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Bachelor Degree Of Science In Mathematics, Bsc in Mathematics | India
Mathematics Programme

Some of humankind’s most powerful and beautiful ideas lie in the field of Mathematics. The wide applicability of these ideas and their deep connection with the natural sciences have made this discipline one of the most fruitful arenas of human inquiry. Combining as it does the greatest creative freedom with the most stringent standards of rigour, Mathematics also happens to be the ideal training ground for learning a broad range analytical and problem-solving skills.

Ashoka University’s programme of study for the Major in Mathematics has been designed to meet two primary goals:

1. Students should be broadly exposed to the primary areas and the central ideas of contemporary Mathematics (as well as their applications); and

2. Students should develop the skills of rigorous analytic reasoning along with a set of useful problem solving skills and strategies.

In addition to the two relevant Foundation Courses (Introduction to Mathematical Thinking, and Critical Thinking in Mathematics), students majoring in Mathematics will take a series of 9 required courses and 3 elective courses within Mathematics. The required courses cover the fields of Analysis, Algebra, Linear Algebra, Differential Equations, Probability and Statistics and Mathematical Modeling. Additional further options are provided by the elective courses.

While elective course offerings will vary semester to semester, all students will have the option of pursuing independent study under the guidance of a faculty member, in which they can study a topic in depth or attempt a research problem.

At the end of their program of study we expect students to be able to read and understand mathematical proofs, learn and apply new mathematical concepts, and construct and communicate a correct and rigorous argument of their own. Students completing this programme will be well prepared to pursue Mathematics further if they wish to, or to take up positions that call for innovative problem solving in concert with strong analytical abilities.

Course Description

The math courses offered at Ashoka include a Foundation Course required for all students, a Critical Thinking Seminar, open to all students, and a sequence of courses that comprise the math major programme.  Note that the Critical Thinking Seminar counts as one of the 12 foundation courses a student must take at Ashoka (as do all CTS courses).

Required Math Major Courses (for students joined in 2017 or before)
• Linear Algebra

• Real Analysis

• Algebra I

• Multivariable Calculus  (Requires Analysis I and Linear Algebra).

• Elementary Differential Geometry

• Probability Theory

• Algebra II

• Complex Analysis

• Mathematical Modeling (including Differential Equations)

• Metric Spaces

Foundation Courses

Introduction to Mathematical Thinking and Quantitative Reasoning

This course aims to give students an experience of contemporary Mathematics.You will see that Mathematics is driven by ideas, not by calculations, it is both beautiful and powerful, and it combines precision with the greatest creativity. En route, you will develop a set of broadly useful problem-solving skills, gain experience in precise thinking and writing, and encounter some of History’s landmark ideas.

Critical Thinking Seminar

Introduction to Proofs

This course aims to make you conversant with the language of mathematics. This means being able to read and write proofs, which are simply careful expressions of reasoning. You will learn how to do so while learning actual mathematics, of course. Topics will be determined by the instructor.

Major Requirements (for students joined in 2018 and afterwards))

Course structure is under revision and might be changed.

Calculus

Syllabus: Number systems. Sequences and series. Functions of a real variable. Graphs of functions. Limits and continuity. Differentiation. Mean value theorem. L’Hospital rule. Maclaurin and Taylor series. Curve tracing. Riemann integral. Definite and indefinite integrals. Fundamental theorem of calculus. Applications of differential and integral calculus in areas such as optimisation and mechanics.

Suggested texts:

1. James Stewart: Calculus, Cenage Publishers, 2012.
2. Kenneth A. Ross: Elementary Analysis, The Theory of Calculus, Second Edition, Undergraduate Texts in Mathematics, Springer, 2013.
3. G. B. Thomas and R. L. Finney: Calculus and Analytic Geometry, Second Edition, Addison-Wesley Publishing, 1998.

"The Calculus course is a prerequisite for further mathematics courses, and students thinking of studying mathematics---and subjects that use mathematics intensively--- are advised to take it as early as possible."

Linear Algebra

Syllabus: Real vector spaces, subspaces, spanning sets, basis sets, dimension of a vector space.  Solution of a system of linear equations. Row space and column space of a matrix, rank of a matrix, elementary row and column operations of a matrix. Inversion of square matrices, rank factorization of a matrix. Properties of determinant. Linear transformations, range and null space of a linear transformation, rank-nullity theorem. Matrix representation of a linear transformation. Inner product spaces, normed linear spaces, examples of different normed linear spaces, orthonormal basis sets. Eigenvalues, eigenvectors, characteristic polynomials. Spectral theorem for real symmetric matrices. Singular value decompositions.

Suggested texts:

1. A. R. Rao and P. Bhimsankaram: Linear algebra, Hindusthan book agency, 2000.
2. Friedberg, Insel and Spence: Linear algebra, Pearson, 2015.
3. David C Lay: Linear algebra and its applications, Pearson, 2014.

Multivariable Calculus

Syllabus: Review of vectors and matrices. Curves and surfaces. Partial derivatives. Maximum and minimum values. Double integrals. Line integrals in the plane. Green’s theorem. Triple integrals and surface integrals in 3-space. Stoke’s theorem. Applications of multivariable calculus.

Prerequisites: Calculus.

Suggested texts:

1. James Stewart: Calculus, Cenage Publishers, 2012.
2. Marsden and Tromba: Vector Calculus, W. H. Freeman, 2003.

Real Analysis

Syllabus: Real and complex number systems. Limits of sequences. Monotonic sequences. Limits superior and limits inferior. Convergence of a series. Absolute and conditional convergence. Power series over real and complex numbers and their radius of convergence. Bolzano-Weirstrass Theorem, Cantor and Heine-Borel Theorems. Point-wise and uniform continuity. Sequences and series of functions. Pointwise and uniform convergence of sequence of functions. Integrals and derivatives of sequences and series of functions. Elementary transcendental functions. Improper integrals, Riemann-Stieiltjes integral. Idea of Lebesgue integral, Weierstrass approximation Theorem.

Prerequisites: Calculus, Linear Algebra.

Desirable: Multivariable Calculus.

Suggested texts:

1. Tom M. Apostol: Mathematical Analysis, Second Edition, Addison    Wesley Publishing Company, 1974.
2. T. Tao: Analysis I, Hindustan Book Agency, 2017.
3. T. Tao: Analysis II, Hindustan Book Agency, 2017.

Algebra 1

Syllabus

Groups: Symmetries of the plane. Groups. The symmetric group S_n. Homomorphisms of groups. Subgroups. Lagrange’s theorem. Conjugacy. Normal subgroups, Quotient groups. Homomorphism theorems. Group actions.

Rings: Rings. Subrings. Ideals. Polynomials.  Integral domains and fields. Roots of polynomials. Symmetric polynomials. Factorization in integral domains. Euclideal and principal ideal domains. Maximal ideals. Prime ideals.

Suggested texts:

1. Michael Artin: Algebra, Second Edition, Pearson Prentice-Hall of India, New Delhi, 2011.
2. David S. Dummit and Richard M. Foote: Abstract Algebra, Third Edition, Wiley, 2005.
3. Joseph A. Gallian: Contemporary Abstract Algebra, Eighth Edition, BROOKS/COLE Cengage Learning, 2013
4. S. Lang: Algebra, Revised Third Edition, GTM 211, Springer, 2002.
5. I. S. Luthar and I. B. S. Passi: Algebra Vols I & 2, Narosa, 1996, 1999.

Probability Theory

Syllabus: Frequency and axiomatic  definition of probability, random experiments with equally likely finite outcomes, Inclusion exclusion principle. General finite sample spaces, infinite sample spaces.  Concept of probability spaces and construction of probability measures. Conditional probability,  Bayes theorem, Independence of events. Random variable (discrete), probability mass function and distribution function. Examples: Bernoulli, Binomial, Poisson, Geometric distributions. Expectation and variance of a random variable, sum law and product law of expectation, moment generating functions. Random vector (discrete), joint distribution, Marginal distributions, joint moment generating functions, covariance, Multinomial distributions. Continuous random variables, density functions, distribution functions, expectation, variance, moment generating function, example: uniform, normal, exponential. Continuous random vector, joint density function, joint distribution function, conditional density,  conditional expectation, conditional variance, example: multivariate normal. Inequalities: Markov, Chebyshev. Different notions of convergence of random variables. Central limit theorem and Strong law of large number (without proof).  Idea of theory of estimations, minimum variance unbiased estimation. Basic testing of hypothesis, Neyman Pearson lemma

Suggested texts:

1. Sheldon M Ross: First Course in Probability, Pearson.
2. V. K. Rohatgi, E. S. Saleh: An Introduction to Probability and Statistics, Wiley-Blackwell, 3rdedition, 2015.
3.  J. L. Devore: Probability and Statistics for Engineering, Cenage, 8th edition, 2012.
4. W. Feller: An Introduction to probability theory and applications (Vol I), Wiley, 2014.

Algebra 2

Syllabus:

Groups: Direct and semi-direct products. Finite abelian groups. Sylow theorems.  Series of subgroups, Solvable groups. Nilpotent groups. Free groups. Generators and relations.

Fields: Algebraic extensions, Splitting fields, algebraic closures and normal extensions.  Roots of unity and finite fields. separable and inseparable extensions. The Galois group and the fundamental theorem. Solubility of equations by radicals

Prerequisite: Algebra 1

Suggested texts:

1. Michael Artin: Algebra, Second Edition, Pearson Prentice-Hall of India, New Delhi, 2011.
2. David S. Dummit and Richard M. Foote: Abstract Algebra, Third Edition, Wiley, 2005.
3. Joseph A. Gallian: Contemporary Abstract Algebra, Eighth Edition, BROOKS/COLE Cengage Learning, 2013.
4. S. Lang: Algebra, Revised Third Edition, GTM 211, Springer, 2002.
5. I. S. Luthar and I. B. S. Passi: Algebra Vols. 1 & 4. Narosa, 1996, 2004.

Complex Analysis

Syllabus: The algebra and geometry of complex numbers, representations of a complex number. Exponential and logarithm functions. Diﬀerentiation, analytic functions, Cauchy-Riemann equations. Contour integrals, Independence of path. Cauchy’s Integral Theorem, Cauchy’s Integral Formula Liouville’s Theorem and its applications. Complex power series, uniform convergence. Removable and isolated singularities, Taylor’s and Laurent’s Theorems The residue theorem and applications.

Prerequisites: Calculus, Multivariable calculus, Real analysis, Linear algebra.

Suggested texts:

1. Ahlfors: Complex Analysis, Mcgraw hill, 1979.
2. Conway: Functions of one complex variable, Springer international students edition.
3. T. W. Gamelin: Complex Analysis, Springer, 2003.

Diﬀerential Equations and Mathematical Modeling

Syllabus: Differential equation associated to real life problems, First order differential equation on R of the form y’(x) = f(x,y(x)), Equivalent integral equation, Existence of approximate solutions of equation upto error $\epsilon$ by Cauchy-Euler method, Existence and uniqueness of solutions when $f$ is Lipshitz continuous in the second variable, Necessary conditions for f(x,y) to be Lipshitz continuous in y,Picard’s method of solutions of equation, Higher order differential equations, Vector valued ordinar differential equations, Reformulation of higher order differential equations as first order vector valued differential equations, Linear vector valued first order differential equation,  Y’(x) = A Y(x) + C(x) — Homogeneous case, C =0, Characteristic values, characteristic vectors of square matrices, Solution when A is independent of x, Linear independence of solutions associated to characteristic values, General solution of the inhomogeneous equation, Peano’s approximation method for existence of solution.

Prerequisites: Calculus, Multivariable calculus, Real analysis, Metric and topological spaces, Linear algebra.

Suggested texts:

1. Earl A. Coddington: An Introduction to ordinary diﬀerential equations, Prentice Hall India, 1968
2. V. I. Arnold: Ordinary Diﬀerential Equations, MIT Press.

Metric and topological spaces

Syllabus:  Metric spaces, open and closed sets. Euclidean spaces, normed linear spaces, examples of different normed linear spaces, sequence spaces. Completeness, Baire category Theorem. Compactness, characterization of compact spaces. Product spaces, Tychonoﬀ’s theorem. Continuous functions, equicontinuous families, Arzela-Ascolli Theorem.  Connectedness, path connectedness. Functions of several variables, Inverse function theorem, Implicit function theorem.

Prerequisites: Calculus, Multivariable Calculus, Real Analysis, Linear algebra.

Suggested texts:

1. M. O. Searcoid: Metric Spaces, Springer, 2007.
2. S. Shirali and H. L. Vasudeva:  Metric Spaces, Springer, 2006.

Elementary Diﬀerential Geometry

Syllabus: Space curves, Curvature and orientability of surfaces, Gauss-Bonnet theorem, Brief introduction to metric geometry Hopf-Rinow theorem.

Prerequisite: Calculus, Multivariable Calculus, Real Analysis, Metric and topologicalspaces.

Suggested texts:

1. Man-fredo P. do Carmo: Diﬀerential Geometry of Curves and Surfaces, Prentice-Hall, (1976).
2. Pressley: Elementary Diﬀerential Geometry, Springer, 2010.
Minor requirements (for students joined in 2017 or before))

The following courses are required:
• CTS in Mathematics (Introduction to Proofs);
• Linear Algebra;
• Real Analysis;
• Algebra 1
• Probability theory
• Any other math course.

Note that the CTS will count both as a foundation course as well as towards the minor and  students taking a course in probability as part of their major may replace the Probability theory course with another course of their choice.

Minor Requirements (for students joined in 2018 or after)
• Calculus
• Linear Algebra
• Algebra I
• Probability Theory
• Multivariable calculus
• Real Analysis

Concentration requirements (for students joined in 2017 or before)

CTS in Mathematics (Introduction to Proofs)

The above course is a required course and any three of the following four courses can be taken.

• Linear Algebra;
• Real Analysis;
• Algebra 1
• Probability theory (students taking a course in probability as part of their major may replace the Probability theory course with another course of their choice)
Concentration Requirements (for students joined in 2018 or after)
• Linear Algebra
• Calculus 1
• Algebra 1
• One more course of your choice (which should not be probability if you are taking a probability course offered by another department.)
ASP diploma rules for an advanced major in Mathematics

Students pursuing an advanced major in Mathematics have to take a minimum of four 400 level maths courses in one year. List of the 400 level courses to be offered may vary from year to year. Additionally, a student may choose to do the 4th-year project under the supervision of some maths faculty. The workload for the project should be equivalent to a 4 credit course. At the end of the project, the student needs to give a presentation on his/her project in front of a three-member committee for evaluation. In monsoon 2019 semester, the following 400 level courses will be offered:

1. Topology
2. Measure theory and integration

Elective Courses

These courses will vary year to year. In the Monsoon 2016 and spring 2017 semester, the electives  offered were:

Statistical Inference
Elective course for 3rd year students. This course will pick up where Probability Theory left off, taking you into statistical inference — the process of drawing conclusions from data.

Fourier Series

Review of Riemann integral and Riemann integrable functions,  Definition of Fourier Series, Fourier Coefficients of functions in R [-pi, pi] - properties, Definition of approximate identities, Abel and Cesaro Convergence, Poisson Kernel, Dirichlet Kernel, Fej\'er Kernel, Convolutions of Riemann Integrable functions on [-pi, pi], Convergence of convolutions with approximate identities, Pointwise convergence of Fourier Series, Pre-Hilbert space structure on L^2( R ), Orthonormality of trigonometric polynomials, Uniform approximation of continuous functions by polynomials,  Mean square convergence of Fourier Series,Parseval, Plancheral relations, Divergence of Fourier series,  Applicaitons — Isoperimetric Inequality, Weyl's Equidistribution Theorem, Nowhere differentiable Continuous functions — Fourier series of functions of any period L, The Wave equation in one dimension, solutions via Fourier Series, The Fast Fourier Transform.

Sample Curriculum Structure

 Batch entering in 2018 and taking Calculus  in Sem 1 Sep-18 Jan-19 Sep-19 Jan-20 Sep-20 Jan-21 Sem 1 Sem 2 Sem 3 Sem 4 Sem 5 Sem 6 FC -- Quant and Mathematical Thinking Multivariable Calculus (If offered) Real Analysis Complex Analysis Diff Eqns& Math. Modeling Calculus Linear Algebra Algebra 1 Algebra 2 Metric and Topological Spaces Elem. Differential Geometry Probability Math Elective Math Elective Batch entering in 2018 and starting Math major in Sem 2 Sep-18 Jan-19 Sep-19 Jan-20 Sep-20 Jan-21 Sem 1 Sem 2 Sem 3 Sem 4 Sem 5 Sem 6 FC -- Quant and Mathematical Thinking Calculus Multivar Calculus Complex Analysis Diff Eqns& Math. Modeling Linear Algebra Algebra 1 Algebra 2 Metric and Topological Spaces Elem. Differential Geometry Probability Math Elective Math Elective Real Analysis

Courses Offered - Monsoon 2019

Rajendra Bhatia  : Real Analysis

Linear algebra and matrix analysis

Probability theory

Maya Saran         : Quantitative reasoning mathematical thinking

Metric spaces

Kumarjit Saha      :  Introduction to Statistical thinking

Multivariable calculus

Pritam Ghosh       :  Complex analysis

Topology

I. B. Passi             : Algebra 1

T. S. S. R. Rao     : Measure theory

Calculus

Faculty