- Set Theory and Functions : Language of mathematics. Understanding sets, logical notation and relations. Building the path towards understanding functions. How to graph functions. The Cartesian plane. Exploring types of functions. Polynomials. Polynomial division. Quadratic equations. Cubic equations. Increasing and decreasing functions. Maxima and minima of functions.
- Basic Algebra, Real Numbers and Complex Numbers : Equations in two and three unknowns. When is a system of equations solvable? Natural numbers. Peano axioms. Introduction to induction. Integers. Rational numbers. Real numbers. Real numbers as solutions to polynomial equations. Transcendental numbers. Introduction to complex numbers. Polar form.
- Elementary Geometry : Triangles. Properties of triangles. Theorems concerning triangles. Some theorems concerning circles.
- Basic Trigonometry : Definitions of trigonometric functions. Their identities. Graphs. Properties. Relationship with complex numbers.
- Coordinate Geometry : Conic sections, their properties.
- Basic Combinatorics : Counting techniques. Pigeonhole principle. Combinations and permutations. Binomial theorem. Binomial coefficients and their summations, identities.
- Further Algebra : Floor function and ceiling function. Their properties. More complex numbers. Basics of inequalities. Some geometric inequalities.
- Preliminaries to calculus : Motivation for calculus. Open and closed sets. Countable and uncountable sets. Axiomatization of real numbers. Well ordering principal. Principle of recursive definition. Principle of mathematical induction. Archimedean property.
- Sequences and Series : Definition of sequences and series. Supremum and infimum. Arithmetic progression. Geometric progression. Arithmetico-geometric progression. Tests of convergence. Radius of convergence. Trigonometrical series.
- Limits : Intuitive description of limit. Epsilon delta definition. Limit laws. Limit theorems. Continuity. Mean value theorems.
- Integral Calculus : Partitions. Step functions. Area function. Calculating integrals.
- Differential Calculus : Rate of change. Derivatives. Formulas of derivatives. Fundamental theorem of calculus. Extremization problems. Introduction to logarithms and exponential functions. Trigonometric functions and their calculus.
- Techniques of Integration : All the standard techniques along with some extra techniques such as Feynman’s rule, etc.
- Vectors and Vector Calculus : Partial derivatives. Lagrange multipliers. Differential calculus of vector fields. Line integrals. Surface integrals. Multiple integrals. Contour integration (a brief introduction). All the crucial theorems.
- Differential Equations : Linear differential equations of first and second order. Some special ODEs and techniques.
- Linear algebra : Linear equations. Matrices. Row operations. Reduced row echelon form. Invertibility of matrices. Determinants. Vector spaces. Linear transformations. Properties of determinants and matrices. Eigenvalues and eigenvectors. Eigenspaces. Cayley Hamilton theorem. Special types of matrices.
- Probability Theory : Conditional probability. Random variables. Discrete distributions. Continuous distributions. Binomial and Poisson distributions. Normal distribution. Bernoulli trials. Random variables. Law of large numbers.
- Genetics : Exploring some applications of probability in biology, with an emphasis on genetics.
- Thermodynamics and Statistical Physics : Further exploring applications of probability in physics. Ideal gas equation. Heat. Temperature. Boltzmann distribution. Maxwell-Boltzmann distribution. Kinetic theory of gases. Pressure. Mean free path. Properties of gases. First law of thermodynamics. Second law of thermodynamics. Carnot’s theorem and Carnot engines. Entropy. Equipartition theorem. Third law.