The schedule of ICTS courses for Aug - Nov 2017 are given below:
- 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
- Serre-Frenet formula for curves, Parametric surfaces, Isothermal parameters, Gauss Map, Gaussian Curvature, Mean curvature, Area functional etc.
- Surfaces that locally minimise area in Euclidean space (minimal surfaces). Harmonic coordinates in isothermal parameters. A lot of examples of minimal surfaces.
- Minimal surfaces with boundary : Plateau’s problem
- The Gauss map for minimal surfaces with some examples.
- Weierstrass-Enneper representation of minimal surfaces. Many more examples of minimal surfaces.
- Conjugate minimal surfaces. One parameter family of isometric minimal surfaces.
- The Björling problem and Schwartz’s solution to it.
*If time permits:
- 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.
- Connection between minimal and maximal surfaces and Born-Infeld solitons.
- Constant mean curvature surfaces of non-zero mean curvature – the optimization problem they solve.
- 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.
- 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
- Introduction to PDEs and well-posedness
- Fourier series and Fourier transforms
- The heat equation
3a. Boundary value problems
3b. Maximum principles
3c. Unbounded domains
- Sturm-Liouville problems
4a. Basic operator theory
4b. Spectral theory
- Laplace's equation 5a Fundamental solution
5b. Maximal principles
5c. Poisson equation
- Linear evolution equations
6a. Wave equation in 1-d
6b. Dispersion relations: consequences and asymptotics
*If time permits:
- A unified approach to boundary value problems for evolution equations
- Transform methods and Riemann-Hilbert problems
- Homeworks: 50% of final grade, about 5 or 6 in all
- Mid-term exam: 20% of final grade, date TBD
- End-term exam: 30% of final grade, date TBD
- Guenther and Lee: Partial Differential Equations of Mathematical Physics and Integral Equations
- G Folland: Introduction to PDEs
- Renardy and Rogers: Introduction to PDES
- L Evans: Partial Differential Equations