{"id":79,"date":"2022-03-08T23:14:46","date_gmt":"2022-03-08T23:14:46","guid":{"rendered":"https:\/\/www.dm.unipi.it\/phd\/phd-courses\/"},"modified":"2022-10-04T12:24:54","modified_gmt":"2022-10-04T10:24:54","slug":"ph-d-courses-2021-2022","status":"publish","type":"page","link":"https:\/\/www.dm.unipi.it\/phd\/ph-d-courses\/past-courses\/ph-d-courses-2021-2022\/","title":{"rendered":"Ph.D. Courses 2021 – 2022"},"content":{"rendered":"\n

Mathematical methods in climate science <\/h4>
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Lecturer:<\/strong> Michael Ghil (UCLA & ENS Paris)<\/p>\n\n\n\n

Schedule:<\/strong> <\/p>\n\n\n\n

The course will be held from 4 to 13 July 2022 at Dipartimento di Matematica, Universit\u00e0 di Pisa, and it will last 10 hours. Streaming Link https:\/\/meetings.dm.unipi.it\/b\/ste-kqb-hw5-q81<\/a><\/p>\n\n\n\n

4-07-2022<\/strong> 10:30-12:30 Lecture 1<\/strong>: <\/em>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 .<\/p>\n\n\n\n

6-07-2022<\/strong> 10:30-12:30 Lecture 2<\/strong>: Energy balance models, paleoclimate & \u201ctipping points,\u201d Preliminary Slides v1.3.1 doi:10.5281\/zenodo.4765734 .<\/p>\n\n\n\n

15:00-17:00 Seminar 1<\/strong>: TBA<\/p>\n\n\n\n

8-07-2022<\/strong> 10:30- 12:30 Lecture<\/strong> 3<\/strong> Nonlinear & stochastic models\u2014Random dynamical systems, , Preliminary Slides v1.3.2 doi:10.5281\/zenodo.4765865 .<\/p>\n\n\n\n

15:00-17:00 Seminar 2<\/strong>: S. Vaienti \u201cThermodynamics and limit theorems for random open dynamical systems\u201d.<\/p>\n\n\n\n

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

15:00-17:00 Seminar 3<\/strong>: S. Vaienti \u201cThermodynamics and limit theorems for random open dynamical systems\u201d.<\/p>\n\n\n\n

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

For further information, please visit the following webpage<\/a><\/p>\n<\/div><\/div>\n\n\n\n

Long-time asymptotic and criticality in random dynamics<\/h4>
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Lecturers:<\/strong> Mauro Mariani and Giacomo Di Ges\u00f9. The first lecturer will cover 18 hours of the course, while the second one 14 hours.<\/p>\n\n\n\n

Schedule:<\/strong> 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.<\/p>\n\n\n\n

Description:<\/strong> 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.<\/p>\n\n\n\n

Syllabus:<\/strong><\/p>\n\n\n\n

  1. Ergodicity and long-time behavior of Markov processes.<\/li>
  2. Potential theory and spectral analysis for processes on graphs.<\/li>
  3. Applications to statistical mechanics models.<\/li><\/ol>\n<\/div><\/div>\n\n\n\n

    A cohomological version of the non-abelian Hodge correspondence<\/h4>
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    Lecturer: <\/strong>Mark De Cataldo (Stony Brook University – New York, USA)<\/p>\n\n\n\n

    Schedule:<\/strong> May 30, June 1, June 3, and June 6, from 10 am to 12 am<\/p>\n\n\n\n

    Description: <\/strong>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.<\/p>\n\n\n\n

    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.<\/p>\n\n\n\n

    Syllabus:<\/strong><\/p>\n\n\n\n