# Ph.D. Courses 2021 – 2022

#### Mathematical methods in climate science

Lecturer: Michael Ghil (UCLA & ENS Paris)

Schedule:

The course will be held from 4 to 13 July 2022 at Dipartimento di Matematica, Università di Pisa, and it will last 10 hours. Streaming Link https://meetings.dm.unipi.it/b/ste-kqb-hw5-q81

4-07-2022 10:30-12:30 Lecture 1: Observations and planetary flow theory. Atmospheric low-frequency variability (LFV) and long-range forecasting (LRF), Preliminary Slides, v1.3.2, eprint doi:10.5281/zenodo.4765825 .

6-07-2022 10:30-12:30 Lecture 2: Energy balance models, paleoclimate & “tipping points,” Preliminary Slides v1.3.1 doi:10.5281/zenodo.4765734 .

15:00-17:00 Seminar 1: TBA

8-07-2022 10:30- 12:30 Lecture 3 Nonlinear & stochastic models—Random dynamical systems, , Preliminary Slides v1.3.2 doi:10.5281/zenodo.4765865 .

15:00-17:00 Seminar 2: S. Vaienti “Thermodynamics and limit theorems for random open dynamical systems”.

11-07-2022 10:30- 12:30 Lecture 4: Advanced spectral methods, nonlinear dynamics, and the Nile River, Preliminary Slides v1.3.1, eprint doi:10.5281/zenodo.4765847 .

15:00-17:00 Seminar 3: S. Vaienti “Thermodynamics and limit theorems for random open dynamical systems”.

13-07-2022 10:30- 12:30 Lecture 5: Applications to the wind-driven ocean circulation, Preliminary Slides v1.3.1, eprint doi:10.5281/zenodo.4765847.

For further information, please visit the following webpage

#### Long-time asymptotic and criticality in random dynamics

Lecturers: Mauro Mariani and Giacomo Di Gesù. The first lecturer will cover 18 hours of the course, while the second one 14 hours.

Schedule: Thursday, May 5th, at 11 am, at Aula Seminari of the Department of Mathematics. Following lectures every Thursday and Friday, 11 am, Aula Seminari, starting from Thursday, May 12th. No lecture will be delivered on Friday the 6th.

Description: The qualitative behavior of random evolutions in the long-time asymptotic is a classical subject, which has recently found new motivations in high dimensional optimization. The course provides an introduction to classical and recent results concerning the long-time behavior of some classes of Markov processes. In the first part, we will introduce some tools typical of potential theory in an elementary context, with a focus on the reversibility non-reversibility paradigm. In the second part of the class, we will focus on establishing recent results for more involved models and infinite-dimensional dynamics.

Syllabus:

1. Ergodicity and long-time behavior of Markov processes.
2. Potential theory and spectral analysis for processes on graphs.
3. Applications to statistical mechanics models.

#### A cohomological version of the non-abelian Hodge correspondence

Lecturer: Mark De Cataldo (Stony Brook University – New York, USA)

Schedule: May 30, June 1, June 3, and June 6, from 10 am to 12 am

Description: Let $X$ be a compact Riemann surface, that is a smooth projective complex algebraic curve. The non-abelian Hodge correspondence establishes a relation between the moduli space of $n$-dimensional representations of the fundamental group of $X$, the moduli space of pairs $(E, D)$, where $E$ is a rank $n$ holomorphic vector bundle and $D$ is a flat holomorphic connection on $E$, and the moduli space of Higgs bundles, i.e., the pairs $(E, \Phi)$, where $E$ is a rank $n$ holomorphic vector bundle on $X$ and $\Phi\colon E\to E\otimes T^\ast X$ is a morphism of holomorphic vector bundles. By imposing suitable stability conditions or irreducible conditions, one can endow these spaces with the structure of algebraic varieties. The non-abelian Hodge correspondence states that these moduli spaces are diffeomorphic. On the other hand, these moduli spaces are not isomorphic as algebraic varieties. The $P=W$ conjecture states that a filtration in cohomology, which arises naturally when one considers the moduli of Higgs bundles, corresponds to the weight filtration of the mixed Hodge structure of the moduli space of representations of the fundamental group.

De Cataldo’s course will introduce the moduli spaces of Higgs bundles (and the Hitchin fibration), the moduli space of flat connections, the $P=W$ conjecture, first over the complex field and later over an arbitrary field, explaining how to readapt the notions and the results in the latter case.

Syllabus:

• Moduli spaces of Higgs bundles and the Hitchin morphism
• Moduli spaces of flat connection and the Hitchin $p$-morphism in positive characteristic
• A cohomological version of the non-abelian Hodge theorem in positive characteristic
• A consequence of $P=W$ for the complex field via finite fields

Bibliography:

#### Mutually enhancing connections between Ergodic Theory, Combinatorics, and Number Theory

Lecturer: Vitaly Bergelson (Ohio State University, USA)

Timetable: The course will last 14 hours (4 lessons of 3h 30min each from 02:30 p.m. to 06:00 p.m.): the first lesson will take place in Aula G (Polo Fibonacci) on May 30, the second and third one (May 31 and June 1) will take place in Aula Magna (Department of Mathematics), and the last one in Aula E1 (Polo Fibonacci) on June 3. It is possible to attend the course online: to register on Teams platform please send an email message to moreno.pierobon@phd.unipi.it

Syllabus:

• Recurrences and multiple recurrences in topological and measurable dynamics.
• Furstenberg’s Principle of Correspondence and Ramsey’s Ergodic Theory.
• Problems and results involving prime numbers.
• A look at Sarnak’s conjecture.
• Some open problems and conjectures.

References

1. Vitaly Bergelson. Ergodic Ramsey Theory – an update, Ergodic Theory of Zd-actions (edited by M. Pollicott and K. Schmidt), London Math. Soc. Lecture Note Series 228 (1996), 1-61.
2. Vitaly Bergelson. Combinatorial and Diophantine Applications of Ergodic Theory (with appendices by A. Leibman and by A. Quas and M. Wierdl), Handbook of Dynamical Systems, vol. 1B, B. Hasselblatt and A. Katok, eds., Elsevier, 2006, 745-841.
3. Harry Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, 2014.

One may find the first two references on Bergelson’s personal website.

#### An introduction to fractional calculus: fundamental ideas and numerics

Lecturer: Fabio Durastante

Description of the course: you may find it on the following page.

#### Model theory

Lecturer: Rosario Mennuni

Timetable: Tuesday and Thursday, 2.00 – 4.00 pm, starting from March 1, 2022.

Venue: Aula Magna, Department of Mathematics.

Duration: 30 hours.

Language: Italian or English, depending on the audience.

Preliminary meeting: 21st February 14:00, Aula Riunioni (Department of Mathematics)
and online (see below).

Prerequisites: Basic properties of ordinals and of cardinal arithmetic. Familiarity with basic
notions in first-order logic are recommended, but not required (necessary definitions and
results will be recalled at the start of the course).

Syllabus:

• Review of first-order structures and theories, compactness, Löwenheim–Skolem, elementary extensions.
• Quantifier elimination, back-and-forth, applications.
• Types, type spaces, saturated models, omitting types, prime models, Ryll-Nardzewski’s
• Theorem.
• One or two final topics, are to be determined depending on the attendees’ interests.
Possible topics include: Fraïssé limits, Morley’s Theorem, basics of stability theory, o-minimality, structures in positive logic, structures in continuous logic, and elimination of imaginaries.

Online: Link to the course’s team on Microsoft Teams: https://tinyurl.com/modelliPisa22

Notes: If you are interested in attending, please send an email to the address below and request to join the team above as soon as possible.

#### Combinatorial topology and group theory

Lecturer: Giovanni Paolini (“Amazon Web Services” at California Institute of Technology in Pasadena, USA).

Description of the course: you may find it on the following page.

Time: The course will last 10 hours (divided into 2-hour lessons),

Preliminary meeting: Thursday, November 4, 2021, from 17:00 to 18:00.