UPPSC Mathematics Assistant Professor Syllabus, Eligibility – GDC Examination 2025

September 15, 2025

The Uttar Pradesh Public Service Commission (UPPSC) conducts the Assistant Professor (Mathematics) Examination for recruitment in Government Degree Colleges (GDCs). To excel in the exam, aspirants must have a strong grasp of the prescribed syllabus and prepare systematically. Below is the complete UPPSC Mathematics Assistant Professor Syllabus 2025, covering all units and key topics.

UPPSC Mathematics Assistant Professor Syllabus 2025

Unit 1: Analysis

  • Set Theory: Finite, countable, and uncountable sets.

  • Real Numbers: Complete ordered field, Archimedean property, supremum, infimum.

  • Sequences & Series: Convergence, limsup, liminf, uniform convergence.

  • Theorems: Bolzano-Weierstrass, Heine-Borel.

  • Metric Spaces: Completeness, connectedness.

  • Integration: Riemann integration, Lebesgue measure & integration.

  • Normed Linear Spaces: Banach spaces, open mapping theorem, closed graph theorem, Hahn–Banach theorem, Hilbert spaces.

Unit 2: Calculus

  • Continuity & Differentiability: Types of discontinuity, uniform continuity, monotonic functions, functions of bounded variation.

  • Mean Value Theorems: Rolle’s, Lagrange’s, Cauchy’s.

  • Multivariable Calculus: Partial derivatives, directional derivatives, maxima & minima, Lagrange multipliers.

  • Multiple Integrals: Double & triple integrals with applications.

  • Improper Integrals: Convergence tests.

  • Vector Calculus: Gradient, divergence, curl, Green’s theorem, Stokes theorem, Gauss divergence theorem.

Unit 3: Algebra

  • Number Theory: Divisibility in Z, Fundamental theorem of arithmetic, Congruences, Chinese Remainder Theorem, Euler’s φ function, Fermat’s theorem.

  • Group Theory: Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equation, Sylow theorems.

  • Ring Theory: Rings, ideals, prime & maximal ideals, UFD, PID, Euclidean domain, polynomial rings.

  • Field Theory: Finite fields, field extensions, Galois theory.

  • Module Theory: Modules, submodules, cyclic modules, Noetherian & Artinian modules, Hilbert basis theorem.

Unit 4: Linear Algebra

  • Vector Spaces: Subspaces, linear dependence & independence, basis, dimension.

  • Linear Transformations: Algebra, matrix representation, rank-nullity theorem.

  • Eigenvalues & Eigenvectors: Diagonalization, Jordan form, Cayley–Hamilton theorem.

  • Inner Product Spaces: Orthonormal basis, quadratic forms, classification & reduction.

Unit 5: Complex Analysis & Topology

  • Complex Functions: Limit, continuity, differentiability, Cauchy-Riemann equations, analytic functions.

  • Complex Integration: Cauchy’s theorem, integral formula, Liouville’s theorem, maximum modulus principle, Schwarz lemma.

  • Series: Taylor & Laurent series, calculus of residues, contour integration.

  • Topology: Basis, dense sets, connectedness, compactness, separation axioms, first & second countability.

Unit 6: Differential Equations & PDEs

  • ODEs: Existence & uniqueness, singular solutions, systems of ODEs, Sturm–Liouville problem, Green’s function.

  • Second Order ODEs: Methods of variation of parameters, changing variables.

  • PDEs: Lagrange’s method, Charpit’s method, classification of second-order PDEs, separation of variables, solutions of Laplace, wave & heat equations.

Unit 7: Numerical Analysis, Calculus of Variations & Integral Equations

  • Numerical Methods: Newton–Raphson, iteration method, Gauss elimination, Gauss–Seidel method.

  • Interpolation: Gregory-Newton, Lagrange, divided difference.

  • Numerical Differentiation & Integration: Newton-Cotes formulae.

  • Numerical Solutions of ODEs: Picard, Euler, modified Euler, Runge–Kutta methods.

  • Calculus of Variations: Euler–Lagrange equation, fixed & variable endpoint problems.

  • Integral Equations: Fredholm & Volterra type, successive approximations, eigenvalues & eigenfunctions, resolvent kernel.

Unit 8: Geometry & Differential Geometry

  • 3D Geometry: Plane, straight line, skew lines, sphere, cone, cylinder, paraboloid, central conicoids.

  • Tensors: Contravariant, covariant tensors, transformation laws, contraction, inner product, quotient law.

  • Differential Geometry: Curves, curvature, torsion, Serret–Frenet formulas, helix, fundamental forms of surfaces.

Unit 9: Operations Research, Statistics & Graph Theory

  • Operations Research: Linear programming, simplex method, duality, transportation & assignment problems, traveling salesman problem, convex optimization, gradient descent.

  • Statistics & Probability: Probability laws, expectation, Bayes theorem, random variables, Binomial, Poisson, Normal distributions, correlation & regression, logistic regression, LDA.

  • Graph Theory: Graph isomorphism, subgraphs, matrix representation, shortest paths, spanning trees, bipartite graphs, planar graphs, Euler’s formula, Hamiltonian graphs.

Unit 10: Mechanics & Fluid Dynamics

  • Mechanics: Moment of inertia, motion of rigid bodies, generalized coordinates, Lagrange’s & Hamilton’s equations, Poisson bracket, principle of least action.

  • Fluid Dynamics: Continuity equation, Euler’s equation of motion, sources & sinks, flow past a cylinder & sphere.

UPPSC Mathematics Assistant Professor Eligibility 2025

Option A: Master’s Degree + NET/SET/SLET

A Master’s degree with 55% marks (or equivalent grade) in Mathematics or a relevant/allied subject from an Indian University,
OR an equivalent degree from an accredited foreign university.

Additionally, candidates must have cleared the National Eligibility Test (NET) conducted by UGC/CSIR or an equivalent accredited test like SLET/SET.

Candidates with a Ph.D. Degree as per UGC Regulations (2009/2016 & amendments) are exempt from NET/SLET/SET, provided they fulfill the following conditions (for Ph.D. registered before July 11, 2009):

  • Ph.D. awarded in regular mode.

  • Thesis evaluated by at least two external examiners.

  • Open Ph.D. viva voce conducted.

  • At least two research papers published from Ph.D. work (one in a refereed journal).

  • At least two papers presented in UGC/ICSSR/CSIR-sponsored seminars/conferences.

Certification must be provided by the Registrar/Dean (Academic Affairs) of the awarding University.

Note: NET/SLET/SET exemption applies to subjects where these tests are not conducted.

Option B: Ph.D. from Top 500 Foreign Universities

A Ph.D. degree from a foreign university/institution ranked among the top 500 in global rankings by:

    • QS (Quacquarelli Symonds)

    • Times Higher Education (THE)

    • Academic Ranking of World Universities (ARWU) by Shanghai Jiao Tong University

Conclusion

The UPPSC Mathematics Assistant Professor Syllabus 2025 is comprehensive and requires a balance between theoretical understanding and practical problem-solving skills. Structured study, practice of past papers, and conceptual clarity will significantly improve your chances of securing a top score and a faculty position in Government Degree Colleges.

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