# Admission to the Graduate Program 2020

Selection process for admission in 2020 to the various programs in Mathematics at the TIFR centers - 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 multiple 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.

More information and resources concerning the nation-wide test, such as old question papers and details on how to apply, can be found in the web page of the TIFR Graduate School, at this link.

** 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. 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.

Sample questions for Part I is given here