Research Statement
My primary interest is analysis and applications of delay differential equations (DDEs). These types of equations are often given less attention by mathematicians due to their unavoidable complexity. Delay differential equations are more mathematically complicated than ordinary or partial differential equations because the presence of delayed arguments makes the associated differential operators nonlocal in time, requiring knowledge of a function’s history over an interval rather than just its value at a single point. I chose to study DDEs because there is a solid foundation of knowledge to build upon, but a great richness of unexplored territory to try and uncover.
Furthermore, delays are common and natural in other areas of science, particularly physics, chemistry, biology, and engineering. A better understanding of this class of equations can improve the interdisciplinary power of mathematical modeling for these fields. That is why my research focuses on understanding delay differential equations, their complexities, and and their applications.
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Areas of Expertise
Mitochondrial Fission
My dissertation research focuses on a model of mitochondrial fission developed in collaboration with Dr. Chad Grueter. The model was originally designed by Dr. Anna Leinheiser and my advisor Dr. Colleen Mitchell. Click here to read more.​​​​
Discrete to Continuous Model Transformation
When attempting to understand the underlying mechanism of a model, a transformation from a discrete space to a continuous one can be enlightening. I specialize in the process of this transformation, especially in regards to the introduction and interpretation of delay terms.
State Dependent DDEs of Threshold Type
The primary focus of my research is on models with delays defined in the following way where tau(t) is the delay function:
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where C(u) is a state variable and N is a parameter. ​​
