Professional Certificates

Who can join?

​BS (CS/SE/AI/Data Science) students who have completed at least 2 years of study
Graduates and professionals who feel that they need to refresh their mathematical knowledge
Whether you are beginning your journey, preparing to enter the job market, or upgrading your technical depth, this course provides a solid grounding in the four most important areas of mathematics that an AI engineer and algorithm designer needs: Discrete Mathematics, Calculus, Linear Algebra, and Probability.

About The Course

Mathematical Foundations for AI and Algorithmic Thinking is a comprehensive course designed to build the core mathematical intuition required for Artificial Intelligence, Machine Learning, and advanced algorithms.
Master the mathematical pillars of AI—Discrete Mathematics, Calculus, Probability, and Linear Algebra. Learn to transform abstract formulas into practical tools for building intelligent technology. Explore the ideas behind intelligent systems through hands-on mathematical modeling connecting theory to neural networks, optimization, probabilistic learning, and high-dimensional data. This journey is designed to turn mathematical principles into clarity, confidence, and creative problem-solving, whether your path leads to Machine Learning, Data Science, competitive programming, or AI research.

Module 1: Discrete Mathematics for AI

Lecture 1 — Foundations of Logic, Proofs & Sets

• Core Concepts: Propositions, truth tables (AND, OR, XOR, NOT), logical implication, equivalence, and quantifiers (∀, ∃). Direct proofs vs. contradiction. Proof of the prime factor bound — why checking divisors only up to √N is sufficient.
• Intuition: Building the “grammar” of mathematics to define truth and falsehood clearly.
• AI/CS Link: Logical rules form the basis of rule-based systems; quantifiers define model assumptions.

Lecture 2 — Mathematical Induction & Recurrence Relations

• Core Concepts: The principle of mathematical induction, induction on sequences, and relating recursion to induction. Key proofs:
  • Arithmetic Series: sum = n(n+1)/2
  • Geometric Series: sum = a(rⁿ−1)/(r−1) and their practical impact
• Intuition: Using a “domino effect” logic to prove that a property holds for all infinite steps of a process.
• AI/CS Link: Induction is used to prove convergence of iterative algorithms and correctness of recursive functions.

Lecture 3 — Growth Rates & Computational Complexity

• Core Concepts: Big O, Big Theta, and Big Omega notation. Comparing function classes: Constant, Logarithmic, Linear, n log n, Quadratic, Exponential.
• Intuition: Understanding how time/space required by an algorithm scales with input size.
• AI/CS Link: Vital for evaluating scalability of training algorithms and efficiency of model inference.

Lecture 4 — Advanced Set Theory & Relations

• Core Concepts: Set operations, Cartesian products, and power sets. 
  • Key relation properties: Closure, Transitivity, Symmetry, Reflexivity.
• Intuition: Organizing data into collections and defining how elements within them interact.
• AI/CS Link: Set operations used for dataset splits (train/test/val) and defining feature sets.

Lecture 5 — Counting, Combinatorics & Binomials

• Core Concepts:
  • Sum, Product, Subtraction, and Division rules
  • Permutations and combinations
  • Binomial Theorem
  • Inclusion-Exclusion principle and its generalization
• Intuition: Calculating the number of possible outcomes or configurations in a finite system.
• AI/CS Link: Model capacity, parameter search space, and number of possible token sequences in NLP.

Lecture 6 — Graph Theory Foundations

• Core Concepts: Nodes, edges, directed vs. undirected graphs, adjacency matrices, paths, and connectivity.
• Intuition: Representing complex relationships between entities as a network.
• AI/CS Link: Neural networks as computational graphs; knowledge graphs for semantic relationships.

Lecture 7 — Trees, DAGs & Topological Sorting

• Core Concepts:
  • Tree structures and Directed Acyclic Graphs (DAGs)
  • Topological sorting
  • Traversal algorithms: DFS and BFS in grids and general graphs
• Intuition: Modeling hierarchical data and processes with a specific execution order.
• AI/CS Link: Backpropagation is a reverse traversal on a DAG; transformers utilize attention graphs.

Lecture 8 — Formal Languages & Automata for AI

• Core Concepts: Finite State Machines (FSM), regular expressions, and the relationship between logic and formal languages.
• Intuition: Defining the rules and structures that generate valid sequences of data.
• AI/CS Link: Essential for understanding tokenization in LLMs and the logic of string processing.

Module 2: Calculus for AI

Lecture 1 — Limits, Continuity & Derivative Foundations

• Core Concepts: Definitions of limits (0/0, ∞/∞), Sandwich Theorem. The 6 basic functions and their derivatives: xⁿ, sin, cos, eˣ (and aˣ), ln x, step/|x| functions.
• Intuition: Measuring how a function changes at an infinitesimal point.
• AI/CS Link: Limits are necessary to define derivatives used in gradient descent.

Lecture 2 — Rules of Differentiation & The Chain Rule

• Core Concepts: The 6 rules of differentiation: Sum, Product, Quotient, Inverse, Chain Rule, L’Hôpital’s Rule.
• Intuition: Calculating the rate of change for complex, nested functions.
• AI/CS Link: The Chain Rule is the mathematical engine behind backpropagation in neural networks.

Lecture 3 — Optimization: Critical Points & Curvature

• Core Concepts:
  • Finding maxima, minima, and inflection points
  • Second derivative test; Newton’s method
  • All Values Theorem and Mean Value Theorem
• Intuition: Identifying the “peaks” and “valleys” of a function to find its best value.
• AI/CS Link: Training a model involves optimizing a loss function to find its global or local minimum.

Lecture 4 — Taylor Series & Linearization

• Core Concepts: Taylor Series expansions, first-order Taylor (linearization), and the Binomial Theorem.
• Intuition: Approximating complex non-linear functions with simpler polynomials.
• AI/CS Link: Linearization lets us understand the local geometry of a high-dimensional loss surface.

Lecture 5 — Integration & The Fundamental Theorem of Calculus

• Core Concepts: Riemann sums, Fundamental Theorem of Calculus I & II, and integral intuition.
• Intuition: Calculating the total accumulated value over an interval (area under a curve).
• AI/CS Link: Integrals define expectations in probability and are core to diffusion models.

Lecture 6 — Multivariable Calculus: Partial Derivatives & Gradients

• Core Concepts: Partial derivatives, the Gradient vector, and the Multivariable Chain Rule.
• Intuition: Understanding how a function of many variables changes as each variable is tweaked individually.
• AI/CS Link: Gradients tell us how to update every parameter in a model simultaneously.

Lecture 7 — The Jacobian & Vector-Valued Functions

• Core Concepts: Defining the Jacobian matrix and its role in multivariable transformations.
• Intuition: Extending the derivative to functions that map multiple inputs to multiple outputs.
• AI/CS Link: Crucial for understanding gradient flow through layers and activations in Deep Learning.

Lecture 8 — Convexity & High-Dimensional Optimization

• Core Concepts: Definition of convexity, the Hessian matrix, and its intuition in loss surface geometry.
• Intuition: Determining if a function is “bowl-shaped,” guaranteeing any local minimum is global.
• AI/CS Link: Convex optimization ensures stable and reliable training for many ML algorithms.

Module 3: Probability Foundations for AI

Lecture 1 — Discrete Probability & Kolmogorov Axioms

• Core Concepts: Basic probability, expectation, variance, and standard deviation. Kolmogorov axioms, sample spaces, and connecting probability to counting with several examples.
• Intuition: Quantifying uncertainty in a mathematically rigorous way.
• AI/CS Link: Provides the foundation for all probabilistic modeling and data science.

Lecture 2 — Conditional Probability & Bayes’ Theorem

• Core Concepts: Conditional probability, Law of Total Probability, Bayes’ Theorem. Application: designing spam filters.
• Intuition: Updating our beliefs about an event based on new incoming evidence.
• AI/CS Link: Used in Bayesian inference and posterior updates.

Lecture 3 — Random Variables: Expectation & Variance

• Core Concepts: Discrete and continuous random variables, PDF vs. PMF, expectation, variance, and standard deviation.
• Intuition: Summarizing a complex distribution with a single average value and a measure of spread.
• AI/CS Link: Expectation is used to calculate the average loss of a model over a dataset.

Lecture 4 — Continuous Distributions & The Gaussian

• Core Concepts: Uniform, Exponential, and Gaussian (Normal) distributions; the Multivariate Gaussian.
• Intuition: Modeling real-world data that clusters around a central mean.
• AI/CS Link: Gaussian distribution is a standard prior in VAEs and the basis for noise in diffusion models.

Lecture 5 — Covariance & High-Dimensional Geometry

• Core Concepts: Covariance and correlation matrices and their geometric interpretation.
• Intuition: Measuring how two or more variables change together.
• AI/CS Link: Covariance matrix is central to latent space modeling and feature redundancy analysis.

Lecture 6 — Limit Theorems & Sampling

• Core Concepts: Law of Large Numbers (LLN), Central Limit Theorem (CLT), and Monte Carlo sampling basics.
• Intuition: Understanding why many small, independent random factors average out into a bell curve.
• AI/CS Link: Provides intuition behind Batch Normalization and stochastic optimization (SGD).

Lecture 7 — Likelihood & Parameter Estimation

• Core Concepts: Likelihood and log-likelihood functions; derivation of Maximum Likelihood Estimation (MLE).
• Intuition: Finding the model parameters that make the observed data most likely.
• AI/CS Link: MLE is the fundamental principle used to derive loss functions like Mean Squared Error.

Lecture 8 — Information Theory: Entropy & Divergence

• Core Concepts: Entropy, Cross-Entropy, KL Divergence, Evidence Lower Bound (ELBO).
• Intuition: Measuring the information content of a distribution or the distance between two distributions.
• AI/CS Link: Cross-entropy is the standard loss for classification; KL divergence is used in VAEs and RLHF.

Module 4: Linear Algebra for AI

Lecture 1 — Vectors, Norms & Dot Products

• Core Concepts: Vector spaces, L1/L2 Norms, dot products, and cosine similarity.
• Intuition: Representing data points as positions and directions in space.
• AI/CS Link: Embeddings (word/image vectors) rely on norms and similarity for retrieval and comparison.

Lecture 2 — Matrices & Linear Transformations

• Core Concepts: Matrix-vector multiplication, linear transformations, rank, and matrix inverse.
• Intuition: Using matrices to rotate, scale, or project data into new coordinate systems.
• AI/CS Link: Each neural network layer is a linear transformation followed by a non-linearity.

Lecture 3 — Systems of Linear Equations

• Core Concepts: Row Echelon Form (REF), Gaussian elimination, and solving Ax = b.
• Intuition: Finding the intersection of multiple hyperplanes.
• AI/CS Link: Used in Linear Regression and weight updates in certain optimization methods.

Lecture 4 — Subspaces, Orthogonality & Projections

• Core Concepts: Basis, subspaces, orthogonality, and projection of a vector onto a subspace.
• Intuition: Finding the “shadow” of a high-dimensional vector in a lower-dimensional space.
• AI/CS Link: Transformer projections and dimensionality reduction rely on these geometric principles.

Lecture 5 — Eigenvalues & Eigenvectors

• Core Concepts: Characteristic equation, Eigendecomposition, and the intuition of invariant directions.
• Intuition: Finding the natural axes of a transformation where data is only scaled, not rotated.
• AI/CS Link: Central to stability of Recurrent Neural Networks (RNNs) and spectral clustering.

Lecture 6 — Singular Value Decomposition (SVD)

• Core Concepts: SVD of a matrix, singular values, and the relationship between SVD and Eigendecomposition.
• Intuition: Decomposing any matrix into three operations: rotation, scaling, and rotation.
• AI/CS Link: Used for matrix compression, denoising, and Low-Rank Adaptation (LoRA) of LLMs.

Lecture 7 — Principal Component Analysis (PCA)

• Core Concepts: Low-rank approximation using SVD and derivation of PCA from covariance.
• Intuition: Identifying directions in a dataset that contain the most variance/information.
• AI/CS Link: A primary tool for data visualization and feature reduction in Data Science.

Lecture 8 — Tensors & Multilinear Algebra

• Core Concepts: High-dimensional arrays (tensors), tensor products, and attention weights geometry.
• Intuition: Generalizing vectors (1D) and matrices (2D) to arbitrary dimensions.
• AI/CS Link: All modern Deep Learning frameworks (PyTorch/TensorFlow) operate on tensors; attention mechanisms use tensor products.