Mykhaylo Shkolnikov Brings His Probability Expertise to CMU

Free boundary problems provide a universal mathematical framework for many phenomena in natural and social sciences and engineering. Mykhaylo (Misha) Shkolnikov wants to advance the understanding of these mathematical models.

"My research is very interdisciplinary," said Shkolnikov, visiting professor of mathematical sciences at Carnegie Mellon University. "It touches on probability theory, mathematical finance and partial differential equations. The three groups at CMU are very strong, and I was very excited to come here and interact with the faculty."

Shkolnikov earned his Ph.D. from Stanford University, and he worked as a postdoctoral fellow at the University of California, Berkeley and at the Mathematical Sciences Research Institute, Berkeley. He came to Carnegie Mellon in January 2023 from Princeton University where he is a member of Operations Research and Financial Engineering Department. He planned to collaborate with faculty in the Center for Nonlinear Analysis and in the Mathematical Finance group on his work in interacting particle systems.

A growing area of probability, the field of interacting particle systems uses equations that describe how small components combine to larger systems. For example, in mathematical finance, his work could be used to see how the default of a small local bank might lead to other banks of the same size or larger also having financial difficulties. Similar equations can apply to neuroscience, where one neuron can lead to a cascade of signaling in the brain, or physics, where "heat particles" give rise to heat propagation.

"You have very powerful models that apply to all these different fields," Shkolnikov said. "It all boils down to similar mathematical equations and problems."

Shkolnikov's research is sponsored by the National Science Foundation (NSF). In February 2023, he published a paper investigating physics' supercooled Stefan problem, which describes the freezing of supercooled liquids. His team used the Euler method, originally invented to solve ordinary differential equations, and in the process made connections with neuroscience and finance.

Shkolnikov also teaches. During the spring 2023 semester, he taught topics in stochastic calculus (21-881), a course for Ph.D. students on the applications of interacting particle systems.

"I had 10 very talented Ph.D. students, and I really enjoyed teaching the class," Shkolnikov said. "Any level of mathematics and technical difficulty was okay with them, and they were excited to get into the mathematical details."

Shkolnikov has enjoyed his time at Carnegie Mellon and in Pittsburgh so much, he will transition to being a full-time tenured track faculty member in January 2024.

Prasad Tetali, head of the Department of Mathematical Sciences, said that Carnegie Mellon is fortunate to have Shkolnikov.

"While several of the faculty in the department had known him for his work in probability at first, his foray into mathematical finance made him an even better fit to our department and CMU," Tetali said. "Besides overlapping in interests with our applied analysis, partial differential equations and math finance groups, Misha can also collaborate with people like Alan Frieze, Konstantin Tikhomirov, Tomasz Tkocz, me and others working in discrete probability."