Note:- TIFR Exam Pattern change every year, so syllabus can be change.

## GS2018: Selection Process for Mathematics

**Part I.** For all the programs, the first part of the selection process will be the nationwide test conducted in various centers in December 2017. This is an objective test of two hours duration, with 25 true/false and 15 multiple choice questions.

The score in the above test will serve as qualification marks for a student to progress to the second step of the evaluation process. The cut-off marks for the various programs can be different.

**Part II.** The second part of the selection process varies according to the program and the center, and is as follows.

**TIFR Mumbai, Int PhD and PhD.** The cutoffs in Part I will be so adjusted that in all a total of about 100 students will qualify to appear for the Part II test. This will be held at TIFR, Mumbai, on a date to be specified later. The test will consist of a written examination of 3 hours duration, with about 15 questions, and the solution to the questions have to be written with detailed justifications. Partial credit will be given for partial solutions. The question paper will be the same for Int PhD and PhD programs. The final selection will be made based on the performances in Part I and Part II, and the criteria can be different for the two programs.

**TIFR CAM, Bengaluru, Int PhD and PhD.** The cutoff scores in Part I will be so adjusted that about 80 applicants to the Int PhD program, and about 20 applicants to the PhD program, qualify for the Part II test. This will be held at TIFR CAM, Bengaluru, on a date to be specified later. The Part II examination will be a written examination of 3 hours duration with separate question papers for Int PhD and PhD candidates. The exam will involve solving problems at the undergraduate level for Int PhD and at the postgraduate level for PhD. Partial credit will be given for partial solutions. The final selection will be made based on the performances in Part I and Part II.

**ICTS, Bengaluru, PhD.** Based on their performances in Part I, about 20 students will be called for Part II at ICTS, Bengaluru, which will consist of a written examination of one hour duration followed by an interview

## Syllabus for Part I

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

**Algebra:** Definitions and examples of group (finite and infinite, commutative and non-commutative), cyclic groups, subgroups, homomorphisms, quotients. Definitions and example of rings and fields. Basic facts about finite dimensional vector spaces, matrices, determinants, ranks of linear transformations, characteristic and minimal polynomials, symmetric matrices. Integers and their basic properties. Polynomials with real or complex coefficients in one variable.

**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).

**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.