In the summer of 2014 the Mathematics Department of the Pennsylvania State University will host a Research Experiences for Undergraduates site. Participants will be selected from qualified applicants. We encourage students to apply for both Summer REU and our MASS program which runs in the Fall semester of each year. Participation in REU followed by MASS provides an opportunity for completing a more substantial long-term research project.
Application deadline is February 28, 2014.
The program will run from July 2 to August 8, 2014. It will combine learning with research and include:
- One mini-course: Lie Groups and Lie Algebras
Instructor: Viorel Nitica, Professor of Mathematics - Weekly seminar run by the program coordinator Misha Guysinsky
- Research projects
- MASS Fest, a two day conference
Each eligible REU 2014 participant will receive a stipend of $2,500, reimbursement for room and board, and travel reimbursement up to $500.
International applicants:
(1) If English is not your native language please ask your recommenders to comment on your English proficiency in their letters;
(2) If admitted to the program, you will need to provide a proof of health insurance that meets the Exchange Visitor regulations or proof of personal funds sufficient to purchase such an insurance, and a proof of additional personal funds of $550 for miscellaneous expenses;
(3) Please make sure that your transcripts are translated into English.
Application Materials
Applications will be considered as long as positions are still available.
- Application Form
- Transcript
- Record of Mathematics courses
- Essay describing student's interest in mathematics
- Two Faculty Recommendations
- Send your applications to:
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REU Summer Program
111 McAllister Building
Department of Mathematics
The Pennsylvania State University
University Park, PA 16802 - Make other inquiries at:
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Phone: 814-865-8462
FAX: 814-865-3735
E-mail: reu@math.psu.edu
Course Description
REU 2014 COURSE DESCRIPTION - Lie Groups and Lie Algebras Our main goal is to discuss the basic structure of Lie groups and Lie algebras. We will assume as known introductory topics in analysis and topology: topological spaces, Euclidean topology, continuous functions, compact spaces, metric spaces, completeness, Heine-Borel theorem, Baire category. For standard references, see 1) and 2) below. If needed, some of these topics will be covered during the weekly seminar, as well as in the individual meetings with the groups. We start with an introduction to differentiable and analytic manifolds. A Lie group is at the same time a group and a differentiable manifold, so some knowledge of differentiable manifolds is required. In particular we will describe here the construction of global solutions for involutive systems of differentiable equations on a manifold. We continue with general results about Lie groups and Lie algebras, the correspondence between Lie groups and Lie algebras, the exponential map, the Campbell-Hausdorff formula, and the fundamental theorems of Lie. The emphasize will be on examples. The last part of the course is devoted entirely to Lie algebras: the theorems of Lie and Engel about nilpotent and solvable Lie algebras; criterion for semisimplicity, namely that a Lie algebra is semisimple if and only if its Cartan-Killing form is nonsingular; Levi’s semisimple decomposition of a general Lie algebra into its radical and a semisimple factor. If time permits, we will try to learn about maximal semigroups with nonempty interior in nilpotent and solvable Lie groups. Standard references for general theory about Lie groups and Lie algebras are 4) and 5). The theory of maximal semigroups in nilpotent and solvable Lie groups is developed in 6). Bibliography 1) Topology without tears, web resource, http://uob-community.ballarat.edu.au/~smorris/topbook.pdf 2) W. Rudin, Principles of real analysis, Elsevier, 1998 3) W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966 4) Varadarajan, Lie groups, Lie algebras and their representations, Springer, 1984 5) M. Postnikov, Lie groups and Lie algebras, 1986 6) J. D. Lawson. Maximal subsemigroups of Lie groups that are total. Proceedings of the Edinburgh Mathematical Society. Series II, 30 (1987) 479–501.