Trimester 05 · Course detail

Measure Theory & Statistical Mechanics

Abstract measure theory and real analysis, classical and quantum statistical mechanics, operator theory and boundary value problems, advanced scientific rhetoric and peer review

Credits: 15
Sequence: 5 of 8
Structure: 4 subjects
Status: Full syllabus

Curriculum focus

Abstract measure theory and real analysis, classical and quantum statistical mechanics, operator theory and boundary value problems, advanced scientific rhetoric and peer review

Evidence of mastery

Proof repository + ensemble simulation

Subject 01

Abstract Measure Theory & Real Analysis

Official syllabus code: MIMUW 1000-135AM2

Measure theory rebuilt from the axioms: outer measures, signed measures, product structures, and the duality theory of the Lebesgue–Stieltjes integral.

Measure Foundations & Convergence

σ-Algebras, Outer Measures & Carathéodory Extension

Construct measures from outer measures, prove the Carathéodory extension theorem, and verify its uniqueness on the generated σ-algebra.

Measurable Functions & the Convergence Theorems

Characterize measurable functions and prove the Monotone Convergence Theorem together with Fatou's Lemma and the Dominated Convergence Theorem.

Products, Signed Measures & Duality

Product Measures, Fubini–Tonelli & Infinite-Dimensional Cylinders

Build product measures, prove the Fubini and Tonelli theorems, and extend the construction to infinite-dimensional cylinder σ-algebras.

Signed Measures, Hahn/Jordan Decompositions & Radon–Nikodym

Decompose signed measures via the Hahn and Jordan decompositions and prove the Radon–Nikodym theorem on the existence of densities.

Lebesgue–Stieltjes Integrals, Total Variation & Riesz Representation

Develop the Lebesgue–Stieltjes integral, master total variation as a norm, and prove the Riesz representation theorem for dual spaces.

Subject 02

Classical & Quantum Statistical Mechanics

Official syllabus code: FUW 1100-3INZ12

Ensemble theory as generating-function machinery: from thermodynamic limits and ensemble equivalence to quantum ideal gases and interacting systems.

Ensembles & the Thermodynamic Limit

Microcanonical, Canonical & Grand Canonical Ensembles

Derive the three classical ensembles and their partition functions from the microstate counting of isolated, thermal, and open systems.

Partition Functions as Generating Functions of Thermodynamic Observables

Show how derivatives of the partition functions generate all thermodynamic observables — energy, entropy, pressure, and particle number.

The Thermodynamic Limit, Fluctuations & Equivalence of Ensembles

Take the thermodynamic limit, bound fluctuations by system size, and prove the equivalence of ensembles away from phase coexistence.

Quantum Gases & Interacting Systems

Quantum Ideal Gases: Fermi–Dirac, Bose–Einstein & Condensation (BEC)

Derive the quantum distribution functions, contrast Fermi–Dirac and Bose–Einstein statistics, and analyze the Bose–Einstein condensation transition.

Interacting Systems: Cluster Expansion, Mean Field Theory & the Ising Model (1D/2D)

Treat interactions via the cluster (virial) expansion and mean-field theory, and solve the one- and two-dimensional Ising models exactly.

Subject 03

Operator Theory & Boundary Value Problems

Official syllabus code: MIMUW / FUW Metody

Banach and Hilbert space geometry with weak topologies, closed by the spectral theory of self-adjoint differential operators and Green's dyadics.

Banach & Hilbert Spaces

Normed Spaces, Banach Spaces & the Classical Lᵖ Spaces (1 ≤ p ≤ ∞)

Study the hierarchy of normed and Banach spaces and establish the completeness and duality structure of the Lᵖ scale.

Duality, Weak Topologies & Weak-* Compactness (Banach–Alaoglu)

Compare norm and weak topologies and prove the Banach–Alaoglu theorem on weak-* compactness of the dual unit ball.

Hilbert Space Geometry, Riesz–Fréchet Isomorphism & the Projection Theorem

Develop inner-product geometry, prove the Riesz–Fréchet identification of a Hilbert space with its dual, and establish the projection theorem for closed subspaces.

Differential Operators & Spectral Bases

Self-Adjoint Sturm–Liouville Operators, Spectral Bases & Green's Dyadics

Analyze self-adjoint Sturm–Liouville differential operators, construct complete spectral eigenfunction bases, and build Green's dyadics for boundary value problems.

Subject 04

Advanced Scientific Rhetoric & Peer Review

Official syllabus code: RHET-301 / Proof Lab

The craft of rigorous mathematical writing: deconstructing the proofs of Gibbs, Kolmogorov, and Lebesgue, then reproducing them inside automated proof assistants.

Proof Culture & Formal Verification

Formal Deconstruction of Seminal Monograph Proofs (Gibbs, Kolmogorov, Lebesgue)

Dissect the argumentative structure of landmark proofs from Gibbs, Kolmogorov, and Lebesgue to extract their reusable proof techniques.

Symbolically Verified Proof Engine in Automated Proof Assistants (Lean / Coq)

Re-verify the trimester's measure-theoretic and statistical results inside Lean or Coq, mastering formal proof state and tactic engineering.

Panel 5 Capstone Artifact: Symbolically Verified Measure-Theoretic Engine

Assemble the capstone: a measure-theoretic engine whose core theorems are machine-checked, submitted for structured peer review.