Overview: 

A thorough review of single-variable and multivariable calculus, focusing on key concepts relevant to advanced studies.

Topics:

• Limits, Continuity, and Differentiation

• Integrals and Theorems of Calculus (Fundamental Theorem, Green's Theorem, etc.)

• Multivariable Calculus: Gradient, Divergence, Curl

• Applications: Optimization, Surface Integrals, Volume Integrals in Engineering

Overview: 

A deep dive into the theory and applications of partial differential equations in science and engineering.

Topics:

• Separation of Variables for PDEs

• The Heat Equation, Wave Equation, and Laplace’s Equation

• Fourier Series Solutions

• Applications: Heat Transfer, Fluid Dynamics, Electromagnetism

Overview: 

Rigorous study of functions, sequences, and limits in real number spaces, forming the foundation of higher mathematics.

Topics:

• Limits, Continuity, and Differentiability in Real Spaces

• The Bolzano-Weierstrass Theorem and Compactness

• Riemann Integration and Measure Theory

• Applications: Convergence of Series, Numerical Analysis, Optimization

Overview: 

An introduction to groups, rings, fields, and other algebraic structures with applications in cryptography and coding theory.

Topics:

• Group Theory: Cyclic Groups, Permutation Groups, Symmetry Groups

• Ring and Field Theory

• Homomorphisms and Isomorphisms

• Applications: Cryptographic Algorithms, Coding Theory, Error Detection

Overview: 

The study of curves, surfaces, and manifolds, with applications in theoretical physics and engineering.

Topics:

• Curves and Surfaces: Curvature, Torsion, and Geodesics

• Manifolds and Riemannian Geometry

• Gauss-Bonnet Theorem and Applications

• Applications: General Relativity, Robotics, Computer Graphics

Overview: 

A high-level abstraction of mathematical structures and their relationships, widely used in theoretical computer science and logic.

Topics:

• Categories, Functors, and Natural Transformations

• Limits, Colimits, and Universal Properties

• Applications in Topos Theory and Logic

• Applications: Theoretical Computer Science, Logic, Algebraic Geometry

Overview: 

The abstract study of spatial relationships through topology, focusing on continuity, compactness, and connectedness.

Topics:

• Topological Spaces: Open and Closed Sets, Continuity

• Compactness and Connectedness

• Homotopy and Fundamental Groups

• Applications: Algebraic Topology, Dynamical Systems, Quantum Field Theory

Overview: 

Mathematical foundations for machine learning and data science, focusing on optimization, probability, and linear algebra.

Topics:

• Gradient Descent and Optimization Techniques

• Probability Distributions and Bayes’ Theorem

• Matrix Factorization and Principal Component Analysis

• Applications: Neural Networks, Clustering, Dimensionality Reduction