The schedule of ICTS courses for **Aug - Nov 2018 **are** **given below:

**Techniques in discrete probability (Elective)**

** Instructor**: Riddhipratim Basu

** Venue**: Math department LH-5, IISc, Bangalore

** Meeting Time**: Tuesdays and Thursdays, 2:00-3:30 pm

** First Class**: Thursday, 2nd August, 2018

**MA 394**: Techniques in discrete probability

**Credits**: 3:0

**Pre-requisites**:

- This course is aimed at Ph.D. students from different fields who expect to use discrete probability in their research. Graduate level measure theoretic probability will be useful, but not a requirement. I expect the course will be accessible to advanced undergraduates who have had sufficient exposure to probability.

We shall illustrate some important techniques in studying discrete random structures through a number of examples. The techniques we shall focus on will include (if time permits)

- the probabilistic method;
- first and second moment methods, martingale techniques for concentration inequalities;
- coupling techniques, monotone coupling and censoring techniques;
- correlation inequalities, FKG and BK inequalities;
- isoperimetric inequalities, spectral gap, Poincare inequality;
- Fourier analysis on hypercube, Hypercontractivity, noise sensitivity and sharp threshold phenomenon;
- Stein’s method;
- entropy and information theoretic techniques.

We shall discuss applications of these techniques in various fields such as Markov chains, percolation, interacting particle systems and random graphs.

**Suggested books**:

- Noga Alon and Joel Spencer, The Probabilistic Method ,Wiley, 2008.
- Geoffrey Grimmett, Probability on Graphs ,Cambridge University Press, 2010.
- Ryan O'Donnell, Analysis of Boolean Functions ,Cambridge University Press, 2014.

The following is the list of courses offered at IISc. For the current list see:

**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 |

The schedule of ICTS courses for **Jan - Apr 2018** are given below

**Introduction to PDEs (Reading)**: Rukmini Dey__Instructor__S N Bose Meeting Room, ICTS Campus, Bangalore__Venue__:: Monday and Friday: 11:30 am - 1:00 pm**Meeting Time****First Class:**: First 5 chapters of**Course contents****Ian Sneddon's book:****Elements of PDEs.**: Ordinary Differential Equations in more than 2 variables; Partial Differential Equations of the first order; Partial Differential Equations of the Second Order; Laplace Equation; Wave Equation. If time permits we will go through some chapters of**Syllabus****"Fourier Series" by Georgi P. Tolstov.**

**Introduction to Dynamical Systems (Reading)**: Vishal Vasan**Instructor**: ICTS Campus, Bangalore**Venue**: Friday: 4:30 pm - 6:00 pm**Meeting Time**: Friday, 19th January, 2018**First Class**: Nonlinear Dynamics and Chaos by S Strogatz. Selected reading from Differential Equations and Dynamical Systems by L Perko and other suitable texts.**Course contents**: The course will cover the entire content of Strogatz' book supplemented with more detailed mathematical treatments of selected theorems from other sources.**Syllabus**