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

__ Course contents__:

- 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.

**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.

**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__:

- 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

__ Grading__:

- 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

__ Reference books__:

- 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