Dr. Kirill Lazebnik, assistant professor in the Department of Mathematical Sciences at The University of Texas at Dallas, doesn’t just solve equations, he seeks to understand the deeper mechanics of how mathematics behaves. A pure mathematician by training, Dr. Lazebnik is driven by a pursuit of clarity, structure and the often-hidden elegance of abstract mathematical systems.
“I became interested in mathematics sometime around high school,” he recalled. “There was a kind of aesthetic satisfaction in making everything work out and understanding it completely.”
That initial spark, fueled by intellectual curiosity and a bit of academic competitiveness, evolved into a lifelong pursuit. Today, Lazebnik’s research resides in the domain of pure mathematics, a field broadly distinguished from applied math or statistics by its focus on understanding mathematical concepts for their own sake, not necessarily for immediate application.
While pure math can seem abstract to outsiders, its influence is both foundational and far-reaching. Lazebnik’s own research, for example, includes the study of Newton’s method – a centuries-old algorithm used to find solutions to equations. “I look at questions such as ‘how well does Newton’s method work?’”
Though research is his primary passion, Lazebnik brings that same spirit of inquiry into the classroom. He teaches undergraduate courses in Differential Equations and Partial Differential Equations, striving to convey not just formulas but the deeper logic and patterns that animate them.
“I enjoy learning and understanding something new, then sharing that understanding with students,” he said. “That process is always rewarding.”
This past summer, Lazebnik mentored three undergraduate students: Aiden Hill, Colton Fisher, and Palmer Thompson on a research project exploring one of mathematics’ most mesmerizing objects: the Julia set.
“Julia sets are fractals (never-ending, self-similar patterns that repeats their structure at different scale) that appear when studying certain mathematical objects called dynamical systems,” he explained. “Roughly speaking, they represent the set of points where a system behaves chaotically.”
The team’s project asks a deceptively simple question: What can a Julia set look like? For instance, can it look like the outline of a dog or a cat, or any figure whatsoever?
“It’s an ideal undergraduate project,” he noted. “It’s accessible enough for students with limited backgrounds but still interesting and non-trivial from a research perspective.”
The project grew out of a question posed by Dr. Malik Younsi, a collaborator at the University of Hawaii. The result will be a co-authored research paper (currently in development) which is a rare and impressive achievement for undergraduate researchers.
The Department of Mathematical Sciences is proud of the recent hiring of Dr. Kirill Lazebnik, along with a few other young, promising mathematicians and statisticians of the highest caliber. The Department continues to build its strengths both in core and applied areas of mathematical sciences and is confident in its future. Dr. Lazebnik has already positioned himself on the world stage with essential results in highly competitive mainstream areas of modern mathematics, like analysis and complex dynamics and received support from the NSF. His papers in Mathematische Annalen, Advances in Mathematics, Geometric and Functional Analysis, and other top journals have already attracted the attention of top specialists around the globe.
Vladimir Dragovic
Department Head
Making the Case for Pure Mathematics
Pure mathematics is a puzzling concept to many: why study mathematics for its own sake?
Lazebnik cites the well-known 1960 essay by the physicist Eugene Wigner: The Unreasonable Effectiveness of Mathematics, which discusses the way in which pure mathematics has often preceded its usefulness in physics and engineering: examples include general relativity, signal/image processing and cryptography.
“Apart from its potential usefulness,” says Lazebnik, “I believe in the innate value of understanding fundamental mathematics.”
Whether through published research, undergraduate mentorship or classroom teaching, Lazebnik’s work strengthens the mathematical community and inspires future generations to dig deeper into the beauty of abstract thinking.
“Many mathematicians don’t really ever retire,” he reflects. “Because they genuinely enjoy what they do.”
In Lazebnik’s case, that joy is unmistakable and possibly even fractal shaped.