ICTS

1.  Data Assimilation and Dynamical Systems (Elective)

Instructor: Amit Apte (ICTS-TIFR) and Somyendu Raha (CDS, IISc)

Course contents:

1. Quick intro (or recap) for nonlinear dynamics: bifurcations, unstable manifolds and attractors, Lyapunov exponents, sensitivity to initial conditions and concept of predictability.
2. Markov chains, the evolution of probabilities (Fokker-Planck equation), state estimation problems.
3. An introduction to the problem of data assimilation (with examples)
4. Bayesian viewpoint, discrete and continuous time cases
5. Kalman filter (linear estimation theory)
6. Least squares formulation (possibly PDE examples)
7. Nonlinear Filtering: Particle filtering and MCMC sampling methods
8. Introduction to Advanced topics (as and when time permits): Parameter estimation, Relations to control theory, Relations to synchronization.

When: Tuesdays and Thursdays 11.30-13.00
Where: room CDS 309 (SERC building), IISc

NOTE:

• This class is joint with CDS, IISc as DS-391.
• First class will be on Tue 10 Jan at 11.30am
• Those interested in attending the course should send an email to apte@icts.res.in expressing their interest and mentioning if they would like to credit/audit the course.

Texts and References:

• Edward Ott, Chaos in Dynamical Systems, Camridge press, 2nd Edition, 2002.(or one of the many excellent books on dynamical systems)
• Van Leeuwen, Peter Jan, Cheng, Yuan, Reich, Sebastian, Nonlinear Data Assimilation, Springer Verlag, July 2015.
• Sebastian Reich, Colin Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, August 2015
• Law, Kody, and Stuart, Andrew, and Zygalakis, Konstantinos, Data Assimilation, A Mathematical Introduction, Springer Texts in Applied Mathematics, September 2015. [First part of the book is available at http://arxiv.org/abs/1506.07825]

2.  Introduction to Minimal surfaces (Elective)

Instructor: Rukmini Dey

Venue: LH-3, Dept of Math, IISc, Bangalore

Timing: 11:00-12:30, Tuesday and Thursday.  The first class will be on Thursday 5th Jan 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 Bjorling problem and Schwartz's solution to it.

If time permits:

1. 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.
2. Connection between minimal and maximal surfaces and Born-Infeld solitons.
3. Constant mean curvature surfaces of non-zero mean curvature – the optimization problem they solve.

Reference Books:

1. Differential geometry of curves and surfaces: Manfredo Do Carmo
2. A survey of minimal surfaces: Robert Osserman
3. Minimal Surfaces I: Dierkes, Hildebrandt, Kuster, Wohlrab
4. Lectures on MInimal surfaces: J. Nitsche
5. Lectures on Minimal Surfaces in R3: Yi Fang
6. Surfaces of constant mean curvature: K. Kenmotsu.
7. Some papers on Minimal and Maximal surfaces and Born-Infeld solitons by various authors including Rukmini Dey, Pradip Kumar and Rahul Kumar Singh.

The following is the list of courses offered at IISc. For the current list see http://math.iisc.ac.in/newcourse.htm

Core Elective Courses

Advanced Elective Courses