PhD Mathematics entrance exam syllabus of DU after M.Sc in Maths

I want to get admission in PhD in Mathematics in DU and for that I want to get the details of PhD Mathematics entrance exam syllabus of DU after M.Sc in Maths so can you provide me that?

[Arun Vats]

As you want to get the details of PhD Mathematics entrance exam syllabus of DU after M.Sc in Maths so here it is for you:

PhD Mathematics entrance exam syllabus of DU after M.Sc in Maths:

Section I – Analysis:
Finite, countable and uncountable sets, bounded and unbounded sets, Archimedean property,
ordered field, completeness of ℝ, sequence and series of functions, uniform convergence,
Riemann integrable functions, improper integrals, their convergence and uniform convergence,
Fourier series. Partial and directional derivatives, Taylor’s series, implicit function theorem, line
and surface integrals, Green’s theorem, Stoke’s theorem.
Elements of metric spaces, convergence, continuity, compactness, connectedness,
Weierstrass’s approximation theorem, completeness, Baire’s category theorem, Bolzano-
Weirstrass theorem, compact subsets ofℝ_, Heine-Borel theorem,
Lebesgue outer measure, Lebsegue measure and Lebsegue integration, Riemann and
Lebesgue integrals.
Complex numbers, analytic functions, Cauchy-Riemann equations, Riemann sphere and
stereographic projection, lines, circles, crossratio, Mobius transformations, line integrals,
Cauchy’s theorems, Cauchy’s theorem for convex regions, Morera’s theorem, Liouville’s
theorem, Cauchy’s integral formula, zero-sets of analytic functions, exponential, sine and cosine
functions, power series representation, classification of singularities, conformal mapping,
contour integration, fundamental theorem of algebra.
Banach spaces, Hahn-Banach theortem, open mapping and closed graph theorem, principle of
uniform boundedness, boundedness and continuity of linear transformations, dual spaces,
embedding in the second dual, Hilbert spaces, projections, orthonormal bases, Riesz
representation theorem, Bessel’s inequality, Parseval’s identity.
Elements of Topological spaces, continuity, convergence, homeomorphism, compactness,
connectedness, separation axioms, first and second countability, separability, subspaces, product
spaces.

Section II – Algebra:
Space of n-vectors, linear dependence, basis, linear transformations, algebra of matrices, rank
of a matrix, determinants, linear equations, characteristic roots and vectors.
Vector spaces, subspaces, quotient spaces, linear dependence, basis, dimension, the algebra of
linear transformations, kernel, range, isomorphism,linear functional, dual space, matrix
representation of a linear transformation, change of bases, reduction of matrices to canonical
forms, inner product spaces, orthogonality, eigenvalues and eigenvectors, projections, triangular
form, Jordan form, quadratic forms, reduction of quadratic forms.
Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups,
permutation groups, Cayley’s theorem, Symmetric groups, alternating groups, simple groups.
conjugate elements and class equations of finite groups, Sylow’s theorem, solvable groups,
Jordan-Holder theorem, direct products, structure theorem for finite abelian groups.
Rings, Ideals, prime and maximal ideals, quotient ring, integral domains, Euclidean domains,
principal ideal domains, unique factorization domains,polynomial rings, chain conditions on
rings,fields, quotient fields, finite fields, characteristic of field, field extensions, elements of
Galois theory, solvability by radicals, ruler and compass construction.

Section III- Differential Equations and Mechanics:
First order ODE, singular solutions, initial value problems of first order ODE, general theory of
homogeneous and non-homogeneous linear ODEs, variation of parameters, Lagrange’s and
Charpit’s methods of solving first order PDEs, PDEs of higher order with constant coefficients.
Existence and uniqueness of solution ( , ) dy
dx f x y , Green’s function, Sturm-Liouville boundary
value problems, Cauchy problems and characteristics, classification of second order PDE,
separation of variables for heat equation, wave equation and Laplace equation,
Equation of continuity in fluid motion, Euler’s equations of motion for perfect fluids, two
dimensional motion, complex potential, motion of sphere in perfect liquid and motion of liquid
past a sphere, vorticity, Navier-Stoke’s equations of motion for viscous flows, some exact
solutions.

All candidates, except those who have been exempted from course work by the DRC, will be required to qualify Pre-Ph.D examination with three courses, selecting at most two from a group listed hereunder.

Group A
• Distribution Theory & Calculus on Banach Spaces
• Operator Theory and Function Spaces
• Geometric Function Theory
• Introduction to Operator Algebras
• Advanced Frame Theory
Group B
• Rings and Modules
• Differential Manifolds
• Group Rings
Group C
• Graph and Network Theory
• Convex and Non smooth Analysis
• Combinatorial Mathematics
• Advanced Compressible Flows

ELIGIBILITY:
Candidates must have passed Master's/M.Phil degree of the University of Delhi or from any other university or any degree recognized as equivalent to Master's/M.Phil degree Mathematics.

She/he must have obtained either a minimum of 50% marks or equivalent grading in the M.Phil degree or a minimum of 55% marks or equivalent grading in the Master's degree.

Hello sir ! I had done M.sc in math , now I want to apply for PhD math. Entrance exams which is conducted by DU , will you please provide me PhD Mathematics entrance exam syllabus ?

[Shaleen]

Hello buddy as you want here we provides you PhD Mathematics entrance
exam syllabus as follows

Duration of the programs.

The duration of the M.Phil. course is one and a half years.

For Ph.D course, the minimum duration is 2 years and the maximum
duration is 5 years.

Selection procedure

Selection Procedure. Admission to the M.Phil. and Ph.D. programmes will be
done on the basis of the relative merit of students' performance at
undergraduate and post-graduate examinations and the written test (of two
hours duration).

The merit lists will be prepared by taking into account 25% weight of marks
scored in each of undergraduate and post-graduate examinations and 50%
weight of marks scored in the test.

The minimum qualifying marks for admission to Ph.D. programme is 60%.

The University/College teachers holding a permanent, temporary or adhoc
positions and having completed two years of service as teacher in a
Department/Constituent Colleges of the University of Delhi and

candidates having fellowships/scholarships instituted by the
University/national

and international agencies under schemes approved/recognized by the
University (as well as certain other category

Syllabus

The questions that are repeatedly asked from the M.Sc (Math/Applied Math)
papers are

Sets,

determinants,

linear transformations,

graphs of functions

group theory,

limits,

rank, nullity,

sequences and coordinate geometry of two and three dimensions etc.

There are still few topics which are most covered by the candidates are

series,

eigen values,

system of linear equations,

integration,

continuity,

differentiation,

vector spaces,

matrices and

elementary probability