### $$\textbf{Chennai Mathematical Institute (CMI)} \\ \text{MSc/PhD Computer Science/Data Science/Mathematics}$$

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

$\textbf{Admission Details:}$ https://www.cmi.ac.in/admissions/

$$\textbf{Chennai Mathematical Institute} \\ \text{MSc / PhD Computer Science}$$

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

$$\textbf{Syllabus}$$

Topics covered in entrance examination

- $\text{Discrete Mathematics}$
- Sets and relations, elementary counting techiniques, pigeon hole principle, partial orders,

- $\text{Elementary probability theory}$
- $\text{Automata Theory}$
- Regular expressions, non deterministic and deterministic finite automata, subset construction, regular languages, non regularity (pumping lemma), context free grammars, basic ideas about computable and noncomputable functions.

- $\text{Algorithms}$
- O notation, recurrence relations, time complexity of algorithms, sorting and searching (bubble sort, quick sort, merge sort, heap sort).

- $\text{Data structures}$
- Lists, queues, stacks, binary search trees, heaps.

- $\text{Graphs}$
- Basic definitions, trees, bipartite graphs, matchings in bipartite graphs, breadth first search, depth first search, minimum spanning trees, shortest paths.

- $\text{Algorithmic techniques}$
- Dynamic programming, divide and conquer, greedy.

- $\text{Logic}$
- Boolean logic, truth tables, boolean circuits - and, or, not, and, nand gates.

$\textbf{Suggested reading material}$

- Frank Harary: Graph Theory, Narosa.
- John Hopcroft and Jeffrey D Ullman: Introduction to Automata, Languages and Computation, Narosa.
- Jon Kleinberg and Eva Tardos: Algorithm Design, Pearson.
- C. Liu: Elements of Discrete Mathematics, Tata McGraw-Hill.

Reference: Computer Science Syllabus (or) https://www.cmi.ac.in//admissions/syllabus/pgcs-syllabus.pdf

$$\textbf{Chennai Mathematical Institute} \\ \text{ M.Sc. Data Science}$$

$\text{Previous Year Papers with solution}:$ https://gatecse.in/cmi-data-science-previous-year-papers-with-solution/

$$\textbf{Syllabus}$$

The entrance examination will primarily check mathematical aptitude and the ability to logically interpret data. Candidates should be familiar with following topics:

- $\text{School Level Mathematics}$
- Arithmetic and geometric progressions; arithmetic, geometric and harmonic mean; polynomials, matrices (basic operations, inverse, transpose), determinants, solving linear equations, prime numbers and divisibility, GCD, LCM, modular arithmetic, logarithms, basic properties of functions (domain, range, injective, bijective, surjective), elementary calculus (differentiation, maxima-minima, integration and its applications)

- $\text{Discrete Mathematics}$
- Sets and relations, combinations and permutations, elementary counting techniques, pigeonhole principle, binomial theorem, mathematical induction, boolean logic and truth tables

- $\text{Probability Theory}$
- Elementary probability theory, conditional probability, and Bayes theorem; random variables, density functions, distribution functions; standard distributions (Gaussian etc.); expectation and variance; data interpretation; summary statistics

- $\text{Programming}$
- Ability to read and interpret algorithms written in simple pseudocode (variables, conditionals, loops)

$\textbf{Suggested textbooks}$

There are many books that cover this material. The questions asked will only test basic concepts. Here are a few suggestions.

- C.L. Liu: Elements of Discrete Mathematics, McGraw Hill (1986)
- Norman Biggs: Discrete Mathematics, Oxford University Press (2002)
- Sheldon M. Ross: A First Course in Probability (9th ed), Pearson (2013)
- Henk Tijms: Understanding Probability, Cambridge University Press (2012)
- R.G. Dromey: How to Solve it By Computer, Pearson (2006)

Reference: Data Science Syllabus (or) https://www.cmi.ac.in//admissions/syllabus/datascience-syllabus.pdf

$$\textbf{Chennai Mathematical Institute} \\ \text{MSc / PhD Mathematics}$$

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

- $\textbf{Important note}$
- The syllabus includes topics for PhD entrants too and so contains material which may often be found only in MSc courses and not BSc courses in the country. Our policy generally has been to have a common question paper for MSc and PhD levels but have separate cut-offs for them.

$$\textbf{Syllabus}$$

- $\text{Algebra.}$
- Groups, homomorphisms, cosets, Lagrange's Theorem, group actions, Sylow Theorems, symmetric group $\text{S}_{n}$, conjugacy class, rings, ideals, quotient by ideals, maximal and prime ideals, fields, algebraic extensions, finite fields
- Matrices, determinants, vector spaces, linear transformations, span, linear independence, basis, dimension, rank of a matrix, characteristic polynomial, eigenvalues, eigenvectors, upper triangulation, diagonalization, nilpotent matrices, scalar (dot) products, angle, rotations, orthogonal matrices, $\text{G L}_{n}, \text{S L}_{n}, \text{O}_{n}, \text{S O}_{2}, \text{S O}_{3}$.

$\textbf{REFERENCES:}$

- Algebra, M. Artin
- Topics in Algebra, Herstein
- Basic Algebra, Jacobson
- Abstract Algebra, Dummit and Foote

$\text{Complex Analysis.}$

Holomorphic functions, Cauchy-Riemann equations, integration, zeroes of analytic functions, Cauchy formulas, maximum modulus theorem, open mapping theorem, Louville's theorem, poles and singularities, residues and contour integration, conformal maps, Rouche's theorem, Morera's theorem

$\textbf{REFERENCES:}$

- Functions of one complex variable, John Conway
- Complex Analysis, L V Ahlfors
- Complex Analysis, J Bak and D J Newman

- $\text{Calculus and Real Analysis.}$
- Real Line: Limits, continuity, differentiablity, Reimann integration, sequences, series, limsup, liminf, pointwise and uniform convergence, uniform continuity, Taylor expansions,
- Multivariable: Limits, continuity, partial derivatives, chain rule, directional derivatives, total derivative, Jacobian, gradient, line integrals, surface integrals, vector fields, curl, divergence, Stoke's theorem
- General: Metric spaces, Heine Borel theorem, Cauchy sequences, completeness, Weierstrass approximation.

$\textbf{REFERENCES:}$

- Principles of mathematical analysis, Rudin
- Real Analysis, Royden
- Calculus, Apostol

$\text{Topology.}$ Topological spaces, base of open sets, product topology, accumulation points, boundary, continuity, connectedness, path connectedness, compactness, Hausdorff spaces, normal spaces, Urysohn's lemma, Tietze extension, Tychonoff's theorem,

$\textbf{References:}$ Topology, James Munkres

Reference: Mathematics Syllabus (or) https://www.cmi.ac.in//admissions/syllabus/pgmath-syllabus.pdf