# Tata Institute of Fundamental Research || TIFR 2022 MATHS PhD SOLUTIONS 2022

TIFR Exam Paper Pattern

Mathematics:

Selection process for admission in 2021 to the various programs in Mathematics at the TIFR centres – 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.

Stages 1: For the PhD and Integrated PhD programs at the Mumbai Centre, this test will comprise the entirety of Stage I of the evaluation process.

The nation-wide will be an objective test of three hours duration, with 20 multiple choice questions and 20 true/false questions.

Stage 2: The second stage of the selection process varies according to the program and the Centre.

TIFR Exam - Syllabus and Pattern

Syllabus of Mathematics:

Stage 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. Definitions 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, examples 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 reasoning with graphs, elementary probability theory.