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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 problem-solving 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
Concepts, methods and applications of linear algebra; systems of linear equations, matrices, determinants, vectors in n-space, and eigenvectors.
Real Analysis
The real number system, functions of a real variable; limits, continuity, derivatives. Formal treatment of Riemann integral, Fundamental theorem of calculus, Sequences of functions and the Infinite series will be explored.
Algebra I
This course will introduce you to all the major structures of abstract algebra. Groups and subgroups, homomorphisms. Polynomials. Rings, subrings, and ideals. Integral domains and fields. Roots of polynomials. Maximal ideals.
Multivariable Calculus (Requires Analysis I and Linear Algebra)
Euclidean space, partial differentiation, multiple integrals, line integrals and surface integrals, the integral theorems of vector calculus.
Elementary Differential Geometry
This is an introductory course on the differential geometry of curves and surfaces in three-dimensional space. Topics include space curves, the curvature and orientability of surfaces, the Gauss-Bonnet theorem, and a brief introduction to metric geometry and the Hopf-Rinow theorem.
Probability Theory
Calculus of probability, random variables, expectation, distribution functions, central limit theorem.
Algebra II
Rings of quotients of an integral domain. Euclidean domains, principal ideal domains. Polynomial rings. Field extensions. Ruler and compass constructions. Finite fields with applications.
Complex Analysis
The theory of functions of a complex variable; topics include Cauchy's theorem, the residue theorem, the maximum modulus theorem, Laurent series, the fundamental theorem of algebra.
Mathematical Modeling (including Differential Equations)
Topics include : 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 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.
Metric Spaces
Introduction to Mathematical Thinking
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.
Calculus
For four centuries Calculus has been at the foundation of mathematics and physics, and more recently of other quantitative sciences.
It is concerned with problems of finding areas and volumes of bodies like spheres, cylinders, cones and frustums; and the laws governing change----of positions of planets, of particles of gases, of prices of commodities, of populations of species. A proper understanding of the laws of probability, optimization and several other areas needs a thorough grasp of calculus.
This course will expose students to the fundamentals of calculus, and at the same time to rigorous mathematical thinking. At the heart of the subject are the concepts of infinity and infinitesimals and one has to learn how to handle them carefully and correctly.
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.
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.
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.
Chaos in Topological Dynamics
Can a butterfly flapping its wings too fast in jungles of Amazonia cause massive flooding in Chennai? Dynamical Systems models various physical, natural phenomenon and attempts to predict its long range asymptotic behaviour. Dynamical Systems is a very broad field which enjoys attention from physicists, engineers, biologists and mathematicians. In mathematics, Dynamical Systems involves a broad range of topics including differential equations, ergodic theory, measure theory, topology, descriptive set theory etc. In this course, we will focus on topological aspects of dynamical systems. We will begin the course by some topology necessary to study topological dynamics. Our focus will be on chaotic topological dynamics. By the end of the course, we will try to make precise the first sentence of this description. Real Analysis is a prerequisite for this course.
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.
Batch entering in 2018 and taking Calculus in Sem 1 |
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Sep-18 |
Jan-19 |
Sep-19 |
Jan-20 |
Sep-20 |
Jan-21 |
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Sem 1 |
Sem 2 |
Sem 3 |
Sem 4 |
Sem 5 |
Sem 6 |
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FC -- Quant and Mathematical Thinking |
Multivariable Calculus |
Analysis |
Complex Analysis |
Diff Eqns& Math. Modeling |
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Calculus |
Linear Algebra |
Algebra 1 |
Algebra 2 |
Metric Spaces |
Elem. Differential Geometry |
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Probability |
Math Elective |
Math Elective |
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Batch entering in 2018 and starting Math major in Sem 2 |
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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 |
Analysis |
Complex Analysis |
Diff Eqns& Math. Modeling |
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Linear Algebra |
Algebra 1 |
Algebra 2 |
Metric Spaces |
Elem. Differential Geometry |
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Probability |
Math Elective |
Math Elective |
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Batch 5: entering in 2018 and starting Math major in Sem 3 -- not recommended but possible: |
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Sep-18 |
Jan-19 |
Sep-19 |
Jan-20 |
Sep-20 |
Jan-21 |
||
Sem 1 |
Sem 2 |
Sem 3 |
Sem 4 |
Sem 5 |
Sem 6 |
||
Calculus |
Multivar Calculus |
Complex Analysis |
Diff Eqns& Math. Modeling |
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FC -- Quant and Mathematical Thinking |
Analysis |
Metric Spaces |
Elem. Differential Geometry |
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Algebra 1 |
Algebra 2 |
Algebra 1 (if not taken in Sem 3) |
Algebra 2 (if not taken in Sem 4) |
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Probability |
Linear Algebra |
Math Elective |
Math Elective |
Courses offered for Spring 2019
Rajendra Bhatia : Calculus 2 (same as the course multivariable calculus)
Krishna Maddaly : Quantitative reasoning mathematical thinking
Elementary differential geometry
Maya Saran : Calculus 1
Introduction to proofs
Jean-Marc Deshouillers : Quantitative reasoning mathematical thinking
Markov chain and finite automata
I. B. Passi : Algebra 2
Kumarjit Saha : Linear algebra
Differential equations and modelling
R. B. Bapat : Statistical Inference (Compulsory for interdisciplinary Maths-CS majors)
Dipti Dubey : Linear programming (Compulsory for interdisciplinary Maths-CS majors)