PhD Mathematics Syllabus & Subjects 2025

PhD Mathematics course has a wide-ranging syllabus that includes both theoretical and applied subjects, ending with a final thesis. Students deeply study areas like Real Analysis, Modern Algebra, Numerical Analysis, Calculus as well as Statistics and Probability. The course also focuses on research methodology to help students improve their research abilities. Furthermore, students get to learn about Differential Equations, Trigonometry and the Theory of Equations. The course duration usually lasts three to six years and you need a Master's degree in Mathematics or related subjects for being eligible. After finishing it, people can work as Mathematicians, Statisticians, Lecturers or Researchers with chances available in different areas.

Here’s a general PhD Mathematics syllabus summary for your reference: 

The syllabus for PhD Mathematics includes complex subjects like algebra, analysis, calculus, differential equations, geometry, number theory and probability among others. This program is designed in a way that merges theoretical learning with hands-on project work to prepare students in the field of mathematical study and problem solving.

Some common specialisations in PhD Mathematics include:

• Mathematical and Computer Sciences
• Mathematical Statistics
• Computational Mathematics
• Applied Mathematics
• Statistics

The syllabus intends to offer comprehensive understanding about mathematical patterns and how they are useful in sectors such as science, engineering, finance and others. The graduate will develop abilities in research, data analysis as well as mathematical modelling.

In PhD Mathematics course, the subjects can change depending on the university and program requirements. However, for general understanding, here is an example of semester-wise subjects for Mathematics PhD course:

Semester 1:

• Compulsory Subjects:Calculus Practical
• Optional Elective Subject:Real Analysis Tutorial

Semester 2:

• Compulsory Subjects:Real Analysis Tutorial
• Optional Elective Subject:Linear Algebra Tutorial

Semester 3:

• Compulsory Subjects:Theory of Real functions Tutorial
• Optional Elective Subject:Partial differential equations and system of ODEs

Semester 4:

• Compulsory Subjects:Numerical Methods and Scientific Computing Practical
• Optional Elective Subject:Ring Theory Tutorial

Semester 5:

• Compulsory Subjects:Multivariable Calculus Tutorial
• Optional Elective Subjects (Choose any two):Linear Programming Tutorial

Semester 6:

• Compulsory Subjects: Independent Research with Seminars
• Optional Subjects: Differential Geometry Tutorial, Number Theory/ Project

Please note that these subjects might differ from institution to another based on their curriculum, so, it’s best to check with your preferred institution for an accurate syllabus & subjects.

Institutions might have different course structures for the PhD Mathematics course, but typically it is planned to give students a deep comprehension of high-level mathematics ideas and research methods. Take a look at the course structure for some of the reputed institutions:

Course Structure at TIFR CAM Bangalore: The initial two years of the Integrated PhD Mathematics course at TIFR CAM Bangalore involve courses such as Real Analysis, Topology, Probability Theory and many others. In the subsequent semesters, Advanced Functional Analysis, Algebra along with Elective courses are covered.

Course Structure at IIT Madras: At IIT Madras, the PhD Mathematics course includes major courses in Advanced Algebra, Analysis, Differential Equations and Discrete Mathematics. Optionally, students can select from a variety of elective courses both within mathematics and outside departments.

Course Structure at at TIFR Mumbai: The PhD Mathematics course in TIFR Mumbai concentrates on creating specific research fields within mathematics, improving abilities in technical writing and communication, learning about ethics related to mathematics and getting ready for research work along with publishing.
 

Usually, a PhD Mathematics course may use teaching methods like lectures, seminars, tutorials and independent research. Lectures are given to explain complex mathematical ideas and theories. Seminars allow students to show their research work and talk about it with others. Tutorials are practical problem-solving sessions where students can use the knowledge from lectures in real-life situations.

 
Research work, which students carry out under guidance of their supervisors, is an important part of the methodology. They learn to identify research problems, create methodology and add into existing understanding in areas they have chosen for study. The method of teaching also focuses on developing critical thinking ability along with skills like solving problems and communication that are very necessary if one wants a good job in mathematics research or academia.
 

PhD Mathematics projects are very varied in nature. They look to push forward the understanding of maths and tackle real-life issues. These projects are usually about doing extensive research, deep analysis, and creating new answers within different maths areas. Often, PhD Mathematics projects focus on themes such as Mathematical Finance, Number Theory, Probability Theory, Differential Equations and more. Students contribute to mathematics by exploring theory, computational modelling and practical application. Students frequently work with advisors on projects that require independent research and presentation of findings at seminars, in papers or during conferences.

Here are some of the best books for your PhD Mathematics course:

• Ordinary and Partial Differential Equations by M.D Raisinghania
• An Introduction to Ordinary Differential Equations by Earl A Coddington
• Advanced Differential Equations by M.D. Raisinghania
• Topics in Algebra by I.N Herstein
• Mathematical Analysis by Walter Rudin
• Introduction To Topology And Modern Analysis by G.F Simmons

These books offer a complete coverage of all essential subjects for PhD Mathematics scholars in India.