### $$\textbf{Tata Institute of Fundamental Research}\\ \textit{(Deemed to be University)}$$

$\textbf{Admission Details:}$ http://univ.tifr.res.in/gs2022/index.html

$\textbf{Frequency:}\;\text{Once in a year.}$

$$\textbf{School of Technology and Computer Science}$$

Instructions for the written test There are two streams in the School of Technology and Computer Science:

1. Computer Science.
2. Systems Science.

The question paper will have three parts. Part A is common to both the streams. It will test the general mathematical aptitude of the candidate. There is no prescribed sylabus for Part A. Part B will be oriented towards the topics listed under ‘Computer Science’ below; and Part C will be oriented towards topics listed under ‘Systems Science’ below. Only one of Parts B, C, should be attempted. The duration of the written test will be three hours. The test will be of multiple choice type, with negative marking for incorrect answers. The use of calculators will not be allowed during the test.

$$\textbf{Syllabus: Computer Science}$$

1. $\text{Discrete Mathematics:}$ Sets and Relations, Combinatorics (Counting) and Elementary Probability Theory, Graph Theory, Propositional and Predicate Logic.
2. $\text{Formal Languages, Automata Theory and Computability.}$
3. $\text{Data Structures and Algorithms:}$ Arrays, Lists and Trees, Sorting and Searching, Graph algorithms, Complexity of problems and NP-completeness.
4. $\text{Fundamentals of Programming Languages and Compilers:}$ Control structures, Parameter passing mechanisms, Recursion, Parsing and type checking, Memory management.
5. $\text{Operating Systems and Concurrency}$
6. $\text{Switching Theory and Digital Circuits}$
7. $\text{Theory of Databases}$

$\textbf{Previous Year Papers with Solution:}$ https://gatecse.in/tifr-previous-year-papers-with-solution/

$$\textbf{Syllabus: Systems Science}$$

1. $\text{Engineering Mathematics:}$ Complex Analysis, Linear Algebra, Elementary Numerical Analysis, Basic Optimization Theory and Algorithms, Introduction to Probability Theory and Statistics.
2. $\text{Electrical and Computer Sciences:}$ Introduction to Signals and Linear Systems Analysis, Control Systems, Digital Signal Processing, Basic Circuit Theory, Introduction to Digital Communications, Digital Computer Fundamentals, Introduction to Computer Programming.

$$\textbf{GS2022 Selection Process for Mathematics}$$

The selection process for admission in 2022 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 CAM-TIFR, Bengaluru and ICTS-TIFR, Bengaluru – will have two stages.

• $\text{Stage I.}$ A nationwide test will be conducted in various centres on December 12, 2021. Performance in this test will be used to decide whether a student progresses to the second stage of the evaluation process. The cut-off marks for a particular program will be decided by the TIFR centre handling that program. The score in the written test may also be used in Stage II.
• $\text{Stage II.}$ The second stage of the selection process varies according to the program and the centre. More details about this stage will be provided at a later date.

$\text{Previous Year Papers with Solution:}$ https://gatecse.in/tifr-mathematics-previous-year-papers-with-solution/

$$\textbf{Syllabus for Stage I}$$

1. $\text{Algebra:}$ 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.
2. $\text{Analysis:}$ 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.
3. $\text{Geometry/Topology:}$ 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.
4. $\text{General:}$ Pigeon-hole principle (box principle), induction, elementary properties of divisibility, elementary combinatorics (permutations and combinations, binomial coefficients), elementary reasoning with graphs, elementary probability theory.
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