Graduate Analysis, 18.156, Spring 2007

In Room 2-146 at 1:00 PM on Tuesdays and Thursdays.

• Chapter 1 -- Distributions:
  • Status: Readable but not complete
  • Last change -- 23 June, 2007: Multiplicativity added, more corrections from Jacob.
  • Chapter1 Pdf
  • Chapter1 Ps
• Chapter 2 -- Elliptic regularity
  • Status: Readable and close to complete
  • Last change --
  • Chapter2 Pdf
  • Chapter2 Ps
• Chapter 3 -- Coordinate invariance and manifolds
  • Status: Partly readable, not complete
  • Last change -- Manifolds section roughly proofread.
  • Chapter3 Pdf
  • Chapter3 Ps
• Chapter 4 -- Invertibility of elliptic operators
  • Status: Readable but not complete
  • Last change --
  • Chapter4 Pdf
  • Chapter4 Ps
• Chapter 5 -- Suspended families and the resolvent:
  • Status: Now readable, although not complete
  • Last change -- 2 August, 2007: Partial evision
  • Chapter5 Pdf
  • Chapter5 Ps
• Chapter 6 -- Compact manifolds with boundary
  • Status: Partly readable, about half done.
  • Last change -- 17 July, 2007: New section on homogeneous operators.
  • Chapter5 Pdf
  • Chapter5 Ps
• Chapter 7 -- Bogomolny equations
  • Status: Not yet readable
  • Last change --
  • Chapter7 Pdf
  • Chapter7 Ps
  • All:- Pdf
  In this course I will discuss one realistic application of analysis to a geometric problem. Namely, the existence and regularity of solutions to a mildly non-linear gauge-invariant system of (transversally) elliptic equations on a non-compact manifold -- the Bogomoly equations for monopoles on $\bbR^3.$
  • Sobolev spaces recalled, coordinate-invariance.
  • Differential operators with variable coefficients, manifolds(\dag)
  • Local elliptic regularity
  • Compact operators, spectral theorem
  • Families of Fredholm operators(*)
  • Non-compact self-adjoint operators, spectral theorem
  • Spectral theory of the Laplacian on a compact manifold
  • Pseudodifferential operators(*)
  • Invertibility of the Laplacian on Euclidean space
  • Lie groups(\ddag), bundles and gauge invariance
  • Bogomolny equations on $\bbR^3$
  • Gauge fixing
  • Charge and monopoles
  • Monopole moduli spaces
  • [*] I will drop these if it looks as though time will become an issue.
  • [\dag,\ddag] I will provide a brief and elementary discussion of manifolds and Lie groups if that is found to be necessary.
  Lecture notes:- Pdf , Postscript (revised 19 Feb 2007).
  Individual Lectures
  • Lecture 1 (pdf) , Lecture 1 (postscript) (revised 19 Feb 2007)
  • Lecture 2 (pdf) , Lecture 2 (postscript) (revised 19 Feb 2007). Photographs: P1, P2, P3, P4, P5, P6, P7, P8, P9, P10, P11, P12, P13, P14, P15.
  • Lecture 3 (pdf) , Lecture 3 (postscript) (revised 19 Feb 2007). Photographs: P1, P2, P3, P4, P5, P6, P7, P8, P9.
  • Lecture 4 (pdf) , Lecture 4 (postscript) (revised 19 Feb 2007). Photographs: P1, P2, P3, P4, P5, P6, P7, P8, P9, P10, P11, P12, P13.
  • Lecture 5 (pdf) , Lecture 5 (postscript) Photographs: P1, P2, P3, P4, P5, P6, P7, P8, P9, P10