Advanced Geometry

The course is an introduction to the combinatorial aspects of set theory. Students will learn to use tools such as Ramsey’s theorem, the Delta-system Lemma, and the pressing-down Lemma, as well as methods for proving consistency and independence results that today find applications in a wide range of areas of mathematics, from topology to measure theory, from group theory to the theory of C*-algebras

The tentative program will be as follows:

• Ramsey theory (finite and infinite versions of Ramsey’s theorem, the Erdős–Rado theorem, Schur’s theorem, and an application of Ramsey theory in number theory).

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

• Cardinal invariants of the continuum, Cichoń’s diagram.

• Club and stationary sets, the pressing-down Lemma. Proof of Silver’s theorem. Jensen’s Diamond principle.

• Trees. Aronszajn and Suslin trees. Proof of the independence of Suslin’s Hypothesis.

• Introduction to Paul Cohen’s forcing method. Proof of the independence of the Continuum Hypothesis.

• Notes by the lecturer of the course.
• T. Jech, "Set Theory: The Third Millennium Edition, revised and expanded", Springer, 2003.
•  K. Kunen, "Set Theory: An Introduction to Independence Proofs", North Holland, 1986.