Listening to Mathematics

The BBC's "In Our Time" radio program has had many fascinating episodes on mathematics and mathematicians. Here are some examples that you can access online:

Archimedes Calculus Cryptography Game Theory Godel's Incompleteness Theorems Imaginary Numbers Indian Mathematics Infinity Mathematics Negative Numbers

Pi Prime Numbers Probability Pythagoras Renaissance Maths Symmetry The Fibonacci Sequence The Observatory at Jaipur The Poincare Conjecture Zero

MAT240 Algebra-1. First Assignment.

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CCC 101 - Mesopotamia & Indus Valley

Notes for Lecture 3 and parts of Lecture 4: Mesopotamia and Indus Civilizations.
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CCC 101 - Course Intro & Ancient Egypt

• Lecture 1 - Course Introduction
• Lecture 2 - Ancient Egypt

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Email list for CCC "Maths in India"

This form is for creating an email list of the students enrolled in this course. Filling it will enable the instructor to send you updates, notes, instructions ...

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The SNU Maths Library

The SNU Library now has a very nice collection of Maths books. Not just the usual textbooks, but books that make the subject accessible, and enjoyable, to a wide audience. Here are some that will show you new aspects of Mathematics, or explain things more lucidly than you might have seen before, or simply entertain. Clicking on the covers will take you to readers' reviews on Amazon.

Writing Maths Online

We have just integrated MathJax into this blog. This allows the use of TeX to represent mathematics. For example, we can display the Laplace transform
\[ \mathcal{L}(f) = \int_0^\infty f(t) e^{-st}\,dt \]
by typing \(\mbox{\[ \mathcal{L}(f)  =  \int_0^\infty f(t) e^{-st}\,dt \]}\)

MAT 684 - Statistics I

Syllabus for MAT 684 – Statistics I

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 201 (Probability & Statistics) or equivalent

Overview: This course builds on a standard undergraduate probability and statistics course in two ways. First, it makes probability more rigourous by using the concept of measure. Second, it discusses more advanced topics such as multivariate regression, ANOVA and Markov Chains.

Detailed Syllabus:

1. Probability: Axiomatic approach, conditional probability and independent events
2. Random Variables – Discrete and continuous. Expectation, moments, moment generating function
3. Joint distributions, transformations, multivariate normal distribution
4. Convergence theorems: convergence in probability, Weak law of numbers, Borel- Cantelli lemmas, Strong law of large numbers, Central Limit Theorem
5. Random Sampling & Estimators: Point Estimation, maximum likelihood, sampling distributions
6. Hypothesis Testing
7. Linear Regression, Multivariate Regression
8. ANOVA
9. Introduction to Markov Chains

References:

• Statistical Inference by Casella and Berger. Brooks/Cole, 2007. (India Edition)
• An Intermediate Course in Probability by Allan Gut. Springer, 1995.
• Probability: A Graduate Course by Allan Gut. Springer India.
• Measure, Integral and Probability by Capinski and Kopp. 2^nd edition, Springer, 2007.

MAT 660 - Linear Algebra

Syllabus for MAT 660 - Linear Algebra

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 240 and 260, or an undergraduate algebra course with basics of groups and fields.

Overview: The theory of vector spaces is an indispensible tool for Mathematics, Physics, Economics and many other subjects. This course aims at providing a basic understanding and some immediate applications of the language of vector spaces and morphisms among such spaces.

Detailed Syllabus:

1. Familiarity with sets: Finite and infinite sets; cardinality; Schroeder-Bernstein Theorem; statements of various versions of Axiom of Choice.
2. Vector spaces: Fields; vector spaces; subspaces; linear independence; bases and dimension; existence of basis; direct sums; quotients.
3. Linear Transformations: Linear transformations; null spaces; matrix representations of linear transformations; composition; invertibility and isomorphisms; change of co-ordinates; dual spaces.
4. Systems of linear equations: Elementary matrix operations and systems of linear equations.
5. Determinants: Definition, existence, properties, characterization.
6. Diagonalization: Eigenvalues and eigenvectors; diagonalizability; invariant subspaces; Cayley-Hamilton Theorem.
7. Canonical Forms: The Jordan canonical form; minimal polynomial; rational canonical form.

Main References:

• Friedberg, Insel and Spence: Linear Algebra, 4^th edition, Prentice Hall India
• Hoffman and Kunze: Linear Algebra, 2^nd edition, Prentice Hall India

Other references:

• Paul Halmos: Finite Dimensional Vector Spaces, Springer India
• Sheldon Axler: Linear Algebra Done Right, 2^nd edition, Springer International Edition
• S. Kumaresan: Linear Algebra: A Geometric Approach, Prentice Hall India

MAT 622 - Topology

Syllabus for MAT 622 - Topology

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 220 (Real Analysis) or equivalent

Overview: This course concerns 'General Topology' which can be characterized as the abstract framework in which the notion of continuity can be framed and studied. Thus topology provides the basic language and structure for a large part of pure and applied mathematics.

We will take up the following topics: Open and closed sets, continuous functions, subspaces, product and quotient topologies, connected and path connected spaces, compact and locally compact spaces, Baire category theorem, separability axioms.

Detailed Syllabus:

1. Review: Operations with infinite collections of sets, axiom of choice, Zorn's lemma, real line, metric spaces.
2. Topological Spaces: Definition and examples of topological spaces, Hausdorff property, fine and coarse topologies, subspace topology, closed sets, continuous functions, homeomorphisms, pasting lemma, product topology, quotient topology.
3. Connectedness and Compactness: Connected spaces and subsets, path connectedness, compact spaces and subsets, tube lemma, Tychonoff theorem, local compactness, one-point compactification, Baire category theorem.
4. Separation Axioms: First and second countability, separability, separation axioms (T₁ etc.), normal spaces, Urysohn lemma, Tietze extension theorem.
5. Topics for Student Presentations: Order topology, quotients of the square, locally (path) connected spaces, sequential and limit point compactness, topological groups, nets, applications of Baire category theorem.

Main Reference:

• Topology by James R. Munkres, 2nd Edition. Pearson Education, Indian Reprint, 2001.

Other References:

• Basic Topology by M. A. Armstrong. Springer-Verlag, Indian Reprint, 2004.
• Topology by K. Jänich. Undergraduate Texts in Mathematics, Springer-Verlag, 1984.
• Introduction to Topology and Modern Analysis by G. F. Simmons. International Student Edition. McGraw-Hill, Singapore, 1963.
• Topology of Metric Spaces by S. Kumaresan. 2^nd edition, Narosa, 2011.

MAT 601 - Mathematical Computing

In the next few posts, we will put up the Master's level courses being taught by the SNU maths department.

Syllabus for MAT 601 – Mathematical Computing

Credits(Lec:Tut:Lab): 1:0:1 (1 lecture and 2 lab hours weekly)

Prerequisites:

Overview: In this course we introduce MATLAB as a platform for scientific computation and simulations; and follow with a brief introduction to C++ as a formal programming language. We also demonstrate how MATLAB and C++ can be integrated to build powerful applications. The course complements other graduate courses like Linear Algebra, Numerical Analysis and Optimization.

Detailed Syllabus:

1. MATLAB:

• Arithmetic expressions, assignment, input and output, Boolean expressions, conditional statements.
• For loop, while loop, nested loops, nested conditionals, vectors, elementary graphics, color schemes.
• Elementary mathematical functions, functions with multiple input parameters, graphics functions.
• Two dimensional arrays, contour plotting, sorting, searching, cell arrays, cell arrays of matrices, functions as parameters, structures.
• Working with image files, acoustic file processing, recursive functions, solving linear programming problems.

2. C++ Programming:

• Fundamental data types, operators, control structures, user defined functions
• Arrays, pointers, function pointers, multi- dimensional arrays
• Classes, constructors & destructors, bitwise operators
• Integrating C++ with MATLAB – calling MATLAB functions within a C++ program.

Main References:

1. Programming in Matlab for Engineers by Stephen J. Chapman. Cengage, 2011.
2. A Guide to Matlab by Brian R. Hunt, Ronald L. Lipsman and others. 2^nd edition, Cambridge, 2011.
3. Insight Through Computing: A MATLAB introduction to Computational Science and Engineering by Charles F. Van Loan and K. Y. Daisy Fan. SIAM, 2009.
4. Introducing C++ for Scientists, Engineers and Mathematicians by D. M. Capper. Springer India, 2001.

Other references:

1. Mastering Matlab 7 by Duane C. Hanselman and Bruce L. Littlefield. Pearson Education, 2005.
2. C++ Programming Language by Bjarne Stroustrup. 3^rd edition, Pearson, 2002.
3. Object Oriented Programming in C++ by R. Lafore. 3^rd edition, Galgotia, 2006.

Ramanujan Mathematical Society Conference

SNU is hosting the 27^th Annual Conference of the Ramanujan Mathematical Society during October 20-23, 2012. The conference venue is the Radisson Blu hotel in Paschim Vihar, Delhi. The main academic programme - twelve plenary talks and five symposia (about 40 invited talks in all!) - is already set. There will also be several sessions of contributed 10-minute talks.

Details of how to register to participate, and apply to give a short talk, will soon be up on the conference website.