Georgia Tech Mathematicians at the Forefront of Research with ICERM at Brown

Georgia Tech’s School of Mathematics is dedicated to exploring the frontiers of computational and experimental research in its discipline, so much so that one of the leading math research centers in the country now has a School of Mathematics professor serving as the chair of its board of trustees.

Rachel Kuske’s new role with the Institute for Computational and Experimental Research in Mathematics (ICERM) is just one Georgia Tech connection to the Center, based at Brown University in Providence, Rhode Island. 

More than 20 School of Mathematics faculty members and graduate students have participated in recent ICERM programs, including a series of seminars during Fall 2022’s semester on harmonic analysis and convexity, mathematical processes that help researchers navigate large collections of data.

“The participation of School of Mathematics members at different levels in ICERM, one of several leading math research institutes in the U.S., is representative of the School’s leadership in the broader research community,” said Kuske, a former School of Mathematics chair. “Georgia Tech’s multi-faceted involvement benefits our research groups as well as research advances and development of talent in the wider research community.”

John Etnyre, a professor specializing in low-dimensional topology, is also involved in various ICERM activities, and helped co-organize a semester-long program on braid theory in Spring 2022. “ICERM is an excellent research center,” Etnyre said. “They provide a great environment to collaborate with others, as well as great conference facilities. They are certainly one of the best mathematical research centers in the world, and they are unique in their focus on bringing computation and experimentation into mathematics.”

Mathematics labs and more

ICERM’s mission is to expand the use of computational and experimental methods in mathematics, support theoretical advances related to computation, and address problems posed by the existence and use of the computer through mathematical tools, research and innovation.

ICERM pursues these goals by supporting what Kuske calls “mathematics labs,” which are typically human resource-intensive, and highly collaborative nationally and globally. 

“ICERM’s goals include catalyzing new directions in research and collaborations, as well as exploiting and expanding the interface between mathematics, computations, and experiments, computational and otherwise,” she said. These intersections have historically been represented in ICERM’s scientific board, with Kuske citing Dana Randall, ADVANCE Professor of Computing in the School of Computer Science, and an adjunct professor in the School of Mathematics, as an example.

Kuske views chairing ICERM’s Board of Trustees as a way to “provide an opportunity to contribute in several directions, including making sure that present and future resources, policies, and procedures support ICERM’s mission.” These include increasing diverse and inclusive participation in mathematical sciences and relevant areas, raising public awareness of the impact of mathematics, and continued service and leadership in the research community.

Etnyre said an important computational aspect to topology — the study of surfaces that can be twisted, bent, or otherwise deformed but never broken — has been around for a while, “but its importance has been increasing over the years,” he said. For example, software called SnapPea/SnapPy “is a program where you can input a three-dimensional space and it will compute a myriad of data about the space. It also has a list of thousands of spaces and data about them. When trying to determine if something you are interested in is true or not, it is always helpful to be able to check its validity on such a large sample of spaces.”

More recently, Etnyre says several teams of people have been using machine learning algorithms to explore relations involving knot theory, the study of closed curves in three dimensional spaces. “There are many other ways in which computation and experimentation is important in topology, and it was great that the ICERM program was able to expose these techniques to a large number of researchers during our program.”

Knot theory and an associated subdiscipline, braid theory, help bring structure to large, complex data problems. Possible applications include finding out more about DNA recombination, Etynre said.  “There are also connections with physics through string theory and gauge theory. There are connections between braids and many areas in mathematics. That was really the focus of the program at ICERM last spring,” referring to the Spring 2022 program he helped co-organize at the center.

Collaboration on convexity 

Galyna Livshyts, associate professor in the Georgia Tech School of Mathematics; Ben Jaye, assistant professor, and postdoctoral researcher and visiting assistant professor Naga Manasa Vempati recently completed the semester-long program on harmonic analysis and convexity at ICERM. Livshyts said the center is one of several institutions around the world that provide such lengthy research opportunities for various areas in mathematics.

“The harmonic analysis and convexity research program presented us with the opportunity to collaborate with other researchers in the area during this time,” Livshyts said. “Also, ICERM often hosts various interesting and stimulating workshops. ICERM is located in the buzzing town of Providence, and it has excellent facilities to allow people to discuss mathematics, and also provide some great views.”

The following Georgia Tech School of Mathematics faculty and students have participated in various recent ICERM programs:

The Combinatorial Algebraic Geometry virtual workshop in early February 2021 included Trevor Gunn, Arvind Ramaswami, Matthew Baker, Cvetelina Hill, and Alperen Ozdemir.

Matthew Baker, Justin Chen, Tianyi Zhang, Anton Leykin, Josephine Yu, and Josiah Park also took part in Collaborate@ICERM projects in 2021.

School of Mathematics students Yvon Verberne, Sudipta Kolay, Justin (Yi-Chang) Chen and Jiaqi Yang were named ICERM Postdoctoral Fellows for 2021-22. 

Professor John Entyre co-organized the Spring 2022 Braids semester program at ICERM, and Professor Dan Margalit participated in Braids in Symplectic and Algebraic Geometry. Postdoctoral students Miriam Kuzbary and Hannah Turner took part in the entire Braids program. Graduate student Sally Collins and undergraduate student Sarah Pritchard participated in the Braids in Low-Dimensional Topology conference in April. 

Georgia Tech faculty and students taking part in the Harmonic Analysis and Convexity program in September 2022 at ICERM include Manuel Fernandez, Orli Herscovici, Galyna Livshyts, Naga Manasa Vempati, Shixuan Zhang, Ben Jaye.

Mohit Singh and Swati Gupta are scheduled to participate in an ICERM program, Combinatorics and Optimization, March 27-31, 2023.