Carnegie Mellon University

Inside modern relational and analytical DBMS internals: relational algebra, slotted-page storage, log-structured merge trees, columnar compression, buffer pool management, and hash/B+Tree indexes.

Rigorous introduction to computability and complexity theory, covering Turing machines, diagonalization, NP-completeness, space complexity, Savitch's theorem, and the polynomial hierarchy.

Graduate systems seminar on modern OLAP execution engines: columnar compression, Parquet/Arrow formats, vectorized SIMD execution, JIT query compilation, worst-case optimal joins, and cloud lakehouses.

Advanced systems course on relational query optimization, analyzing System R bottom-up dynamic programming, Volcano/Cascades top-down rule engines, cardinality estimation, and join ordering.

Intensive PhD-level complexity theory sequence examining circuit complexity, randomized computation, interactive proofs, PCP theorem foundations, pseudorandomness, and hardness amplification.

Research-level course developing Fourier analysis over the Boolean hypercube, hypercontractivity, noise stability, invariance principles, and applications to hardness of approximation and social choice.

Mathematical development of quantum circuits, entanglement, quantum query complexity, phase estimation, Shor's factoring algorithm, quantum channels, and quantum error correction.

Essential mathematical machinery for theoretical computer science research: asymptotic tail bounds, spectral graph theory, expanders, linear and semidefinite programming duality, and Fourier analysis.