Locally Compact Abelian Groups

Instructor: Amritanshu Prasad      Room 217; Wednesdays and Fridays at 9:30am.
The theory of duality and Fourier transforms for locally compact abelian groups generalizes the classical theory of Fourier series and Fourier transforms on real vector spaces. This theory finds diverse applications including differential equations, quantum mechanics, representation theory, algebraic geometry and number theory. This course will focus on
  1. Pontryagin duality
  2. Structure of locally compact abelian groups
  3. Fourier transforms
  4. Heisenberg groups
For the first two topics, we loosely follow Morris's book (listed below, copies available in the library). More references will be added later.

Grading scheme

There will be one homework assignment every week, due before class the following Wednesday. Four randomly selected problems will be corrected. There will be a mid-semester exam and a final exam. The aggregate score will be computed giving homework a weight of 30%, mid-semester exam a weight of 20%, and final exam a weight of 50%. Letter grades will be determined as follows:
Grade A: 90 - 100
Grade B: 80 - 89
Grade C: 70 - 79
Grade D: 60 - 69
Grade F: 0 - 59
First year students will be given 10 points bonus.

Homework

Listed by due date
August 12, 19 (Solution to no. 5), 26.
September 2, 9, 16.
October 7, 14, 28

Topics covered

5th August: Topological groups, the dual of a discrete group, the dual of a compact group, the dual of T.
10th August: weak* topology, strong operator topology, Banach-Alouglu, statement of the Riesz representation theorem.
12th August: Proof of von Neumann's theorem on the existence of an invariant probability measure for a compact topological group à la Kakutani via Zimmer.
17th August: The Peter-Weyl Theorem.
19th August: Peter-Weyl interpreted for compact abelian groups, duality for compact abelian groups.
24th August: Existence of continuous functions in an invariant space of L2(G), duality for discrete groups, Fourier transforms and Plancherel theorem for discrete and compact groups, behavior of subgroups under duality.
26th August: Injectivity of divisible groups in the category of discrete abelian groups. Projectivity of torsion-free groups in the category of compact abelian groups.
2nd September: The open mapping theorem. Stone-von Neuman-Mackey theorem for discrete and compact abelian groups.
4th September: The field of p-adic numbers is an extension over the Prüfer group with kernel as the p-adic integers.
9th September: Pontryagin duality for locally compact abelian groups which have a compact open subgroup.
11th September: The Pontryagin dual of Qp. Baer sum.
16th September: The Baer group and its computation.
23rd September: Midsemester Examination (with solutions).
30th September: Primary decomposition for topological torsion groups. An orthonormal basis for L2(L) when L admits a compact open subgroup.
7th October: Plancherel theorem, Fourier inversion formula and Mackey-Stone-von Neumann theorem for locally compact abelian groups which admit a compact open subgroup.
9th October: Beginning of the theory for R; Schwartz space S of rapidly decreasing functions. Position and Momentum operators. Fourier transform and inversion formula in S.
14th October: Creation and annihilation operators. The harmonic oscillator. Hermite functions. Make-up midsemester exam.
21st October: The Wigner transform. Proof of first part of the Stone-von Neumann theorem for R.
23rd October: Stone-von Neumann theorem for R. Deduction of Stone-von Neumann theorem for products of real vectors spaces and groups with compact open subgroups.
28th October: Automorphisms of the Heisenberg group which fix its centre. The Weil representation.
30th October: The Weil representation of SL2(Fq).

References

  1. Pontryagin duality and the structure of locally compact abelian groups by Sidney A. Morris, Cambridge University Press, 1977.
  2. On the role of the Heisenberg group in harmonic analysis by Roger E. Howe. Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 2, 821--843.
  3. Sur certains groupes d'opérateurs unitaires by André Weil. Acta Math. 111 (1964), 143--211.
  4. Fourier Analysis in Number Fields and Hecke's Zeta functions by John Tate (Princeton University PhD thesis, 1950; can be found in Algebraic Number Theory, edited by Cassels and Fröhlich, Academic Press 1967).
  5. Essential results of functional analysis by Robert J. Zimmer, The University of Chicago Press, 1990.
  6. Harmonic Analysis in Phase Space by Gerald B. Folland, Princeton University Press, 1989.
  7. The Stone-von Neumann-Mackey Theorem by Amritanshu Prasad (Unpublished Notes).
  8. On character values and decomposition of the Weil representation associated to a finite abelian group by Amritanshu Prasad (Submitted).