April - July 2021,

**"Around finite basis results for quasi-orders"**, R. Carroy (Turin).*Course duration:*30 h.*CFU:*6.Course InfoMore information are available at the following link .

LecturesA finite set of minimal elements in a class of objects, for a given quasi-order, is the typical example of a finite basis. Many important results, notably in descriptive set theory, consist in giving the existence of a finite basis. The objective of this series of lectures is to first see some examples of such results, then to introduce the quasi-orders which always admit finite bases, also called well-quasi-orders, and their theory. We will also discuss various ways to prove the impossibility of a finite basis result.

- Lecture 1: Raphaël Carroy, April 12th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 2: Raphaël Carroy, April 16th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 3: Raphaël Carroy, April 19th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 4: Raphaël Carroy, April 23rd, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 5: Raphaël Carroy, April 26th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 6: Raphaël Carroy, April 30th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 7: Raphaël Carroy, May 3rd, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 8: Raphaël Carroy, May 7th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 9: Raphaël Carroy, May 10th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 10: Raphaël Carroy, May 14th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 11: Raphaël Carroy, May 17th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 12: Raphaël Carroy, May 21st, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 13: Raphaël Carroy, May 24th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 14: Raphaël Carroy, May 28th, 2021, 10.45-12.30 (In remoto al seguente link )
- Lecture 15: Raphaël Carroy, May 31st, 2021, 10.45-12.30 (In remoto al seguente link )

March - April 2021,

**"Logic methods for Banach spaces"**, J. Iovino (UTSA Texas).*Course duration:*15 h.*CFU:*3.Course InfoMore information are available at the following link .

Lectures- 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.

- Lecture 1:
~~José Iovino, March 12th, 2021, 10.45-12.30 (Aula 4 Palazzo Campana (in remoto al seguente link ))~~**CANCELED** - Lecture 1: José Iovino, March 15th, 2021, 10.45-12.30 (Aula 4 Palazzo Campana (in remoto al seguente link ))
- Lecture 2: José Iovino, March 19th, 2021, 10.45-12.30 (Aula 4 Palazzo Campana (in remoto al seguente link ))
- Lecture 3: José Iovino, March 22nd, 2021, 10.45-12.30 (Aula 4 Palazzo Campana (in remoto al seguente link ))
- Lecture 4: José Iovino, March 26th, 2021, 10.45-12.30 (Aula 4 Palazzo Campana (in remoto al seguente link ))
- Lecture 5: José Iovino, March 29th, 2021, 10.45-12.30 (Aula 4 Palazzo Campana (in remoto al seguente link ))
- Lecture 6: José Iovino, April 2nd, 2021, 10.45-12.30 (Aula 4 Palazzo Campana (in remoto al seguente link ))
- Lecture 7: José Iovino, April 6th, 2021, 10.45-12.30 (Aula 4 Palazzo Campana (in remoto al seguente link ))
- Lecture 8: José Iovino, April 9th, 2021, 10.45-12.30 (Aula 4 Palazzo Campana (in remoto al seguente link ))

May 2020,

**"Logic methods for Banach spaces"**, R. Carroy (Turin), J. Iovino (UTSA Texas).*Course duration:*24 h.Course InfoCourse shared between Milan and Turin university.

LecturesJosé 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.

- 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.

- Lecture 1:
~~José Iovino, May 8th, 2020, 11.15-12.30 (Milano, TBA)~~**CANCELED** - Lecture 1:
~~José Iovino, May 8th, 2020, 14.00-15.15 (Milano, TBA)~~**CANCELED** - Lecture 1:
~~Raphaël Carroy, May 8th, 2020, 15.30-16.45 (Milano, TBA)~~**CANCELED** - Lecture 1:
~~José Iovino, May 15th, 2020, 11.15-12.30 (Torino, TBA)~~**CANCELED** - Lecture 1:
~~José Iovino, May 15th, 2020, 14.00-15.15 (Torino, TBA)~~**CANCELED** - Lecture 1:
~~Raphaël Carroy, May 15th, 2020, 15.30-16.45 (Torino, TBA)~~**CANCELED** - Lecture 1:
~~José Iovino, May 22nd, 2020, 11.15-12.30 (Milano, TBA)~~**CANCELED** - Lecture 1:
~~José Iovino, May 22nd, 2020, 14.00-15.15 (Milano, TBA)~~**CANCELED** - Lecture 1:
~~Raphaël Carroy, May 22nd, 2020, 15.30-16.45 (Milano, TBA)~~**CANCELED** - Lecture 1:
~~José Iovino, May 29th, 2020, 11.15-12.30 (Torino, TBA)~~**CANCELED** - Lecture 1:
~~José Iovino, May 29th, 2020, 14.00-15.15 (Torino, TBA)~~**CANCELED** - Lecture 1:
~~Raphaël Carroy, May 29th, 2020, 15.30-16.45 (Torino, TBA)~~**CANCELED**

April 2020,

**"Generalized Descriptive Set Theory"**, L. Motto Ros (Turin).*Course duration:*30 h.*CFU:*6.Course Info**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).LecturesGeneralized 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.

- 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).

- Lecture 2:
~~Luca Motto Ros, April 3rd, 2020, 10.30-14.30 (Online)~~**CANCELED** - Lecture 2:
~~Luca Motto Ros, April 6th, 2020, 10.30-14.30 (Online)~~**CANCELED** - Lecture 2:
~~Luca Motto Ros, April 7th, 2020, 12.30-16.30 (Online)~~**CANCELED** - Lecture 2:
~~Luca Motto Ros, April 16th, 2020, 10.30-14.30 (Online)~~**CANCELED** - Lecture 2:
~~Luca Motto Ros, April 17th, 2020, 10.30-14.30 (Online)~~**CANCELED** - Lecture 2:
~~Luca Motto Ros, April 23rd, 2020, 10.30-14.30 (Online)~~**CANCELED** - Lecture 2:
~~Luca Motto Ros, April 24th, 2020, 10.30-14.30 (Online)~~**CANCELED**

- Lecture 1: Luca Motto Ros, April 2nd, 2020, 10.30-14.30 (Online) Abstract
April - June 2019,

**"Iterated Forcing"**, M. Viale (Turin).

Course Info*Course duration:*30 h.*CFU:*6.*Materiale didattico:*pdf.LecturesThe 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- 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 4th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)~~**CANCELED** - Lecture 11:
~~Matteo Viale, June 7th, 2019, 14.30-16.30 (Palazzo Campana, Aula 5)~~**CANCELED** - 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)

Titolo: Model theory and groups

Relatrice: Annalisa Conversano (Massey, NZ)

Data: **8 gennaio 2021 ore 9:30 ^{(*)}**

Abstract: Model theory is a branch of mathematical logic with strong connections to most areas of mathematics and theoretical computer science. The last few decades have seen very interesting interactions between model theory and various classes of groups, in particular topological groups and Lie groups. Lie groups are smooth manifolds with a compatible group operation, and they are found everywhere in mathematics and all areas of science. If you have ever encountered a group of matrices during your studies, it was most likely a Lie group.

In this talk recent work on the connections between groups and model theory will be explained, and many examples will be presented, with a view towards future developments.

Zoom Meeting ID: 886 2603 1968

**(*)** La riunione si apre alle ore 9:20 con una chiacchierata informale in italiano. I seminario vero e proprio, in inglese, comincia alle 9:30 e durerà circa un'ora.

**Seminario Math-Lab - Per il riconoscimento vedi pagina Moodle.**