This short course

This course begins with foundational structures in number theory, tracing key developments from early mathematicians through the work of Fermat and Legendre. It then explores modern applications of these ideas in coding theory and cryptography.

• Foundational Concepts: Divisibility, greatest common divisors, prime numbers, the fundamental theorem of arithmetic, and number-theoretic functions (e.g., divisor counting).
• Chinese Remainder Theorem: Theory and applications.
• Cryptography: Classical systems (shift, digraph, matrix) and public-key cryptosystems, primality testing, factorization, error-correcting codes, and analysis of efficiency and security.
• Congruence and Powers: Use of primitive roots and indices to solve congruences. Quadratic residues, Legendre symbols, and Jacobi symbols.
• Diophantine Analysis: Quadratic forms, continued fractions, and Pell's equation.

Vector Spaces: Basic definitions and examples, subspaces, dimension and basis, the vector spaces and subspaces.  Inner Product spaces: Basic definitions with many examples, the notions of norm (length) and distance (emphasis will be on the Euclidean inner product). Linear Transformations: Basic definitions and results, including images and kernels, with examples; matrix representations of linear transformations

This course extends the fundamentals of calculus—differentiation, integration, and infinite series—to the domain of complex variables, where classical real analysis evolves into a deeper and more elegant theory. Developed by pioneers such as Gauss, Cauchy, Riemann, and Weierstrass, complex analysis provides essential tools for modern mathematics and diverse scientific disciplines, including fluid dynamics, potential theory, heat conduction, electromagnetism, and elasticity. This course equips students across mathematics, engineering, physics, and applied sciences with indispensable analytical techniques.

• Complex differentiation and the Cauchy–Riemann equations
• Complex integration: the Cauchy–Goursat theorem and its implications
• Cauchy’s integral formulas and key results:
  Cauchy’s inequality, Liouville’s theorem, the fundamental theorem of algebra
• Morera’s theorem, maximum and minimum modulus principles
  The argument principle and Rouché’s theorem
  Practical applications of these results
• Series expansions: Taylor and Laurent series
• Singularities, residues, and the residue theorem
• Applications of the residue theorem to evaluating real integrals and infinite series

software due to its clear syntax, advanced mathematical functions, and strong graphical capabilities, with dedicated tutorials and assignments to build programming proficiency.

• Computing Foundations: Basic programming principles, error analysis, rounding errors and their impact, numerical stability.
• Linear Systems: Numerical solution via Gaussian elimination with partial pivoting; Gauss–Jordan method and efficiency considerations; LU decomposition and matrix inversion; iterative methods (Gauss–Seidel and Jacobi).
• Nonlinear Equations: Root-finding for single equations using bisection, Newton’s, and secant methods; Newton’s method for systems of nonlinear equations.
• Curve Fitting: Least squares polynomial and general function fitting; polynomial interpolation with error analysis; spline interpolation for curve design.
• Differentiation and Integration: Numerical differentiation with truncation and rounding error analysis; composite trapezoidal and Simpson’s rules for integration.
• Ordinary Differential Equations: First-order problems solved by Euler, Taylor, and Runge–Kutta methods; extension to systems of first-order equations and second-order initial value problems; boundary value problems via the shooting method and finite differences.

PDE of mathematical physics and an introduction to their study. Classification of second order PDE’s in two independent variables; Derivation of the wave, Laplace and Poisson equations;

Fourier series theory; Application to BVP.

This course provides an introduction to the fundamental concepts of graph theory, emphasizing both intuition and rigorous proof. Students will explore the structure of graphs, learn to identify similarities and differences between graphs, and investigate the principles that underlie graph classification and representation. Topics progress from basic definitions and constructions to more advanced themes, including graph operations, connectivity, planarity, graph coloring, and special substructures. The course aims to equip students with both theoretical insight and practical techniques for analyzing graphs, with connections to applications in mathematics and computer science.

Lecture Outline

• Graph Construction and Families: Valency sequences, constructing graphs from degree sequences, and introduction to families of graphs.
• Graph Operations and Relations: Graph unions, intersections, complements, and an overview of the graph reconstruction conjecture.
• Graph Structure and Connectivity: Distance, connectedness, bipartite graphs, trees, and an introduction to the use of matrices in graph theory.
• Planarity: Planar graphs, planarity testing (Kuratowski’s theorem), and embedding graphs in a “nice” planar drawing.
• Graphs on Surfaces: Introduction to drawing graphs on higher-genus surfaces (beyond the plane).
• Graph Coloring: Vertex coloring, planar coloring theorems (Four Color Theorem introduction), and the chromatic polynomial for counting colorings.
• Special Subgraphs: Eulerian circuits and Hamiltonian cycles, including necessary and sufficient conditions for their existence

This course introduces widely-used mathematical software packages, emphasizing their application to mathematical problem-solving, visualization, and computation. Students will gain hands-on experience with key tools for symbolic algebra, numerical analysis, data plotting, and algorithm development. The focus is on developing practical skills for implementing mathematical models, automating calculations, and presenting results effectively across academic and professional contexts. 

Lecture Outline

• Overview of mathematical software: Symbolic, numerical, and graphical environments
• Introduction to MATLAB: Basic syntax, matrix operations, plotting, and scripting
• Introduction to Python with NumPy/SciPy/Matplotlib: Setup, numerical arrays, scientific computing, and visualization
• Introduction to R: Statistical computation, data frames, and graphical output
• Symbolic computation with Mapleor Mathematica: Algebraic manipulation, calculus, and equation solving
• Comparative use cases: Choosing the right tool for different mathematical tasks
• Integrating software outputs into reports and presentations (LaTeX, Jupyter, live scripts)
• Automation, debugging, and best practices in computational workflow

This course offers a rigorous and applied introduction to Newtonian mechanics, the foundational mathematical framework for analyzing physical systems. Students will develop the skills to model and solve problems involving motion, forces, and energy in real-world contexts—from planetary orbits to engineering applications like accident reconstruction and satellite launch dynamics. Emphasis is placed on both theoretical principles and practical problem-solving, preparing students to apply mechanics across physics, engineering, and applied mathematics.

• divergence, curl. Conservative and solenoidal fields.
• Particle Kinematics
  Velocity and acceleration vectors. Constant acceleration and rectilinear motion. Relative motion and pursuit problems. Circular motion.
• Particle Dynamics
  Newton’s laws of motion. Types of forces (action-at-a-distance, reactive, etc.). Projectile motion with and without air resistance. Variable mass systems: rocket motion. Impulse and momentum.
• Oscillatory Systems
  Simple harmonic motion (SHM) and the simple pendulum. Damped, forced, and coupled oscillations. Resonance phenomena.
• Work, Energy, and Power
  Work done by constant and variable forces. Conservative forces and potential energy. Line integrals. Principle of conservation of energy.
• Central Forces and Orbits
  Conservative force fields and equations of motion. Orbital mechanics: inverse-square law, Kepler’s laws, and polar equations of conic sections (elliptic, parabolic, hyperbolic orbits).
• Rigid Body Mechanics
  Equations of motion for rigid bodies. Mass distribution and center of mass. Forces, couples, and moment of momentum.

This course offers an introduction to the theory of partial differential equations (PDEs) and presents methods for solving boundary and initial value problems arising in mathematical physics. Students will learn to classify PDEs, derive equations satisfied by given functions, and solve standard forms of first- and second-order equations. Special attention is given to key second-order PDEs that model physical phenomena such as wave propagation and heat transfer. The course emphasizes practical solution techniques, considering relevant initial and boundary conditions applicable to real-world problems.

• Classification of PDEs: identification of types and derivation from given functions
• Analytical solution methods for first-order PDEs in standard forms
• Second-order PDEs: detailed examination of key equations
• Derivation of PDEs from physical models (wave equation, heat equation, etc.)
• Solution techniques for second-order PDEs, incorporating initial and boundary conditions
• Application of methods to problems in wave propagation, heat transfer, and related phenomena
• Focus throughout on practical approaches to solving PDEs in applied contexts

This advanced topics course covers several key themes in modern graph theory, focusing on both classic results and active areas of research.

• Graph Covering Problems
  Primal graphs and related covering problems.
• Planar Graphs and the Four Colour Theorem
  Study of planar graphs and an overview of the Appel–Haken proof of the Four Colour Theorem.
• Perfect Graphs
  Exploration of perfect graph theorems and major conjectures.
• Matching Theory and Reconstruction
  Matching theory, with possible revisitation of the graph reconstruction conjecture.
• Algebraic Graph Theory
  Introduction to algebraic methods in graph theory, including distance-regular and distance-transitive graphs.