## Department of Mathematics and Philosophy

### Undergraduate Research - Differential Equations

#### Faculty

##### Samson Adeleke

**Areas of Interest:** Algebra, Combinatorics, Differential Equations

**Description:** The material could be on any topic beyond the contents of Math 355 (Combinatorics), Math 421 (Abstract Algebra), and Math 333 (Ordinary Differential Equations). In more detail, the Math 355 course introduces applications of counting and graphs to computer science, chemistry, operations research, etc. The Math 421 course covers the basics of abstract algebra of groups, ranges, and fields, an aesthetically satisfying subject while Math 333 on Differential Equations models down-to-earth phenomena across a broad spectrum.

##### Fedor Andreev

**Areas of Interest:** Web-based education, Mathematics-Physics interrelation, Differential Equations, Computer programming

**Description:** I'd like to invite students to choose a topic from my areas of interest and work on it. It could be a participation in development of an online homework system for the department, or if a student has an interest in physics, we could find a mathematical idea or theorem and look at its applications to physics. Students may be expected to work with different textbooks and/or special literature. As a researcher, I could suggest some topics from Differential Equations theory to study. For this area, students with knowledge of Differential Equations and Complex Variable are welcome. Students with interests in computer areas could develop a program for graphical representation of paths in plane or, actually, another program for some other mathematical discipline.

##### Amy Ekanayake

**Areas of Interest:** Biomathematics, Difference and Differential Equations, Mathematical Modeling, Stochastic Processes

**Description:** I welcome those interested in modeling real-world phenomena. We would consider all stages of the modeling process: determining the most appropriate model type (continuous/discrete, deterministic/stochastic, etc), forming the model by determining appropriate assumptions, analytically studying the model's behavior (equilibria, stability, etc), and using software (such as Matlab, Maple) to find numerical solutions. Finally, students would write a report explaining the model and analysis and its implications for the real-world phenomenon. This type of modeling experience would benefit those considering a career in industry or graduate studies in applied mathematics, and pre-service teachers wishing to be able to promote the practical applications of mathematics in their classrooms.

I am most interested in modeling and computation analysis for biological systems, such as birth-and-death processes, epidemics, genetics, predator-prey and competition relationships, pharmacokinetics, and cellular and other medical topics, among others. No biology prerequisite is required.

##### Dinesh Ekanayake

**Areas of Interest:** control and system theory, optimal control and optimization, differential equations, functional and integral equations

**Description**: I would guide students interested in applied mathematics. Research projects would include modeling and control problems arising from interdisciplinary areas. Examples include problems pertinent to smart materials and structures, wind energy, imaging, and some of finance and biology. Students would study one or more aspects of modeling and control approaches, such as investigating possible models, studying existence, uniqueness and stability of feedback, robust or adaptive control. Students would be expected to study relevant literature, participate in analyzing and simulating, and write a report discussing the findings. The goal of such research would be for students to learn to apply their mathematical skills to complex real-life problems.

##### Boris Petracovici

**Areas of Interest:** Differential Equations, Mathematical Modeling, Finite Element Methods, Applied Analysis

**Description:** I would help the interested students explore topics from the above areas. The research project could come from another field of study, such as economics, physics, engineering, biology, meteorology. However, mathematical ideas and developments must play an essential role. The students would be expected to consult relevant literature and write a report discussing their findings. My main focus in this course is helping students understand new mathematical ideas and communicate them clearly.

##### Lia Petracovici

**Areas of Interest:** Real Analysis, Complex Analysis, Dynamical Systems, and Differential Equations

**Description:** Some possible topics for undergraduate research would be chaotic dynamical systems (what are some different definitions of a chaotic dynamical system? how can we prove that these definitions are equivalent?), fractals (how do they arise? how can we produce fractals on the computer?), and visualizing conformal mappings of the complex plane. I would be happy to work with interested students to find other projects of interest to them.

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