Probability theory is now considered an integral part of mathematics. The mathematics programmes of most Indian Universities, however, do not include probability theory in much detail. The aim of this school is to give a comprehensive training in probability and stochastic processes to students of the undergraduate and postgraduate programmes. Besides, giving them a chance to interact with researchers in these topics is also a goal of this instructional school.

In addition, over the two weekends, introductory lectures on research topics will be given, which will be followed by interactions with the students.

We propose to run three courses:

- Basic probability theory
- Measure free Stochastic Processes
- Modules of Linear Algebra and Real Analysis

In addition, over the two weekends, introductory lectures on research topics will be given, which will be followed by interactions with the students.

• SPONSORS & CONTACTS •

To attend the school, please fill up the Google form.

Last date of application was: May 14, 2023 (Sunday).

*Application process has been closed!*

NISER Bhubaneswar is located at Jatni, which is around 30 km from **Bhubaneswar City/Railway station (BBS)**. The nearest railway station is: **Khurda Road Junction (KUR)**, which is around 4 km from NISER. One can avail auto-rickshaw service or state run road transportation buses from both the train stations (towards Khurda) to reach NISER. The nearest airport is: **Biju Patnaik International Airport (BBI)**, which is around 28 km from NISER. One can avail *Ola* or pre-paid taxi service from the airport to reach NISER. See Map

Any EIGHT of the following speakers will deliver talks according to their availability and mutual understanding.

IACS, Koklata

ISI, Delhi

ISI, Delhi

ISI, Kolkata

ISI, Kolkata

CMI, Chennai

NISER, Bhubaneswar

NISER, Bhubaneswar

Ashoka University, Delhi

IISER, Pune

NISER, Bhubaneswar

IIM, Bengaluru

ISI, Kolkata

**Basic Probability Theory:** Orientation, Elementary concepts: experiments, outcomes, sample space, events. Discrete sample spaces and probability models. Combinatorial probability and urn models; Conditional probability and independence; Random variables – discrete and continuous; Expectations, variance and moments of random variables; Markov inequality, Chebycheff inequality, First and second moment method (in tutorial with applications), Transformations of univariate random variables; Jointly distributed random variables; joint density and change of variables formula, independence and conditional distribution; probability transforms (if time permits), Modes of convergence, Borel-Cantelli lemma, WLLN under 2nd moment assumption and SLLN under 4th moment assumption, CLT (statement with applications)

**Stochastic Processes:** Simple symmetric Random Walk – reflection princliple, return to origin, Arc-sine law, Discrete Markov chains with countable state space, example of two state Markov chain, Recurrence, transience, criterion for recurrence, positive and null recurrence; Irreducibility and decomposition of state space, class property, stationary distributions, ratio limit theorem, periodicity, limit theorems, reversible chains. Generating functions; Several illustrations including the Gambler's ruin problem, queuing chains, birth and death chains etc., Branching process, Poisson process, continuous time Markov chain with countable state space, continuous time birth and death chains.

**Analysis:** Sequences, limits, limsup and liminf; infinte series, convergence and absolute convergence, Limit of a function at a point, continuity, intermediate value theorem, differentiation, Rolle's theorem, Taylor's theorem Power series – radius of convergence, term by term differentiation and integration, Abel's theorem, Cauchy-Schwarz Inequality for infinite series and integrals.

**Linear Algebra:** Matrices, Matrix Operations, Determinants, Matrix Representation of system of linear Equations, Vector Spaces and Subspaces, Basis of a Vector Space, Linear transformations, matrix representation, Inner product spaces and orthogonality, Cauchy-Schwarz Inequality for finite sum Eigenvalues, Classification and Transformation of Quadratic Forms, Diagonalisation of real symmetric matrices.

**References:**

- A. Ramachandra Rao and P. Bhimasankaram: Linear Algebra.
- G. F. Simmons: Introduction to Topology and Modern Analysis
- S. M. Ross: A first course in Probability
- Jacod & Protter: Probability Essentials
- W. Feller: Introduction to the Theory of Probability and its Applications, Vol. 1.
- P.G. Hoel, S.C. Port and C.J. Stone: Introduction to Stochastic Processes.
- (Lecture notes from the speakers, if available)

Accommodation will be provided to all the participants in NISER hostels during the period of the school. **Participants may plan to reach NISER preferably on Sunday, June 25, 2023**. The name of the hostel allotted to the participants will be notified here.

- Sleeper class train fare will be reimbursed to all the outstationed participants.
- The competent authority of NISER has approved free hostel accommodation from June 25, 2023 to July 22, 2023 for participants. Participants may bring their bed cover and other personal use materials.
- Food will be arranged for the participants from 25th June evening till 22nd July Morning.

• Aakriti | Deen Dayal Upadhyaya College |

• Aindrila Rakshit | Lady Brabourne College |

• Alisha Rani Sahu | Larambha College |

• Alok Naik | Veer Surender Sai University of Technology |

• Anandi Roy | IISER Berhampur |

• Arif Ali | Ramakrishna Mission |

• Ashutosh Mahanta | Fakir Mohan Autonomous College |

• Avik Shakhari | Institute of Mathematics and Applications |

• Devansh Mishra | Kisan P G College |

• Dishari Bhaduri | Madras Christian College |

• Gaurav Kumar | Indian Statistical Institute |

• Ipsita Panda | Veer Surendra Sai University of Technology |

• Isha Gupta | Deen Dayal Upadhyaya College |

• Ishita Mandal | Hooghly Mohsin College |

• Lekhraj | Kisan Post Graduate College |

• Lopamudra Barad | Banki (Autonomous) College |

• Nidhi Kumari | Banaras Hindu University |

• Parmita Das | Deen Dayal Upadhyaya College |

• Priyanka Vishnudas Ghuge | NIT Rourkela |

• Rishith Reddy V | IISER Tirupati |

• Samaroha Chatterjee | Institute of Mathematics and Applications |

• Santu Sarkar | IIT Palakkad |

• Satyajit Dhadumia | IIT Palakkad |

• Shubham Dahiya | Deen Dayal Upadhyaya College |

• Sonu Verma | South Asian University |

• Thaneshwar Prasad Choudhary | Mohanlal Sukhadia University |

• Vishnu Unnikrishnan | St. Thomas College (Autonomous) |

• Yash Gupta | Raj Rishi Govt. Autonomous College |

**Click here** to see the Time-Table with names of speakers and course associates/tutors.

**Speakers:**

**Arijit Chakrabarty**(A.C.), ISI, Kolkata (Stochastic Process II & III)**B V Rao**(B.V.R.), CMI, Chennai (Basic Probability Theory IV)**Dinesh Keshri**(D.K.), NISER, Bhubaneswar (Analysis)**Kaushik Majumdar**(K.M.), NISER, Bhubaneswar (Linear Algebra)**Kumarjit Saha**(K.S.), Ashoka University, Delhi (Basic Probability Theory I)**Moumanti Poddar**(M.P.), IISER, Pune (Basic Probability Theory II)**Rishideep Roy**(R.R.), IIM, Begaluru (Stochastic Process I)**Nabin Kumar Jana**(N.K.J.), NISER, Bhubaneswar (Basic Probability Theory III)

**Sukrit Chakraborty**, ISI, Kolkata**Priyanka Sen**, ISI, Kolkata

[Parent] : Department of Atomic Energy, Govt. of India

[Host] : National Institute of Science Education and Research

[In collaborartion with ISI Kolkata]

Postal contact address:

Dr. Nabin Kumar Jana (Readder-F) | Dr. Kaushik Majumder (Asst. Professor)Organisers, BISSP 2023

School of Mathematical Sciences

National Institute of Science Education and Research, Bhubaneswar

PO: NISER, Via: Jatni, District: Khurda, Odisha - 752050

- +91-674-2494090 / 2494109
- nabinjana | kaushikmajumder [@niser.ac.in]

- School of Mathematical Sciences, NISER

- +91-674-2494090 / 2494109

- aissp@niser.ac.in