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Introduction to the PhD Program in Mathematics and Statistics

The Department of Mathematics at IIT Bombay offers a PhD program in Mathematics and Statistics, providing students with the opportunity to pursue a career in research. The program is designed to equip students with advanced knowledge and skills in their chosen area of specialization.

Admission to the PhD Program

Admission to the PhD program is based on a written test and interview. Students are required to have a first-class master's degree in Mathematics, Statistics, or Computer Science, and a valid GATE score or an award of NBHM/CSIR/UGC Research Fellowship. The admission process takes place twice a year, in June and December.

PhD Course Requirements

All PhD students with an MSc or equivalent qualification must acquire a minimum of 34 credits within the first three semesters of joining the PhD program. The students must also maintain a minimum CPI of 6.0 in each of these semesters. The requirements include:

• Completing at least 3 core PhD courses
• Earning at least 24 credits through PhD courses (core/elective)
• Earning up to a maximum of 8 credits through seminar courses (MAS801/MAS802)
• Crediting up to two 500-level MSc courses to partially satisfy the credit requirement

PhD Courses

The department offers a range of PhD courses, including:

Autumn Semester Core Courses

• MA 811: Algebra I
• MA 813: Measure Theory
• MA 815: Differential Topology
• MA 817: Partial Differential Equations I
• MA 833: Weak Convergence and Martingale Theory
• MA 861: Combinatorics-I
• MA 863: Theoretical Statistics I

Spring Semester Core Courses

• MA 812: Algebra II
• MA 814: Complex Analysis
• MA 816: Algebraic Topology
• MA 818: Partial Differential Equations II
• MA 820: Stochastic Processes
• MA 823: Probability I
• MA 824: Functional Analysis
• MA 862: Combinatorics-II
• MA 867: Statistical Modelling- I

PhD Course Contents

The course contents for each PhD course are as follows:

MA 811 Algebra I

• Field extensions
• Galois theory
• Ring extensions
• Transcendental extensions
• Dedekind domains
• Valuations and completions

MA 812 Algebra II

• Modules over a PID
• Noetherian modules and rings
• Semisimple and simple rings
• Representations of finite groups
• Categories and functors
• Homological algebra

MA 813 Measure Theory

• Review of measure theory
• Borel measures
• Complex measures
• Differentiation
• Product measures
• Content on a locally compact Hausdorff space

MA 814 Complex Analysis

• Review of basic complex analysis
• Harmonic functions
• Maximum modulus principle
• Analytic continuation
• Little Picard theorem

MA 815 Differential Topology

• Review of differentiable manifolds
• DeRham complex
• Vector bundles
• Transversality
• Sard's theorem

MA 816 Algebraic Topology

• Paths and homotopy
• Fundamental groups
• Covering spaces
• Singular homology
• Cohomology of spaces

MA 817 Partial Differential Equations I

• Distribution theory
• Sobolev spaces
• Second-order linear elliptic equations
• Second-order linear parabolic equations
• Second-order linear hyperbolic equations

MA 818 Partial Differential Equations II

• Nonlinear first-order scalar equations
• Calculus of variations
• Hamilton-Jacobi equations
• System of conservation laws

MA 820 Stochastic Processes

• Discrete time Markov chains
• Continuous time Markov chains
• Poisson process
• Random walk

MA 823 Probability I

• Probability space
• Random variables
• Expectation
• Convergence of random variables
• Characteristic functions

MA 824 Functional Analysis

• Review of normed linear spaces
• Hahn-Banach theorems
• Uniform boundedness principle
• Open mapping theorem
• Closed graph theorem
• Riesz representation theorem

Qualifying Examination Requirement

The qualifying examination requirement has been discontinued for PhD students admitted from July 2025 onwards. Instead, students must fulfill the following three criteria by the end of the third semester of their enrollment in the PhD program:

1. Successfully complete a minimum of four core PhD courses offered by the Department of Mathematics.
2. Successfully complete one seminar course (MAS801 or MAS802) under the supervision of a faculty member from the Department of Mathematics.
3. Attain a minimum Grade Point Average (GPA) of 7.0, computed from the four core PhD courses with the highest grades completed during the first three semesters.

Research Areas

The department has research groups in various areas, including:

• Algebra
• Analysis
• Geometry and Topology
• Number Theory
• Probability and Statistics
• Partial Differential Equations

Faculty

The department has a diverse faculty with expertise in various areas of mathematics and statistics. The faculty members are actively involved in research and guide PhD students in their research work.

Facilities

The department has well-equipped facilities, including a library, computer labs, and seminar rooms. The department also has a mathematics research center, which provides a platform for researchers to interact and collaborate.

Conclusion

The PhD program in Mathematics and Statistics at IIT Bombay is a prestigious program that provides students with the opportunity to pursue a career in research. The program has a strong focus on academic excellence, and the department has a diverse faculty with expertise in various areas of mathematics and statistics. The program is designed to equip students with advanced knowledge and skills in their chosen area of specialization, and the department provides well-equipped facilities to support research work.