I have been interested in problems concerning deterministic fractal functions: e.g. nowhere differentiable but continuous functions, self-affine functions and space-filling curves. In other words, I like to study strange and irregular functions! Since recent great progress of computer science and observation technology started showing such a strange function as real data (for example, brain waves, the distribution of galaxies in the universe, fluctuations of stock prices, the growth of plants and so on), it has been recognized that irregular functions actually provide a much better representation of many natural phenomena than functions to which the methods of classical calculus can be applied.
The goal of my research is to find new techniques to analyze fractal functions since several powerful methods of classical calculus are clearly unsuited to them. For instance, a basic application of a derivative is finding extreme values of a differentiable function. However, if a function is nowhere differentiable, how can we find the extreme values?
UNDERGRADUATE / GRADUATE RESEARCH MENTORING AND ADVISING:
- Mike Trenfield RTG Undergraduate Research Summer 2013, Spring 2014
- Aiden Young TAMS Undergraduate Research Spring 2014, Summer 2014
- Ben Schwaighofer M.S. Graduate study (with Dr. Allaart) Spring 2014
- Paul Teszler TAMS Undergraduate Research Summer 2014
- Chia Ting Han RTG Undergraduate Research Summer 2014
- Noah Fredrickson Undergraduate Research Fall 2019 - Summer 2021
- Michalis Paizanis Undergraduate (Honors) Research Summer 2021 - Fall 2021
- Tobey Mathis Undergraduate (Honors) Research Fall 2019 - Spring 2022
RESEARCH COLLABORATION (with PhD graduate students)
- Andrew Allen PhD student Spring 2018 - Summer 2019
- Nathan Dalaklis PhD student Summer 2021- Present Matthew
- Ortiz PhD student Spring 2023 - Present