AMHERST, Mass. – Mathematician Alexei Oblomkov of the University of Massachusetts Amherst has won a five-year, $420,000 faculty early career development (CAREER) grant from the National Science Foundation, the agency’s most prestigious award to support junior faculty who exemplify the role of teacher-scholars through outstanding research, excellent education and integrating education and research with their institution’s mission.
As a representation theory mathematician, Oblomkov works at the intersection of several different areas including topology, algebraic geometry, knot theory, braid groups and number theory. Topology, which grew out of set theory and geometry, is the mathematical study of shapes, spaces, boundaries and properties of space preserved in deformations such as stretching and bending. “It’s very exciting that my work synthesizes or bridges many fields and looks for connections between them,” Oblomkov says.
At present he works mainly with geometric questions motivated by string theory in physics. He explains, “Theoretical physicists investigate strings, that is an object moving in three-dimensional space, and they produce formulae about these objects. In particular, they propose different approaches for understanding the complexity of knots.”
“Mathematicians like me come up with computer-based algorithms to measure the complexity of knotted strings or braids,” Oblomkov adds, pulling a shoelace out of one shoe to show how a physicist or mathematician might pose a simple question about an open string with one knot in it, tying the ends together and asking whether the knot can be untied. In this case it could not, but that’s not always true, as Oblomkov demonstrates next by putting a complicated-looking braid in the open shoelace, tying the ends together again, and easily untying the braid.
One of Oblomkov’s goals is to develop new algorithms and discover new properties of algorithms already in use. Other mathematicians will be interested in the results, because sometimes knots with symmetries turn out to be related to other branches of mathematics, he points out. “Some knots have symmetry, so one of the questions you can answer is whether symmetries imply something about the outcome of the algorithms. This can become useful for solving other knotted problems such as you might find in DNA or similar strings of proteins,” he explains.
Another component of the CAREER grant that is very important to the mathematician is working with undergraduates. He plans to launch a 10-week summer institute for UMass Amherst, Smith and Mount Holyoke students to attract and nurture a new group of “brilliant young minds” into the field. These students will be benefit from meeting guest experts, cutting-edge researchers from other institutions, he adds.
“In my field there are many problems that are very elementary to state but very hard to answer,” Oblomkov notes. “I would like to introduce these to undergraduates to solve. Their fresh and fast minds might discover unexpected ways to tackle these problems. To be honest, this may be one of the greatest benefits to society, to train a new generation of mathematicians. Perhaps equally important, those who don’t become mathematicians will be introduced to math culture. I use a lot of computer simulations in this work and I will pass along this understanding to young people, which will be useful in whatever field they choose to pursue.”