previous work, Dr. Miasnikov and Dr. Kharlampovich developed an algorithm to state categorically with first-order theory that certain mathematical objects (in that instance free groups) held a definitive property. This algorithm opened the gates to answering a host of other unsolved problems, such as the first-order theory of natural numbers, which they will confront in their new research project.

Dr. Miasnikov and Dr. Kharlampovich’s work extends a legacy of research that stretches back through the ages. Equations have long established a universal language to describe scientific problems in all their variety. As a result of centuries of work, equations and their solutions have become very complex. In order to understand their hidden structure, one must use very elaborate techniques from algebra and geometry.  Equations alone can no longer sufficiently describe subtleties of scientific phenomena; new problems require much more powerful means and more powerful languages. Dr. Miasnikov and Dr. Kharlampovich’s work empowers the language of first-order logic to formalize and study previously unapproachable mathematical questions.

Their contributions will open more perspectives and avenues for research. “The process of solving certain problems requires the resolution of many smaller questions,” says Dr. Miasnikov. “You build a tremendous theoretical procedure or framework that can turn out to be more important than the solution itself, because it can be applied to a myriad of other problems to dismantle formerly daunting mathematical obstacles.”

Dr. Miasnikov received the prestigious and competitive Marsden Fund Award in 2010 for the project, “From automatic groups to automatic structures and beyond,” an international collaboration with Dr. Bakh Khoussainov from the University of Auckland in New Zealand, and Dr. Olga Kharlampovich from McGill University. Along with Dr. Robert Gilman of the Stevens Department of Mathematical Sciences, he received a grant from the National Security Agency's (NSA) Mathematical Sciences Program to test a novel approach to the Andrews-Curtis conjecture.

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