Professional Certificates

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.