Advanced Geometry

Academic Year 2026/2027 - Teacher: SANTI DOMENICO SPADARO

Expected Learning Outcomes

The course provides an introduction to the combinatorial aspects of set theory. Students will learn to use tools such as Ramsey’s theorem, the Δ-system lemma, and the pressing-down lemma, as well as principles independent of ZFC, such as the Continuum Hypothesis and Martin’s Axiom. These tools and principles have applications in a wide range of areas of mathematics, from topology and measure theory to group theory and the theory of C*-algebras.

Course Structure

The course consists of theoretical lectures delivered by the instructor and problem-solving sessions in which exercises assigned during the course will be discussed. Students will be encouraged to participate actively by presenting their solutions at the board.

Should the course be delivered in a blended or online format, the necessary adjustments to the arrangements described above may be introduced in order to ensure that the programme set out in this syllabus is completed.

Required Prerequisites

Basic knowledge of the ZFC axiom system for set theory, familiarity with the basics of ordinal and cardinal arithmetic. A sound knowledge of general topology. To that aim it's enough to have taken (or be taking at the same time) the first module of the course "Istituzioni di Geometria Superiore" (Set-theoretic Topology).

Attendance of Lessons

Strongly recommended.

Detailed Course Content

The provisional syllabus will be the following:

- Ramsey theory: the finite and infinite versions of Ramsey’s theorem. Determination of a few Ramsey numbers. The Erdős–Rado theorem. Schur’s theorem and an application of Ramsey theory to number theory. Applications of Ramsey Theory to general topology.

- Martin’s Axiom and some of its consequences in topology (a proof of the consistency of the Suslin Hypothesis) in set theory (the construction of a Ramsey ultrafilter and the regularity of the continuum) and in measure theory.

- Cardinal invariants of the continuum and Cichoń’s diagram.

- Club and stationary sets. The pressing-down lemma. A proof of Silver’s theorem. Jensen’s diamond principle.

- Trees. Aronszajn and Suslin trees. A proof of the independence of the Suslin Hypothesis.

Course Planning

	Subjects	Text References
1	Ramsey Theory.
2	Martin's Axiom.
3	Cardinal invariants of the continuum.
4	Club and stationary sets.
5	Trees.

Learning Assessment

Learning Assessment Procedures

There will be an oral exam which will test both knowledge of the main course content and mastery of the main techniques covered by the course. Learning assessment may also be carried out on-line, should the conditions require it. To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l'integrazione Attiva e Partecipata – Servizi per le Disabilità e/o i DSA) referring teacher within their department (https://www.cinap.unict.it/content/referenti).

Examples of frequently asked questions and / or exercises

- Clarify the relationship between separability and countable chain condition both in the realm of a general topological space and in that of linearly ordered topological spaces.

- Explain why the statement of MA(k) is true in ZFC if k is countable and false in ZFC if k is the continuum.

- Clarify the impact of Martin's Axiom on the countable chain condition of a finitely supported product of partial orders.

- Derive the finite version of Ramsey Theorem from its infinite version using the compactness principle.

- Exhibit an example of a stationary subset of a regular uncountable cardinal which is not a club.

Degree Course in

Mathematics