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)
- 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
Where: room CDS 309 (SERC building), IISc
- 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 firstname.lastname@example.org 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.
- 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.
- 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 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|