Basic Linear Algebra and Algebraic Structures, Calculus 3, and Differential Equations.
Basic knowledge of Lie algebras and Lie groups (as groups of transformations) is one of the main prerequisites of any professional mathematician or physicists. Our students will learn about these and other topics necessary for the project during the first phase of the project. A special course on this subject has been designed last year, and will be used this year too. Once they master the notions of complex, hyper-complex, symplecitic and quaternion structure on a vector space, further understanding of smooth manifolds and Lie groups with emphasis on particular examples of nil-manifolds will be obtained. A short introduction to Graph Theory will be presented, if necessary. This year’s group of students, having juniors among them, is well prepared for the project. Based on the achievements of their colleagues last year, we expect the research part of the project to go faster and more effectively than a year ago. Faculty members who work actively in this area will be invited to give additional talks to the students on the subject. This will serve as a further motivation for the students to do the project, and will provide an advanced insight in what they are supposed to do.The stated above Problems 1 and 2 are natural and important for the emerging theory of Lie algebras associated with simple graphs. We expect the students to find interesting examples, and develop a theory explaining them. We expect also the results to be publishable in peer reviewed journals.
Phase 1 - Teaching sessions (first 3 weeks): lectures on Linear Algebra, Category Theory, Graph Theory, and basic Differential Geometry (10:00 –12:00 and 2:00 –4:00); discussion/exercises 4:30 –6:00; examinations on the material learned during the week: Friday 10:00 a.m. –2:00p.m.
Phase 2 - Research (last 3 weeks) : 10:00 –11:00 discussion on research; research time: 11:00 –5:00; discussion: 5:15 –6:00. Library and Internet resources and use of them: a couple of sessions with a representative of the University library (for 50 min in Phase 1, in the interval 4:30 p.m. -6:00 p.m.); End of 4th week –reports on results; End of 6th week: presentations of the students.
The first phase achievements will be assessed via monitoring the students' work in class on a daily basis, and on regular Fridays’ examinations. The second phase will be assessed via midterm reports on the progress of the participants in their research, and the final presentations. The planned every day discussions with the students on the project will help in that too.