The schedule of ICTS course for **Jan -Apr 2017 **is given below:

**1. Data Assimilation and Dynamical Systems (Elective)**

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

__ Course contents__:

- Quick intro (or recap) for nonlinear dynamics: bifurcations, unstable manifolds and attractors, Lyapunov exponents, sensitivity to initial conditions and concept of predictability.
- Markov chains, evolution of probabilities (Fokker-Planck equation), state estimation problems.
- An introduction to the problem of data assimilation (with examples)
- Bayesian viewpoint, discrete and continuous time cases
- Kalman filter (linear estimation theory)
- Least squares formulation (possibly PDE examples)
- Nonlinear Filtering: Particle filtering and MCMC sampling methods
- 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

__: room CDS 309 (SERC building), IISc__

**Where**__ 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**:

- 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 Bjorling 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, Kuster, Wohlrab
- Lectures on MInimal surfaces: J. Nitsche
- Lectures on Minimal Surfaces in R3: 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.

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**

Course No. |
Course Title |

MA 212 | Algebra I |

MA 219 | Linear Algebra |

MA 221 | Analysis I: Real Anaysis |

MA 231 | Topology |

MA 261 | Probability Models |

MA 223 | Functional Analysis |

MA 232 | Introduction to Algebraic Topology |

MA 242 | Partial Differential Equations |

MA 213 | Algebra II |

MA 222 | Analysis II : Measure and Integration |

MA 224 | Complex Analysis |

MA 229 | Calculus on Manifolds |

MA 241 | Ordinary Differential Equations |

**Advanced Elective Courses**

Course No. |
Course Title |

MA 215 | Introduction to Modular Forms |

MA 277 | Advanced PDE and Finite Element Method |

MA 361 | Probability Theory |

MA 368 | Topics in Probability and Stochastic Processes |

MA 278 | Introduction to Dynamical Systems Theory |

MA 313 | Algebraic Number Theory |

MA 314 | Introduction to Algebraic Geometry |

MA 315 | Lie Algebras and their Representations |

MA 317 | Introduction to Analytic Number Theory |

MA 319 | Algebraic Combinatorics |

MA 320 | Representation Theory of Compact Lie Groups |

MA 332 | Algebraic Topology |

MA 364 | Linear and Nonlinear Time Series Analysis |

MA 369 | Quantum Mechanics |