Resources

Seminar and course notes, preprints, and other material shared by the group.

Student seminar notes

• Combinatorial and topological games

  Baire spaces and applications in classical mathematics

  José Iovino

  A gentle introduction to topological games prepared for researchers and students. Part I presents a self-contained introduction to the Banach–Mazur and strong Choquet games, Baire spaces and complete metrizability; the later parts tell how these games, and their relatives, solved longstanding problems in number theory, dynamics, functional analysis, set theory, Ramsey theory, combinatorics, logic and the theory of computation. The last part tells the story of the Scottish Book.
• Tame and wild theories

  An introduction to the logic of dependence and independence

  José Iovino

  A gentle introduction to classification theory (in the sense of logic) written for students and researchers. It includes stable theories, NIP theories, and simple unstable theories. The text is accessible to undergraduate and graduate students. It presupposes familiarity with the basic of first-order logic, including the compactness theorem, ultraporoducts, and saturated extension. The text contains many exercises and illustrations.
• Topological and logical aspects of the problem of hidden variables in quantum mechanics

  Lecture notes

  José Iovino

  Notes for students and researchers on the the problem of hidden variables in quantum mechanics and the Bell–Kochen–Specker theorem. Bell's proof and the Kochen-pecker's proof are linked by the compactness theorem of propositional logic. The history of the problem and its connection to logi are traced in detail.

Student research notes

• Gleason, Kochen–Specker, and the model existence game

  Olivia Aubone and José Iovino

  The Kochen–Specker theorem shows that certain finite sets of rays admit no assignment of the values 0 and 1 compatible with orthogonality. We prove Gleason's theorem and a strengthening of the Kochen–Specker theorem in infinitary real-valued logic. In this logic the probability of a quantum proposition is the value of a formula. The hypothesis that every probability is 0 or 1 fails in every model of a countable theory. The standard models of this theory are the separable infinite-dimensional complex Hilbert spaces with a density operator. In them, every hypothesis that the probabilities avoid a nonempty Borel or σ-projective subset of [0,1] fails as well. In place of compactness we use model-theoretic forcing, cast as a two-player topological game. The game yields finite obstructions that constrain the probabilities themselves. They identify the axioms on which each refutation rests. They exist even when every finite set of axioms has a model.
• Omitting types for infinitary real-valued logic: a topological perspective

  Olivia Aubone, Eduardo Dueñez and José Iovino

  We prove an omitting types theorem for countable fragments of infinitary logic for real-valued structures in the sense of Keisler, structures on which no metric is given. The proof is topological: for a countable vocabulary and a countable fragment, the Henkin types form a Čech-complete space, and the required models are obtained from the Baire category theorem.

Course notes

• An introduction to quantum error correction for mathematicians

  Eduardo Dueñez

  Notes for the Fall 2026 Quantum Error Correction Seminar: stabilizer codes, surface codes and their decoding, belief propagation, and quantum LDPC codes, assuming only undergraduate linear algebra and elementary probability.
• Introduction to quantum mechanics: lecture notes

  José A. Morales Escalante

  Introductory notes on the mathematical formalism of quantum mechanics.
• Quantum computing & information: lecture notes (Fall 2022)

  José A. Morales Escalante

  Selected lessons from a Fall 2022 course on quantum computing and information.

Preprints

• Classification of bipartite observables I: stable and uniformly stable observables

  Eduardo Dueñez and José Iovino

  Part I of a series with Eduardo Dueñez. Stability of a bipartite observable as independence from the order of two idealized limits, with Grothendieck's double limit criterion as the unifying theorem.
• Classification of bipartite observables II: NIP observables and Todorčević's trichotomy

  Eduardo Dueñez and José Iovino

  Part II of a series with Eduardo Dueñez. NIP observables, Rosenthal compacta, and Todorčević's trichotomy, with physical orders in each class and the transverse-field Ising model.
• Classification of bipartite observables III: simplicity, mixtures and quantum error correction

  Eduardo Dueñez and José Iovino

  Part III of a series with Eduardo Dueñez. The Rado graph, the paradigm of a simple unstable theory, and its counterpart in quantum error correction, the Pauli group: logical errors as forking, Pauli channels as probes, and the two routes by which mixing produces the tree property of the second kind.
• Shelah's dividing lines in quantum measurement: 32 variations in three parts

  José Iovino

  A survey of the three-part series: stability, NIP and simplicity as a classification of bipartite observables, written for model theorists and operator theorists.