- [Cr:3, Lc:2, Tt:1, Lb:0]

- Basic operations, row reduction, determinant and trace, Cramer’s rule.
- Basic notions of groups, permutation groups.
- Group actions, prime-power groups.
- Finite Abelian groups.
- Matrix groups, SO(3) and rotations.
- Groups of symmetries of real plane and Platonic solids.
- Vector spaces and linear transformations, eigen values and eigen vectors, diagonalization.

- M. A. Armstrong, Groups and Symmetry, Springer (1988).
- M. Artin, Algebra, Prentice-Hall of India, New Delhi (1994).
- I. S. Luthar and I. B. S. Passi, Algebra Vol. I, Narosa Publishing House, New Delhi (1996).
- J. A. Gallian, Contemporary Abstract Algebra, D. C. Heath Canada (1986).

- [Cr:3, Lc:2, Tt:1, Lb:0]

- The real number system, completeness axiom, complex numbers.
- Sequences, limits, convergence, series.
- Polynomials, rational functions, continuous functions.
- Trigonometric, exponential, logarithmic and hyperbolic functions.
- Differentiation, mean value theorem, Taylor’s theorem.
- Uniform convergence, power series.
- Riemann integral, fundamental theorem of calculus.
- Fourier series.

- R. R. Goldberg, Methods of Real Analysis, Wiley (1970).
- K. A. Ross, Elementary Analysis, The Theory of Calculus, Springer (2004).
- T. M. Apostol, Calculus, Blaisdell Publishing Company, 1961.
- S. Shirali and H. L. Vasudeva, Mathematical Analysis, Alpha Science International Ltd. (2006).

- [Cr:3, Lc:2, Tt:1, Lb:0]

- Differentiation of vectors.
- Curves in the plane and in space, arc length, reparametrization.
- Curvature, torsion, Serret-Frenet formulae.
- Fundamental theorem of curves in plane and space.
- Surfaces in three dimension (2-manifolds), smooth surfaces.
- Tangents, normals, quadratic surfaces.
- Change of variable formula, surfaces of revolution.
- First and second fundamental forms, isometries, conformal mappings.
- Normal and principal curvatures, Gaussian curvature and the Gauss map.
- Geodesics, geodesic curvature, Gauss’ theorema egregium.

Additional Topics

- Isoperimetric inequality, four vertex theorem.
- Area and volume integrals, surface area.

- L. Brand, Vector Analysis, Dover Publications (2006).
- A. Pressley, Elementary Differential Geometry, SUMS, Springer (2001).

- [Cr:3, Lc:2, Tt:1, Lb:0]

- Recapitulation: Counting (urn, coins, cards).
- Axiomatic approach to probability, conditional probability, independence of events.
- Discrete random variables, probability mass function, some standard discrete distributions and examples.
- Continuous random variables, probability density function, some standard continuous distributions and examples.
- Bivariate distributions (discrete and continuous), marginal and conditional distributions, covariance, correlation coefficient.
- Moments, Markov’s inequality, Chebychev’s inequality.
- Sums of independent random variables, law of large numbers, central limit theorem
- A glimpse into estimation theory (maximum likelihood estimation, method of moments) and testing of hypothesis.

- K. L. Chung and F. AitSahila, Elementary Probability Theory, Springer (2004).
- R. Isaac, The Pleasures of Probability, Springer (Undergraduate Texts in Mathematics) (1995).
- S. Ross, A First Course in Probability, Pearson Education Inc. (2006).