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Course Co-ordinated by IIT Madras
Coordinators
 
Prof. Sankaran Vishwanath
Institute of Mathematical Sciences

 

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This course is an introduction to the ideas and methods of mathematics.The only prerequisite is some familiarity with topics in a typical high school mathematics curriculum. We will revisit many of these from a more conceptual viewpoint and through numerous examples. Special emphasis will be laid on the interconnections between seemingly disjoint topics. This course seeks to go beyond the “procedures-to-solve-routine-problems” approach of a typical school curriculum to offer a glimpse of what mathematics is really about. It should be suitable for high school students, lower undergraduates, teachers at various levels, or others with a keen interest in mathematics.

◦     Polynomials: Interpolation, Taylor's formula, Polynomials with integer values, Polynomials in several variables, Counting monomials.

◦     Counting Principles: Basic methods, the Pigeonhole principle, the Binomial theorem, Permutations, Graphs, Recurrence relations, Bijective proofs.

◦     Functions: Continuous functions, the Intermediate value property and its applications, Fixed points, Linear transformations of the plane, the Derivative and dilations, Higher order derivatives and the binomial theorem, Polynomial approximation to functions.

◦     Matrices: Matrices and transformations, Multiplication vs composition, Determinants as dilation factors, Polynomials applied to matrices, Matrices in probability theory. Matrices in Polynomial interpolation, the Vandermonde determinant.

◦     Conservation laws: Invariants of transformations, Discrete transformations, and applications, Transformations in Euclidean Geometrycircle inversions.

◦     Elementary number theory: Modular arithmetic, Divisibility, Prime numbers.

◦     Exploratory project suggestions.

Books

Here is a list of books that offer very interesting perspectives on a range of topics in mathematics, beyond what has been touched upon in this course.

  1. What is Mathematics?by Richard Courant and Herbert Robbins.
  2. Combinatoricsby N. Ya. Vilenkin.
  3. Number, Shape and Symmetryby Diane Herrmann and Paul Sally Jr.
  4. The Pleasures of Countingby T. W. Korner.
  5. Calculus for the Ambitiousby T. W. Korner.
  6. Integer partitionsby George E. Andrews and Kimmo Eriksson.

These are highly recommended as further reading material for anyone taking this course.

Suggested references.

  1. artitions:In our course, we encountered partitions when we talked about permutations and cycle types. The theory of partitions is an important subject in its own right, with applications to many different parts of mathematics.The book titled "Integer partitions" by George E. Andrews and Kimmo Eriksson gives a very accessible introduction to this field, assuming only minimal prerequisites.With numerous exercises that include some mini research projects, this book is ideal for students wanting to engage with some interesting mathematics on a longer term on their own.

  2. Circle Inversions:For those interested in Euclidean geometry, a study of "inversions in a circle" is a nice project. Circle inversions are a certain kind of transformation of the plane. Unlike the examples in the course, these are not linear or affine transformations. They have many interesting properties, specifically related to how they transform points, lines and circles, and this can be used to derive many results in Euclidean geometry.TheWikipedia articleon inversions is a good place to start. The book titled "Geometry Revisited" by H.S.M. Coxeter and Samuel L. Greitzer has a very nice exposition of inversion, in addition to other topics in Euclidean geometry.


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