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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 problemsolving skills.
Ashoka University’s programme of study for the Major in Mathematics has been designed to meet two primary goals:
Students should be broadly exposed to the primary areas and the central ideas of contemporary Mathematics (as well as their applications); and
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.
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).
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
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 problemsolving skills, gain experience in precise thinking and writing, and encounter some of History’s landmark ideas.
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.
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:
"The Calculus course is a prerequisite for further mathematics courses, and students thinking of studying mathematicsand 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, ranknullity 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.
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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 3space. Stoke’s theorem. Applications of multivariable calculus.
Prerequisites: Calculus.
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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. BolzanoWeirstrass Theorem, Cantor and HeineBorel Theorems. Pointwise 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, RiemannStieiltjes integral. Idea of Lebesgue integral, Weierstrass approximation Theorem.
Prerequisites: Calculus, Linear Algebra.
Desirable: Multivariable Calculus.
Suggested texts:
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.
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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
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Algebra 2
Syllabus:
Groups: Direct and semidirect 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
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Complex Analysis
Syllabus: The algebra and geometry of complex numbers, representations of a complex number. Exponential and logarithm functions. Diﬀerentiation, analytic functions, CauchyRiemann 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.
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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 CauchyEuler 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.
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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, ArzelaAscolli Theorem. Connectedness, path connectedness. Functions of several variables, Inverse function theorem, Implicit function theorem.
Prerequisites: Calculus, Multivariable Calculus, Real Analysis, Linear algebra.
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Elementary Diﬀerential Geometry
Syllabus: Space curves, Curvature and orientability of surfaces, GaussBonnet theorem, Brief introduction to metric geometry HopfRinow theorem.
Prerequisite: Calculus, Multivariable Calculus, Real Analysis, Metric and topologicalspaces.
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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.
CTS in Mathematics (Introduction to Proofs)
The above course is a required course and any three of the following four courses can be taken.
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 4thyear 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 threemember committee for evaluation. In monsoon 2019 semester, the following 400 level courses will be offered:
1. Topology
2. Measure theory and integration
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, PreHilbert 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.
Batch entering in 2018 and taking Calculus in Sem 1 

Sep18 
Jan19 
Sep19 
Jan20 
Sep20 
Jan21 

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 

Sep18 
Jan19 
Sep19 
Jan20 
Sep20 
Jan21 

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 
Rajendra Bhatia : Real Analysis
Linear algebra and matrix analysis
Krishna Maddaly : Calculus
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