# Alexei Miasnikov

Dr. Alexei Miasnikov: Innovative Approaches to Mathematics

*Stevens faculty member recognized as an Inaugural Fellow of the American Mathematical Society*

Professor Alexei Miasnikov, Director of the Department of Mathematical Sciences at Stevens Institute of Technology, has been named an inaugural Fellow of the American Mathematical Society (AMS). The AMS, the primary organization of American research mathematicians, has named a small percentage of its approximately thirty thousand members in the first year of the Fellows program. The program recognizes members who have made outstanding contributions to the creation, exposition, advancement, communication, and utilization of mathematics.

“This great honor affirms the enormous impact of Dr. Miasnikov’s contributions to the mathematical sciences,” says Dr. Michael Bruno, Dean of the Charles V. Schaefer, Jr. School of Engineering and Science. “Under his leadership, the Department of Mathematical Sciences is fostering innovative approaches to both research and education that are reverberating throughout the mathematical sciences community and beyond.”

Dr. Miasnikov continues to lead groundbreaking research into previously unapproachable mathematical topics. In collaboration with Professor Olga Kharlampovich of Hunter College, Dr. Miasnikov was awarded consecutive grants by the National Science Foundation to develop a framework using first-order logic that allows mathematicians to reconsider previously unsolvable problems. The researchers developed an algorithm to state categorically with first-order theory that certain mathematical objects held a definitive property. This algorithm opened the gates to answering a host of other unsolved problems, such as ongoing work on the first-order theory of natural numbers.

Dr. Miasnikov is also conducting important research in the field of algorithmic algebra, with applications in both theoretical and applied areas of mathematics. In mathematics and its applications, and in particular algebra, there are many problems for which there is no all-encompassing algorithm—i.e., it is impossible to write a program that will solve all instances of the problem. In these situations, Dr. Miasnikov has found that the impossibility of finding a total algorithm does not preclude the possibility of finding an effective algorithm. Furthermore, a focus on most cases could result in optimized algorithms. Dr. Miasnikov says, “Sometimes, we can develop an algorithm that proves to be useful for most instances of a problem, and the occasions in which the problem cannot be solved are rare. Indeed, we often find a total algorithmic solution for some problems which turns out to be very slow, in which case it might be more useful to implement fast algorithms that work for most instances.”

This research has substantial implications in cryptography. For example, there may be a problem used to encrypt data for which no total algorithm can be written, but for which a partial algorithm solves most instances relatively painlessly. In such a case, encryption that relies on the difficulty of that problem becomes susceptible to being broken. According to Dr. Miasnikov, “New types of mathematical problems related to cryptography require new, more practical types of algorithmic thinking."

Along with Dr. Robert Gilman of the Stevens Department of Mathematical Sciences, Dr. Miasnikov received a grant from the National Security Agency's (NSA) Mathematical Sciences Program to test a novel approach to the Andrews-Curtis conjecture. Dr. Miasnikov also 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.

The AMS Fellows program allows mathematicians to be recognized by their peers as distinguished for their contributions to the profession. The program also seeks support the advancement of more mathematicians to leadership positions in their own institutions and in society at large.