PhD Courses

ATTENTION:

PhD courses may suffer last minutes modification due to coronavirus.

Some courses may be given online: interested students must contact the responsible professor for information and to schedule dates and modality of the course.

Ongoing courses:

TBA

Past:

  • May 2020, "Logic methods for Banach spaces", R. Carroy (Turin), J. Iovino (UTSA Texas).

    Course duration: 24 h.

    Course shared between Milan and Turin university.

    Course Info

    José Iovino:

    • Metric structures. Ultraproducts of metric structures and compactness. Keisler-Shelah theorem for metric structures.
    • Topologies on classes of structures. Model-theoretic forcing and Baire category. Omitting types and Cech completeness.
    • Tao’s metastability. The Uniform Metastability Principle and connections with compactness.
    • Type definability. Connections with combinatorics and the Grothendieck-Ptak double limit theorems.
    Raphaël Carroy:
    • Infinite combinatorics, fronts and barriers, the proof of Ramsey's Theorem and the Nash-Williams extension thereof.
    • Relating infinite combinatorics to Topology: proving topological Ramsey results: the Galvin-Prikry Theorem and Ellentuck's result.
    • Relating infinite combinatorics to Banach space theory: Rosenthal's result and extension thereof.
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    Lectures

      More lectures will be announced.

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    • April 2020, "Generalized Descriptive Set Theory", L. Motto Ros (Turin).

      Course duration: 30 h.
      CFU: 6.

      ATTENTION: ONLINE COURSE. Interested students must send an email to the lecturer.
      On April 2 there will be the first trial meeting on Webex to discuss the organization of the course (including the calendar). Lectures will start on April 6.
      A continuation of the course will be offered by prof. Dimonte in Udine (10 hours - 2CFU).

      Course Info

      Generalized Baire and Cantor spaces: definition and basic properties. Generalized Borel and analytic sets. Codings for uncountable structures, nonseparable spaces and alike. Some classification problems whose complexity can be settled in this framework. The regular case. Links with stability from model theory. The singular case: $\lambda$-Polish spaces.

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      Lectures

      • Lecture 1: Luca Motto Ros, April 2nd, 2020, 10.30-14.30 (Online) Abstract

        First trial meeting on Webex e discussion about the organization of the course (including calendar).

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    • April - June 2019, "Iterated Forcing", M. Viale (Turin).

      Course duration: 30 h.
      CFU: 6.
      Materiale didattico: pdf.

      Course Info

      The teaching material here linked is a draft of a book that will serve as the main reference for the course. I plan to cover in detail chapter 1, then depending on the background of the people attending I may just skim through chapters 3 and 4 or spend more time on them. Then I will make with great care chapters 6 and 7. If time remains we could either try to look at Woodin’s generic absoluteness results for second order arithmetic (chapters 2, 5, 12) or at the iteration theorem for semiproper forcings (chapters 2, 9, 10, and maybe parts of 8 or 11).
      As a reference for the background material, I suppose the audience should be familiar with this: link

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      Lectures

      • Lecture 1: Matteo Viale, April 9th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)
      • Lecture 2: Matteo Viale, April 12th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)
      • Lecture 3: Matteo Viale, April 30th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)
      • Lecture 4: Matteo Viale, May 3rd, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)
      • Lecture 5: Matteo Viale, May 7th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)
      • Lecture 6: Matteo Viale, May 10th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)
      • Lecture 7: Matteo Viale, May 14th, 2019, 14.00-15.30 (Palazzo Campana, Aula 5)
      • Lecture 8: Matteo Viale, May 17th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)
      • Lecture 9: Matteo Viale, May 31st, 2019, 10.30-12.30 (Palazzo Campana, Aula 5)
      • Lecture 10: Matteo Viale, June 3rd, 2019, 10.30-12.30 (Palazzo Campana, Aula 5)
      • Lecture 11: Matteo Viale, June 14th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)
      • Lecture 12: Matteo Viale, June 18th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)
      • Lecture 13: Matteo Viale, June 19th, 2019, 09.30-12.30 (Palazzo Campana, Aula 5)
      • Lecture 14: Matteo Viale, June 20th, 2019, 09.30-12.30 (Palazzo Campana, Aula 5)

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