Meet Nathan Dalaklis, a UNT doctoral student studying mathematical theory. Nathan's current research is advised by Dr. Mariusz Urbański, but the nature of his interests have given him opportunities to work with several faculty and researchers both in the department and abroad.

"Most of the time I find myself staring at papers and chalkboards -- face contorted -- trying to make sense of it all," said Nathan, whose dissertation research concerns the development of a theory of multifractal analysis for time averages of a certain general class of dynamical systems called conformal graph directed Markov systems. "Multifractal analysis can be used to better understand image processing, fluid turbulence, neural networks, and more. There are also some statistics applications as well since these multifractal questions can be framed as questions about distributions. I don't know exactly what will come of continuing this research but hope it will become clearer to me as I continue to work. I think that is part of the joy of mathematics; many theories end up applied in ways that mathematicians don't necessarily expect them to be."

Because his research interest is expansive and he is able to be creative with the questions he works to answer, Nathan has collaborated with researchers near and far, including Dr. Jason Atnip, who is currently at the University of Queensland in Australia. Together they analyze different problems in the multifractal setting.

"I have also had the opportunity to work with UNT mathematics faculty Dr. Kiko Kawamura on advising undergraduate research," said Nathan. "Our paper regarding our work on Okamoto's functions was published last year."

"Nathan was very helpful during the intensive eight-week summer undergraduate research project," said Dr. Kawamura. "While we officially met twice per week during the summer, he kept discussing the project with two excellent undergraduate students almost every day and helped them find new results."

Nathan initially became interested in research as an undergraduate working on a project at Carleton College under the direction of Dr. Elizabeth Sattler regarding Symbolic Dynamics. "I have enjoyed thinking about symbolic representations of dynamical systems ever since," he said. "Later my doctoral advisor introduced me to how these symbolic systems are used in the dynamics, ergodic theory, and fractal geometry. I always knew I wanted to do research and so it seemed beneficial to move into a more mathematically interdisciplinary area that was still related to the math that introduced me to this research in the first place."

Last summer Nathan was invited to present his research at the 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications hosted at UNCW. "It was a great experience and opened a lot of doors," he said. "I strongly encourage others to attend conferences in their area if feasible."

"It is generally rare for graduate students to receive an invitation to present at this conference," said Dr. Kawamura. "Not only is he a very hard-working, disciplined, bright, outgoing student, but he is also an excellent speaker!"

Because of his interest in Dr. Mariusz Urbański's and Dr. Kiko Kawamura's research, UNT was an easy choice for his doctoral degree once accepted. "I don't remember opening any of my other applications to see if I got in anywhere else, and I'm quite happy that I made the decision to come to UNT," Nathan said, citing the mentorship of his now-advisor as a key to some of his research success. "Dr. Urbański has really helped me better identify when certain assumptions are necessary, and when assumptions are helpful for seeing the overall scheme of an argument for a more general case and can then, therefore, be adjusted to fit that more general framework. I think this is an important skill for mathematicians to develop and I am glad to have an advisor that pushes me in this way."

Like all research, mathematics has its unique challenges, and Nathan admits that sometimes analyzing new information can be overwhelming. "But many of the walls I have hit in research are also some of the most interesting," said Nathan. "A lot of times I must turn off the inner monologue of "I don't understand" to take a step back and intentionally say to myself "I don't understand this right now because I need to know what [blank] means." That has often helped alleviate my research anxiety."

Nathan runs a YouTube channel called CHALK with over 17,000 followers, where he shares his life as a graduate student and math problems he finds interesting. After earning his PhD at UNT, Nathan hopes to continue his research either in a post-doctoral position or at a liberal arts college.