TIFR GS Syllabus and Sample Paper Mathematics

GS2020: Selection Process for Mathematics

Selection process for admission in 2020 to the various programs in Mathematics at the TIFR cen-
ters - namely, the PhD and Integrated PhD programs at TIFR, Mumbai as well as the programs
conducted by TIFR CAM, Bengaluru and ICTS, Bengaluru - will be held in two stages.

Part I. A nation-wide test will be conducted in various centers on December 8, 2019. For the
PhD and Integrated PhD programs at the Mumbai Center, this test will comprise the entirety of
Part I of the evaluation process. For more precise details about Part I of the selection process at
other centers (TIFR CAM, Bengaluru, and ICTS, Bengaluru) we refer you to the websites of those
centers.
The nation-wide test on December 8 will be an objective test of three hours duration, with 20 mul-
tiple choice questions and 20 true/false questions. The score in this test will serve as qualification
marks for a student to progress to the second step of the evaluation process. The cut-off marks for
a particular program will be decided by the TIFR center handling that program.
Additionally, some or all of the centers may consider the score in Part I (in addition to the score in
Part II) towards making the final selection for the graduate program in 2020.

Part II. The second part of the selection process varies according to the program and the center.
More details about this part will be provided at a later date.

Syllabus for Part I

Part I of the selection process is mainly based on mathematics covered in a reasonable B.Sc. course.
This includes:

Algebra: Definitions and examples of groups (finite and infinite, commutative and non-commutative),
cyclic groups, subgroups, homomorphisms, quotients. Group actions and Sylow theorems. Defini-
tions and examples of rings and fields. Integers, polynomial rings and their basic properties. Basic
facts about vector spaces, matrices, determinants, ranks of linear transformations, characteristic
and minimal polynomials, symmetric matrices. Inner products, positive definiteness.

Analysis: Basic facts about real and complex numbers, convergence of sequences and series of
real and complex numbers, continuity, differentiability and Riemann integration of real valued
functions defined on an interval (finite or infinite), elementary functions (polynomial functions,
rational functions, exponential and log, trigonometric functions), sequences and series of functions
and their different types of convergence.

Geometry/Topology: Elementary geometric properties of common shapes and figures in 2 and 3
dimensional Euclidean spaces (e.g. triangles, circles, discs, spheres, etc.). Plane analytic geometry

(= coordinate geometry) and trigonometry. Definition and basic properties of metric spaces, exam-
ples of subset Euclidean spaces (of any dimension), connectedness, compactness. Convergence in
metric spaces, continuity of functions between metric spaces.

General: Pigeon-hole principle (box principle), induction, elementary properties of divisibility,
elementary combinatorics (permutations and combinations, binomial coefficients), elementary rea-
soning with graphs, elementary probability theory.

Sample Questions for Part I

Sample multiple choice questions

1. Let f : R → R be a continuous bounded function. Then

(a) f has to be uniformly continuous
(b) there exists an x ∈ R such that f (x) = x
(c) f can not be increasing
(d) lim f (x) exists.
x→∞

2. Define a function (
x + x2 cos( πx ), if x 6= 0
f (x) =
0, if x = 0
Consider the statements:

I. f is differentiable at x = 0 and f 0 (0) = 1.
II. f is differentiable everywhere and f 0 (x) is continuous at x = 0.
III. f is increasing in a neighbourhood around x = 0.
IV. f is not increasing in any neighbourhood of x = 0.

Which one of the following combinations of the above statements is true.

(a) I. and II.
(b) I. and III.
(c) II. and IV.
(d) I. and IV.

Sample true/false questions

1. If A and B are 3 × 3 matrices and A is invertible, then there exists an integer n such that
A + nB is invertible.

2. Let P be a degree 3 polynomial with complex coefficients such that the constant term is
2010. Then P has a root α with |α| > 10.

3. The symmetric group S5 consisting of permutations on 5 symbols has an element of order 6.

4. Suppose fn (x) is a sequence of continuous functions on the closed interval [0;1] converging
to 0 pointwise. Then the integral
Z 1
fn (x)dx
0
converges to 0.

5. There are n homomorphisms from the group Z/nZ to the additive group of rationals Q.

6. A bounded continuous function on R is uniformly continuous.