Carnegie Mellon University
October 10, 2024

Low-Dimensional Topology Offers High Rewards for Simone

By Tricia Miller Klapheke

Heidi Opdyke
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Jonathan Simone explores the landscape of topology. The construction of 4-dimensional objects — in particular exotic 4-manifolds — intrigues him.

“If you think about shapes in dimensions one, two or three, you can typically draw them and visualize them,” said Simone, who joined Carnegie Mellon University’s Department of Mathematical Sciences in August 2024 as an assistant professor. “When you get to dimension four then all of that goes out the window.”

In topology objects can be molded, shrunk, or stretched, as long as they’re not broken. Simone said, as the joke goes, a topologist cannot tell the difference between a doughnut and a coffee mug. The objects, called manifolds, are widely studied in topology and have applications to areas such as physics and computer science. Manifolds of dimension 1, 2, and 3 are well-understood, as are manifolds of dimension at least 5.

“Dimension 4 is the sweet spot in which you have phenomena that do not occur in lower dimensions because there is not enough room,” Simone said. “And the phenomena do not occur in larger dimensions because there is plenty of room to dissolve any such phenomena. So, the study of 4-dimensional manifolds is a big field, and there are a lot of unknown questions there, especially related to exotic 4-manifolds, which are manifolds with interesting additional geometric structures.”

Simone said he cares deeply about topology and teaching. He came to Carnegie Mellon largely because of the freedom and time the university allows to develop courses and work with undergraduate students.

“Jonathan brings his unique technical expertise in topology to broaden and strengthen the research portfolio of the department, said Prasad Tetali, Alexander M. Knaster Professor and head of the Department of Mathematical Sciences. “His considerable experience in mentoring undergraduates in research is an added bonus!”

A Brooklyn, New York, native, Simone earned his bachelor’s degree in mathematics from New York University. He also earned his master’s degree at Hunter College and his Ph.D. at the University of Virginia. He recalls a class early on in abstract algebra that was really interesting to him.

“My love of math got stronger and stronger the deeper into it I got,” he said.

In his first semester at Carnegie Mellon, he is teaching two sections of linear algebra for computer science students, and in the future, he said he hopes to continue that and add topology classes to his workload. Simone said he enjoys when anyone who wants to talk math visits his office hours. In class, he said he will mention more advanced questions related to the subjects covered and encourage people interested in more to visit.

“I want all students to know they can always come bother me about math,” he said. “My door’s open, and I’m here to help them, whether that be on a specific homework problem or to learn about something that goes beyond the course curriculum.”

He held visiting assistant professor roles at the University of Massachusetts, Amherst, and the Georgia Institute of Technology. He has received recognition for his teaching at three institutions. At Georgia Tech in 2023, he earned the Student Recognition of Excellence in Teaching, which goes to faculty members with high scores in students feedback. At UMass he was a two-time distinguished teaching award finalist, and at the University of Virginia he was an all-university graduate teaching award winner.

He said he hopes to continue organizing research experiences for undergraduates over the summer for math majors. He conducted research with undergraduates using high-tech linear algebra to answer questions about knots during the summers of 2022 and 2023 at Georgia Tech. The results of those are in the process of being reviewed for publication.

Tetali, who taught at Georgia Tech for more than 20 years before joining Carnegie Mellon, said Simone and his wife, Nadiya Kostyuk, a new faculty member in the Carnegie Mellon Institute for Strategy and Technology, could have chosen to stay in Atlanta, but said he was grateful for their move to Pittsburgh.

“It is a truly win-win-win situation to all parties involved,” Tetali said.