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Manjunath Krishnapur

 Department of Mathematics, Indian Institute of Science, Bangalore 560 012

Topics in analysis (Spring 2014)

Tue, Thu 9:30-11:00, LH-5, Mathematics department
Description: This course is somewhat experimental and aimed at students who have already studied real and complex analysis. Usually in a course one learns techniques and sees interesting results as "applications". Here we state several interesting results first, and prove them one by one, in each case drawing upon all the techniques already learned (and perhaps learn a few new ones). The emphasis is on learning techniques of proofs in analysis.
  • Isoperimetric inequality in Euclidean space
  • Weyl's equidistribution theorem
  • Picard's theorem for entire functions
  • Shannon's source coding theorem
  • Hall's marriage theorem
  • Dirichlet's theorem on infinitude of primes in arithmetic progressions
  • Existence of Haar measure on compact groups
  • Wigner's semicircle law for random matrices
  • Uncertainty principles in Fourier analysis

Prerequisites: Real analysis, complex analysis, measure theory, basic probability and linear algebra, topology and basics of groups. It is strongly recommended to take the functional analysis course simultaneously, if not already taken. UG 4th year and Int. PhD. (Math) 2nd year students are most suited to take this course.

Grading: There will be a final exam (50%) and one or two mid-term exams. Many problems will be given and homeworks will carry at least 15% of the final score.

Texts and other resources: In no particular order (I may sample material from many places) -
  1. T.W. Korner Fourier analysis, Cambridge university press, 1989.
  2. W. Rudin Real and complex analysis, Tata McGraw-Hill, 2006, 3rd ed.
  3. P.D. Lax and L. Zalcman Complex proofs of real theorems, American Mathematical Society, 2012.
Notes for specific topics: