The FRAP Directory allows students to identify UCSB faculty who are looking for undergraduate students to participate in their research projects or creative activities. Please use the links below to find opportunities by discipline. Students, if your desired discipline is not listed, please contact the Undergraduate Research Initiatives office at 805-893-3090 or firstname.lastname@example.org for assistance. Faculty, if you would like to post your research or creative activity opportunity, please complete the online submission form.
"Analysis in almost Abelian homogeneous spaces"
Almost Abelian groups form a class of 1-step solvable Lie groups generalizing many properties of the 3-dimensional Heisenberg group. As the Heisenberg group itself (and even more so), these groups together with their homogeneous spaces are well suited for development of new, non-commutative methods of analysis. The following are parts of this large project:
* Classification of algebraic, geometrical and topological properties of these spaces
* Explicit description of invariant geometrical structures, including geometric PDEs
* Explicit unitary representation theory and harmonic and Fourier analysis in these spaces The 3-dimensional representatives of this class play an important role in mathematical cosmology.
This project is in collaboration with Zhirayr Avetisyan 6515 South Hall, Department of Mathematics email@example.com
Explicit faithful matrix representations for all almost Abelian groups can be found. This makes all derivations on these spaces completely explicit. All PDEs in question can be solved by a judicious separation of special variables. This makes the present project technically amenable to undergraduate students, who can learn advanced mathematically concepts by working on explicitly given examples.
Strictly speaking, a student doesn't need to have attended a course X in order to have some background in X. Therefore, instead of listing course prerequisites, I will list subject prerequisites.
Upper division level linear algebra and matrix theory, including canonical forms and spectral theory
Upper division level real analysis and analysis in normed vector spaces
Basics of differential geometry and Lie theory
Mathematical modeling and computational simulation of problems arising in fluid mechanics, soft materials, and biophysics. See the research website for more details at http://www.atzberger.org/
Mathematical modeling and computational simulation work in collaboration with research members. Development of models, implementation of numerical methods, and performance of simulation studies. Present results in workshops and group meetings.
Some experience with programming would be helpful, but not strictly required. Overall, a strong motivation and enthusiasm to use mathematical approaches to tackle problems arising in the sciences and engineering.
Mathematics, Mechanical Engineering
Applied mathematical research presents exciting opportunities to develop analytic and computational approaches to tackle interesting problems arising in the natural sciences, engineering, and information sciences (machine learning / data sciences). We work in an area called "stochastic analysis" which we use to develop novel models and computational methods for simulation of problems arising in fluid mechanics, soft materials, and biophysics. In related work, we also use mathematical approaches to develop novel ways to analyze data developing further techniques in machine learning and statistical inference. See our research website for more details on projects at http://www.atzberger.org/
Mathematical modeling and computational simulation work in collaboration with research group members. This could include development of new models, implementations of numerical methods, performing simulation studies of physical systems, or development of machine learning methods for data analysis. Present results at workshops and group meetings. See our research website for more details: http://www.atzberger.org/
Some experience with programming would be helpful, but not strictly required. Overall, a strong motivation and enthusiasm to use mathematical approaches to tackle exciting problems arising in the natural sciences, engineering, or information sciences.