Courses for Aug - Nov 2017
  1. Introduction to Minimal Surfaces (Elective)

Instructor: Rukmini Dey

Prerequisites: Some very basic knowledge of complex analysis will be assumed.

Venue: S N Bose Meeting Room, ICTS Campus, Bangalore

Meeting Time: Tuesday, Thursday from 9:45 - 11:15 AM

First Class:  Thursday, 17th August 2017

Course contents:

  1. Serre-Frenet formula for curves, Parametric surfaces, Isothermal parameters, Gauss Map, Gaussian Curvature, Mean curvature, Area functional etc.
  2. Surfaces that locally minimise area in Euclidean space (minimal surfaces). Harmonic coordinates in isothermal parameters. A lot of examples of minimal surfaces.
  3. Minimal surfaces with boundary : Plateau’s problem
  4. The Gauss map for minimal surfaces with some examples.
  5. Weierstrass-Enneper representation of minimal surfaces. Many more examples of minimal surfaces.
  6. Conjugate minimal surfaces. One parameter family of isometric minimal surfaces.
  7. The Björling problem and Schwartz’s solution to it.
     

    *If time permits:


  8. Surfaces that locally maximise area in Lorenztian space (maximal surfaces). A lot of examples and analogous results, as in minimal surface theory, for maximal surfaces.
  9. The connection between minimal and maximal surfaces and Born-Infeld solitons.
  10. Constant mean curvature surfaces of non-zero mean curvature – the optimization problem they solve.
     

Reference Books:

  • Differential Geometry of curves and surfaces: Manfredo Do Carmo
  • A survey of minimal surfaces: Robert Osserman
  • Minimal Surfaces I : Dierkes, Hildebrandt, Küster, Wohlrab
  • Lectures on MInimal surfaces: J. Nitsche
  • Lectures on Minimal Surfaces in R 3 : Yi Fang
  • Surfaces of constant mean curvature: K. Kenmotsu.
  • Some papers on Minimal and Maximal surfaces and Born-Infeld solitons by various authors including Rukmini Dey, Pradip Kumar and Rahul Kumar Singh.

 

  1. Introduction to PDEs (Core)

Instructor: Vishal Vasan

Prerequisites: Analysis-I and some complex analysis

Venue: Raman Building IISc, Bangalore

Meeting Time: Tuesday and Thursday 11:30 am - 1:00 pm

Commencement date:  Thursday, 17th August 2017

Course contents:

  1. Introduction to PDEs and well-posedness
  2. Fourier series and Fourier transforms
  3. The heat equation
    3a. Boundary value problems
    3b. Maximum principles
    3c. Unbounded domains
  4. Sturm-Liouville problems
    4a. Basic operator theory
    4b. Spectral theory
  5. Laplace's equation 5a Fundamental solution
    5b. Maximal principles
    5c. Poisson equation
  6. Linear evolution equations
    6a. Wave equation in 1-d
    6b. Dispersion relations: consequences and asymptotics
     

    *If time permits:


  7. A unified approach to boundary value problems for evolution equations
  8. Transform methods and Riemann-Hilbert problems

 

Grading:

  • Homeworks: 50% of the final grade, about 5 or 6 in all
  • Mid-term exam: 20% of the final grade, date TBD
  • End-term exam: 30% of the final grade, date TBD

 

Reference books: 

  1. Guenther and Lee: Partial Differential Equations of Mathematical Physics and Integral Equations
  2. G Folland: Introduction to PDEs
  3. Renardy and Rogers: Introduction to PDEs
  4. L Evans: Partial Differential Equations