Ph.D. in Mathematics: Overview, Course, Eligibility Criteria, Admission, Syllabus, Fees, Scope, Result and Cutoffs, & FAQs

PhD in mathematics

Best Ph.D. in Mathematics Colleges in India 2023: Ph.D. in Mathematics stands for ‘Doctor of Philosophy in Mathematics.’ The doctoral degree deals with various concepts of mathematics like space, change, quantity, structure, etc. The Ph.D. in Mathematics program takes 3 to 6 years to complete. After successfully completing the program, the student receives a doctorate in mathematics. The ‘Thesis’ has to be submitted by the student & approved by the higher authorities to be eligible to take home this doctorate. They also become experts in theory, topology, formulating research methodologies, logical interpretation, analysis, formulation, and statistical concepts in mathematics.

During this degree, a student gets well-versed in the overall structure of mathematics giving birth to new laws and theories for the overall development of mathematics as a core subject. The candidate should hold an M.Tech degree, MSc in Mathematics, or physical science to proceed with their application for a Ph.D. in mathematics. After getting this degree, you will have many different career roles to choose from in the academic field and the research sector, such as lecturer, Math expert, or subject matter expert.

Course Highlights

Name of DegreeDoctorate Level Course/Ph.D. in Mathematics 
Degree Full FormDoctor of Philosophy in Mathematics
Examination TypeSemester-based Examination
Age LimitNo Age Bar
Course DurationThree to Six Years
Entrance ExamGATE, NBHM, and CSIR-UGC NET
Eligibility CriteriaM.Sc. Degree in Maths Subject or Physical Sciences/ B.Tech degree in any field with a minimum 55% aggregate score
Total Course FeesINR 50,000 to INR 4 lakh per annum
Average Expected SalaryINR 21,000/- to INR 1,50,000/- per month
Skills Required Mathematical skills,  Researching,  Qualitative Understanding, Interpretation skills, Logical Reasoning, Quantitative Data Interpretation
Minimum PercentageGeneral Category: Minimum 55% aggregate in 12th Standard & bachelor/Master degrees 
SC/ST/OBC: Minimum 50% aggregate in 12th Standard & bachelor/Master degrees
Cut-off MarksVaries University Wise
Admission ProcessMerit-Based Or Entrance Exam Based
OpportunitiesInvestment Firms, Universities,  Statistics,  Commerce Industry, Banks, Business Industry, Colleges, Inventory Management,  Insurance Agencies, Research and Development Firms, Indian Civil Services, etc. 
Career RolesResearcher,  Mathematician, Cryptographer,  Statistician, Personal Banker,  Loan Officer, Professor,  Accountant, Economist, Demographer, etc.
Top Recruitment FirmsInnovation Lab, Ernst & Young, Weizmann Institute,  Tiger Analytics, TCS, etc. 

Eligibility Criteria

The eligibility criteria set for a Ph.D. in Mathematics in India vary according to the different terms and conditions set by Indian universities. Students need to conduct a thorough background check on their desired Ph.D. university to ensure that they meet its admission eligibility criteria. It will help them have a smooth onboarding process at the university. The best university for Ph.D. mathematics in India has preset protocols designed for the upcoming students, i.e, to have prior relevant experience as a lecturer, teacher, or in some research sector. On the other hand, other universities have direct admission criteria with no hard and fast rules. Let’s learn them from the below section –

  • The student should have an aggregate score of at least 55% in his 12th standard class, graduation degree, and post-graduation degree. The score should be at least 50% for ST, SC, or OBCs.
  • The student should score at least passing marks in the admission entrance exam set by the particular university.
  • The student should have completed their master’s degree or postgraduate degree in mathematics, physical science, or any relevant field. The widely accepted degrees are MSC chemistry, MSc statistics, MSc Mathematics, Mtech, or MA economics.

Admission Process

The admission process for a Ph.D. in applied mathematics is straightforward and very simple. The students can apply for Ph.D. admission in 2023 either through their merit records or by clearing the entrance examination. Although in either of these two ways, a student has to follow the stringent step-by-step process to be accepted by their dream university. If your desired university has a policy for entrance examination then you should have clarity on the starting & ending date of filling out forms for the entrance examination and what all syllabus is going to be included in the exam. It will help you achieve good scores and make your admission process very easy. Now let’s understand the two ways of obtaining a Ph.D. in Mathematics admission in 2023 in detail below –

  • Merit-Based Admission: The universities that follow the merit-based admission process don’t conduct any entrance examination but require solid marks, say at least 55% aggregate marks in their previous years of education. Various universities in India provide direct admission to students
    based on their previous year’s degree and school marks. Apart from this, most of these universities, like AMU and Lovely Professional University, prefer to conduct personal interview rounds with the Ph.D. candidates. Such interview rounds are solely based on understanding the mindset of a candidate, his plans, the level of interest he has in research, and how he is going to make a change in this field.
  • Entrance Exam-based admission: Certain top-tier universities follow the entrance-based examination process to filter out eligible & non-eligible students. Here, a candidate needs to clear the entrance examination conducted by the university and score a preset passing mark to be eligible to be a student for the doctorate in mathematics. Such entrance exams have a core mathematics subject that includes topics from various chapters of math like discrete mathematics, advanced algebra, theorems, analysis, etc. The questionnaire you get is in the form of multiple-choice questions where you have to select the right option from the given choices.

Candidates appearing for the Ph.D. in Mathematics admission 2023 are shortlisted based on their entrance examination scores. The process of taking admission to top math Ph.D. programs based on the entrance examination starts in January every year. Further to this, the classes commence in August. Most of the best universities for Ph.D. in Mathematics in India accept GATE and CSIR-Net examinations.

Syllabus for Ph.D. in Mathematics

The syllabus stated by the top universities after taking admission to a Ph.D. in Mathematics is subjective, and some of the subject topics might vary based on the university. Even the specialization chosen by the candidate plays a major role in deciding the syllabus. All the candidates are going to learn about the practical aspects of mathematics, and its theoretical nature, followed by a thesis covered in the Ph.D. in Mathematics syllabus. Based on your current year and the specialization you have chosen, you can find the in-depth Ph.D. in Mathematics syllabus in the beneath section –

1st Year Syllabus

Computational TechniquesComputer Science

2nd Year Syllabus 

Differential EquationDifferential GeometryDiscrete Mathematics
English LiteratureLinear Programming

3rd Year Syllabus

Mathematical FinanceMechanics Metric Space
Number Theory Probability Theory

Fee Structure for Ph.D. in Mathematics

In India, a Ph.D. in Mathematics is very reasonably priced. Depending on the university they want to enroll in, students may have to pay a fee structure that ranges from INR 50,000 to INR 1.5 lakh annually. Your overall costs for completing the Ph.D. in Mathematics degree program could reach Rs. 4 lakhs. The costs associated with earning a math Ph.D. at a public university will probably be considerably less than at a private university. If the student has been a meritorious student or has made an international achievement, he may also be eligible for a reduction in his overall tuition and use the scholarship. But once more, this will depend on the requirements set by the university. A candidate who performs well on the entrance exam for the Ph.D. in Mathematics program is eligible to receive a fellowship title.

Top Entrance Examinations for Ph.D. in Mathematics

Students need to clear the entrance examination to be admitted to the top math Ph.D. programs. Some of the most commonly accepted entrance examinations are the NBHM, CSIR-UGC NET, and GATE.

The GATE entrance examination is held by the Indian Institute of Technology for a candidate to take admission to the Ph.D. degree & M-Tech programs. Candidates can also get a job in the government sector by qualifying for the GATE entrance exam & personal interview sessions.

Syllabus for Entrance Exam

This exam is conducted by the UGC to offer assistant professorship and junior fellowship awards to the candidates who qualify for it with high scores. The syllabus for this examination is mentioned in the below table.

TopicsSub Topics
AnalysisElementary set theory, finite, countable and uncountable sets, Real number system as acomplete ordered field, Archimedean property, supremum, infimum.

Sequences and series, convergence, limsup, liminf.Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, uniform continuity, differentiability, mean value theorem.

Sequences and series of functions, uniform convergence.

Riemann sums and Riemann integral, Improper Integrals.

Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral.
Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse, and implicit function theorems.

Metric spaces, compactness, connectedness. Normed linear Spaces. Spaces of continuous functions as examples.
Linear AlgebraVector spaces, subspaces, linear dependence, basis, dimension, algebra of linear Transformations.

Algebra of matrices, rank and determinant of matrices, linear equations.

Eigenvalues and eigenvectors, Cayley-Hamilton theorem.

Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms.

Inner product spaces, orthonormal basis.

Quadratic forms, reduction and classification of quadratic forms
Complex AnalysisAlgebra of complex numbers, the complex plane, polynomials, power series, transcendental functions such as exponential, trigonometric and hyperbolic functions.

Analytic functions, Cauchy-Riemann equations.Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem.Taylor series, Laurent series, calculus of residues.

Conformal mappings, Mobius transformations.
AlgebraPermutations, combinations, pigeon-hole principle, inclusion-exclusion principle, Derangements.

Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ø- function, primitive roots.

Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equations, Sylow theorems.

Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain.

Polynomial rings and irreducibility criteria. 

Fields, finite fields, field extensions, Galois Theory.
Topologybasis, dense sets, subspace and product topology, separation axioms, connectedness and compactness.
Ordinary Differential Equations (ODEs)Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs.

General theory of homogenous and non-homogeneous linear ODEs, variation of parameters, Sturm-Liouville boundary value problem, Green’s function.
Partial Differential Equations (PDEs)Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs.

Classification of second order PDEs, General solution of higher order PDEs with constant coefficients, Method of separation of variables for Laplace, Heat and Wave equations.
Numerical AnalysisNumerical solutions of algebraic equations, Method of iteration and Newton-Raphson method, Rate of convergence, Solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical differentiation and integration, Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods.
Calculus of VariationsVariation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema.

Variational methods for boundary value problems in ordinary and partial differential equations.
Linear Integral EquationsLinear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel.
Classical MechanicsGeneralized coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s principle and principle of least action, Two-dimensional motion of rigid bodies, Euler’s dynamical equations for the motion of a rigid body about an axis, theory of small oscillations.
Descriptive statistics, exploratory data analysis


The National Board for Higher Mathematics (NBHM) is accepting applications for the entrance exam of Ph.D. program. Candidates must take the Joint Screening Test for Admission and Scholarship for Ph.D./Integrated Ph.D. Programs in order to be considered for admission and financial aid. Students pursuing a Mathematics Ph.D. are eligible for scholarships from the NBHM.

Apart from these, there are several universities conducting entrance examinations on their main campuses. Most universities accept candidates based on state or central-level entrance exams. The declaration of the final merit list is often based only on these entrance examinations. These entrance examinations act as a direct gateway to the best university for a Ph.D. in Mathematics in the world. Students need to meet the cutoff requirements by obtaining good scores in the university entrance examinations. All such details can be easily found on the official website of your desired university. The total time taken by the entrance examination to get completed is approximately 3 hours. The candidate needs to appear for such examinations in offline mode. The questions asked in such examinations are multiple choice questions (MCQs). The exam is held to assess qualitative ability, logical reasoning, data interpretation, and math subject-related knowledge.

Scholarship and Fellowship

The candidates wishing to pursue a Ph.D. in Mathematics have a great opportunity to support their finances through scholarships and fellowships. as the government takes care of the upcoming scientists and values their research papers by sponsoring them with financial support. The candidates can avail of the fellowship scheme by appearing in the CSIR-UGC NET exam, and some of the top students are provided with financial support as mentioned below.

For the first two years, a JRF (Junior Research Fellowship) selected via the CSIR-UGC National Eligibility Test (NET) will receive a stipend of Rs 31,000 per month. Additionally, a yearly contingency grant of Rs. 20,000 will be given to the university or institution for each fellow. After two years, the candidate is eligible to apply for the SRF (Senior Research Fellowship) program, in which the stipend amount is increased to Rs 35,000 per month.

Result and Cutoffs

Cut-off scores are the minimum marks that a candidate must have earned on his or her previous years of graduation, education, and entrance exams in order to be considered for admission to the university and course of their choice. Each Indian university has a different cutoff score for mathematics Ph.D. candidates. Such cut-off points are established in accordance with the number of seats available at the university in question and the popularity of that institution among its students.

Future Scope and Career Options after Ph.D. in Mathematics

The future is bright for Ph.D. in Mathematics degree holders, especially after 2023. These students mostly get roles in the educational industry as lecturers, teachers, or heads of departments. Furthermore, they have the choice to work in the government sector, the private sector, or research-based sectors. Some of the career options mentioned below can be pursued by students after completing Ph.D. in mathematics.

Loan OfficersAccountantsMathematicians
Personal bankersProfessorsEconomists

Employment Sectors 

military schoolsconsultanciesaccounting
defense servicesbudget planningcorporations
market researchprivate banksstock exchanges
government banksmath research centersprivate financial institutions
governmental financial companiestrade and transportation centersAnalysis Group
Complexity and NetworksDynamical SystemsFluid Dynamics
Computational Complex

Top Recruiters

AirtelITC InfotechIBM Global Services
NEXT Techno EnterprisesBrainfuseAquis

Advantages of Studying Ph.D. in Mathematics

Whether you opt for a Ph.D. in Mathematics online or offline, it can turn out to be a very rewarding career for you. After getting a doctorate in mathematics degree from the best university for Ph.D. in Mathematics in India, you can become an affiliate researcher. Henceforth, your role will be to find the answer to the current mathematical problems. If not the exact solution, then at least the precise solution that is acceptable to the existing mathematical question. The best part is that you can go for the Ph.D. in Mathematics distance learning programs without disturbing your current work if you are lacking time to go ahead with the offline programs. Here is the list of advantages of taking admission to the top math Ph.D. programs –

  • Pursuit of excellence: The subject of math is known as the pursuit of excellence. The researchers working in the field of mathematics are more inclined towards understanding all types of complications faced in this mathematical field and providing solutions to them. The Solutions can be formed after designing the appropriate theorems and patterns by conducting thorough research on the problem statement.
  • Research-oriented skills: The Math Ph.D. program helps an individual to get incredible knowledge about the subject of mathematics. Moreover, the course curriculum supports refining the research-related skills of the student. The students this way can go ahead and perform better in real time based on complex research-related tasks. Not only this the students can also contribute to accomplishing the social and economical goals of our country.
  • Part-time opportunities: The Ph.D. in Mathematics students not only get good opportunities in full-time jobs but they can also work part-time from the comfort of their homes and make a good amount of money. You can try freelancing or take part in skill-based jobs where you need to solve mathematical problems by devoting 2 to 3 hours or less every day without leaving your home.
  • Work with different industries: Being a Ph.D. in Mathematics degree holder your career pathways are open to various industries & sectors with no limitations. Once you successfully become a Ph.D. mathematician, you can successfully take roles & responsibilities in various industries like stock exchange, academics, banking, defense, etc.

Salary Package After Completing Ph.D. in Mathematics

The Ph.D. in Mathematics graduate salary ranges between 2.5 lakhs to 10 lakhs per annum. The fresher in this field can expect to have a salary of 2.5 lakhs to 3 lakhs per annum based on his knowledge. Moreover, your salary gradually increases every year as soon as you gain expertise in real-time scenarios & can go up to 9 to 10 lakhs yearly. It comes down to INR 21,000 to 84,000 per month. Although, in private sector companies, you can even expect a monthly package between INR 65,000 to INR 1,50,000 per month. Your home takes in hand salary and upgrades in it solely depend on your expertise, education, grades, graduation from university, performance, and various other factors. On average, the salary of a Ph.D. mathematics graduate is INR 3.84 LPA in the Indian market. Your pay scale is also heavily affected by the role that you are applying for, whether it’s as a stock broker, professor, subject matter expert, etc.

Top Cities for Ph.D. in Mathematics

Candidates can find numerous top universities in India for a Ph.D. in Mathematics. Different universities are popular for different offerings. Although some of the universities in India are more popular than others, this is due to their location, research centers, and practical exposure to mathematical details. Most of these cities are tier-1 and tier-2 but they are number one in facilitating all the resources for Ph.D. in Mathematics candidates. The exposure you will get in these cities is hard to find in other parts of the nation due to the top-notch research-equipped curriculum and well-experienced facilities. Here is the list of top cities that offer Ph.D. in Mathematics in India –

Bangalore Kolkata Delhi 
Mumbai Kanpur Jaipur 
Jalandhar Madras Noida

Top university for Ph.D. in Mathematics in India

India is a country that is well known for top-notch math Ph.D. universities that are popular not only among Indian candidates but also foreign applicants. The best university for Ph.D. mathematics in India is recognized for its offerings, such as research proposals, experienced faculty, real-time-based industry exposure, quality education, and decent infrastructure. Candidates can apply to private universities or government universities for admission to the Ph.D. mathematics programs.

1. Private Ph.D. in Mathematics Universities in India

Ph.D. in Mathematics programs at private universities in India have gained immense attention from candidates who wish to pursue Ph.D. in Mathematics in the future. These universities are well known for providing research scope, hands-on experience with live projects, and fellowship rewards. Even they help students through placement assistance & the best pay scale in the industry. That’s one of the motivating factors for pursuing doctorate degrees from private universities. Here is the list of Ph.D. in Mathematics private universities in India –

Amity University, NoidaGalgotias University, Greater NoidaGIET University, Gunupur 
LPU Jalandhar Kalinga University, RaipurSiksha ‘O’ Anusandhan, Bhubaneswar
LNMIIT Jaipur Bennett University, Greater NoidaNIMS University, Jaipur
JS University, ShikohabadJain University, Bangalore

2. Government Ph.D. in Mathematics Universities in India

Government colleges often get popular among students due to their minimal fee structure and the bright minds that take admission there. The admission criteria to government colleges for Ph.D. math might be quite difficult as compared to private colleges in India, but once you clear the entrance exam, they prove to be the best colleges in terms of gaining subject-relevant knowledge. They also provide the best scholarships, fellowships, and research facilities for their Ph.D. students. Whether you wish to go for Ph.D. in Mathematics distance learning or from the best university for Ph.D. in Mathematics in India in the offline mode, these top-tier government universities never fail to make you satisfied. Here is the list of top-notch Ph.D. in Mathematics government universities in India –

MNIT JaipurIIT Delhi IIT Madras
NIT JalandharIT Roorkee – Indian Institute of TechnologyHBTU Kanpur
NIT CalicutIIT BombayNIT Trichy
DU Delhi – University of DelhiPanjab University, ChandigarhIIT Kanpur – Indian Institute of Technology
PEC Chandigarh

3. Top University for Ph.D. in Mathematics in World

If money isn’t the issue, then the sky’s the limit when it comes to choosing Ph.D. in Mathematics universities. Although, many universities across the world provide scholarships to make the fees minimum for a student having excellent scores in his previous years of education. Candidates wish to opt for Ph.D. programs outside their country to explore more about a particular subject. Undoubtedly, there are so many Ph.D. mathematics universities in the world that provide top-notch research facilities and well-equipped knowledge from highly experienced faculties. Apart from this, a student with a Ph.D. in mathematics from a renowned university across the world is eligible to get incredible salary packages across multiple MNCs in the world. Here is the list of the best university for Ph.D. in mathematics in the world –

Université de Paris, France ETH Zurich, SwitzerlandUniversity of Cambridge,  United Kingdom
University of California Berkeley, United StatesUniversity of Oxford, United KingdomUniversity of Electronic Science & Technology of China, China
Fudan University,  ChinaTon Duc Thang University, VietnamKing Saud University,  Saudi Arabia
University of Toronto, CanadaIslamic Azad University, IranNational University of Singapore, Singapore
University of Vienna, Austria

FAQ’s Related to Ph.D. in Mathematics

Q. What entrance exams are conducted for admission to Ph.D. mathematics 2023?
Ans. The candidate needs to clear GATE and UGC NET entrance examinations for admission to the best university for a Ph.D. in Mathematics in the world.

Q. What all job roles are available for the best mathematics graduate?
Ans. The Ph.D. in applied mathematics graduate can take up the job role of a scientist, statistician, research associate, lecturer, demographer, economist, etc.

Q. How much time does it take to complete a Ph.D. in mathematics in India?
Ans. The overall time to complete the Ph.D. mathematics syllabus is at least 3 years. Whereas, a student is allowed to take up to 6 years to complete his math Ph.D. degree.

Q. Do big MNCs hire Ph.D. mathematics graduates?
Ans. Yes, several top-notch companies hire Ph.D. mathematics graduates. Some of these companies are TCS Innovation Lab, Ernst & Young, the Weizmann Institute, and Tiger Analytics.

Q. What is the monthly salary of a Ph.D. mathematics graduate degree holder?
Ans. If you have been a student at the best university for Ph.D. mathematics in India and successfully cleared all the examinations, then you can expect your monthly take-home between INR 65,000 and INR 1,50,000.

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