DescriptionThis project will lay out the elementary properties of the the Fourier series and then move to an application such as the isoparametric problem or Jacobi's identity for the theta function, the heat flow, wave motion (e.g. Huygen's principle), Poisson's summation formula, Heisenberg's inequality, Minkowski's theorem in the geometry of numbers, Gauss law of quadratic reciprocity. The wikipedia article on Fourier series is an excellent source for an introduction to the subject, and there is considerable scope for the students to choose from applications in pure and applied mathematics. The classic treatise by Dym and McKean has many applications, and a more recent student-friendly source is the book by Pereyra and Ward.Mode of Operation and Evidence of Learning for the group project
The group project will revolve around learning basic concepts and results in the field of convex analysis. By the end of the group project we would have learned Individual project The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats. PrerequisitesAnalysis in Many Variables II , Complex Analysis IICorequisiteAnalysis IIIResourcesH. Dym, H.P. McKean : Fourier Series and Integrals, Academic Press 1974 Enrique A. Gonzalez-Velasco (1992). "Connections in Mathematical Analysis: The Case of Fourier Series". American Mathematical Monthly 99 (5): 427 - 441 Evans, 1998: Partial differential equations Elias M. Stein and Rami Shakarchi, Fourier analysis, Princeton lectures in analysis, 2003 Cristina Pereyra and Lesley A. Ward, Harmonic analysis: from Fourier to wavelets, AMS. Student mathematical library (2012) Links |
email: Wilhelm Klingenberg