The school is two-week long, with free Wednesday afternoons. There are three introductory courses and three advanced courses. Typically, an introductory course consists of 9-hour lectures including exercise sessions and an advanced course consists of 6-hour lectures including exercise sessions.  There will be at least one tutorial session every day.

 Introductory courses

Course 1

Title: Some topics in Diophantine geometry

Duration: 9h
Lecturer’s NAME: LORENZO GARCIA   
Lecturer’s given name: Elisa
Lecturer’s gender: Female
Lecturer’s institution: Université de Neuchâtel
Lecturer’s country: Switzerland

Abstract of the course: The goal of this introductory course is to give an overview of some classical tools and results in diophantine geometry, starting from the theory of heights to Falting's theorem and some open problems. Because of lack of time, certain results will be assumed without proofs (but precise references will be provided).

Detailed plan:

  • Absolute values on number fields and the product formula.
  • Heights in projective spaces.
  • Some results on the geometry of curves and abelian varieties.
  • The Néron-Tate height on abelian varieties.
  • The (Weak) Mordell-Weil theorem.
  • Falting's theorem and proof strategy.
  • Some open problems about points of small height.

References:
There are a lot of very good online notes containing all the above standard topics. Some classic books are:
[HS 00] M. Hindry, J.H. Silverman, Diophantine Geometry: An Introduction, GTM 201, Springer-Verlag, New York, 2000.
[BG 06] E. Bombieri, W. Gubler, Heights in diophantine geometry, New Mathematical Monographs, 4. Cambridge University Press, Cambridge, 2006.

Course 2

Title: Galois theory of linear difference equations and applications

Duration: 9h
Lecturer’s NAME: DREYFUS
Lecturer’s given name: Thomas
Lecturer’s gender: Male
Lecturer’s institution: Université de Strasbourg
Lecturer's country: France


Abstract of the course: The aim of this course is to develop theory and algorithms that will allow us to understand the algebraic relations between the solutions of linear discrete equations. We shall focus on examples such as holonomic sequences, M-functions, q-difference equations. We assume the reader is familiar with basic notions in algebra (groups, rings, fields, ideals, etc.)

Detailed plan:

  • Historical survey: classical Galois theory, Picard-Vessiot theory for linear differential equations.
  • Algebraic varieties and linear algebraic groups: varieties and ideals, morphisms, coordinate rings, linear algebraic group, Lie-Kolchin theorem, Torsors.
  • Picard-Vessiot extensions: difference algebra, Picard-Vessiot extensions, pseudofields.
  • Galois groups and Galois correspondence: algebraic structure of the Galois group, Torsor Theorem, Rank one equations and Liouvillian solutions.
  • Applications:  automatic sequences, transcendence of special functions, differential transcendence.

References:
[vdPS 97] M. van der Put, M. Singer, Galois theory of difference equations, Lecture Notes in Mathematics (LNM), 1666. Springer-Verlag, Berlin, 1997.
[Spr 98] T. A. Springer, Linear algebraic groups. Second edition. Progress in Mathematics, 9. Birkhäuser Boston, Inc., Boston, MA, 1998.
[HSS 16] C. Hardouin, J. Sauloy, M. Singer, Galois theories of linear difference equations: an introduction. Mathematical Surveys and Monographs, volume 211. American Mathematical Society, Providence, RI, 2016.

Course 3

Title: Introduction to transcendental number theory

Duration: 9h
Lecturer’s NAME: WALDSCHMIDT
Lecturer’s given name: Michel
Lecturer’s gender: Male
Lecturer’s institution: Institut de Math. de Jussieu, Université Paris 6
Lecturer’s country: France

Abstract of the course: The first part of the course will be a historical survey of transcendental number theory: what is known? what is expected? Next, we introduce the basic tools for the proofs. A full proof of the Schneider--Lang Criterion will be given. The proofs of more recent results will be outlined, emphasis will be on the ideas and not on technical details.

Detailed plan:

  • Historical survey. Hermite's method: transcendence of e.
  • Lindemann's Theorem: transcendence of \pi. Linear independence: Baker's Theorem; lower bounds for linear forms in logarithms of algebraic numbers. Algebraic independence. State of the art. Conjectures.
  • Tools: criteria for irrationality, transcendence, algebraic independence. Liouville's inequality, heights. Schwarz's lemma. Zero estimates.
  • Schneider--Lang criterion: transcendence of e, \pi, e^{\pi}, . Proof.
  • Baker's method; sketch of proofs.
  • Algebraic independence; transcendence of \Gamma(1/4).
  • E- and G- functions.

- Mahler's method.

Reference:
[Mas 16] Masser, David. Auxiliary polynomials in number theory. Cambridge Tracts in Mathematics, 207. Cambridge University Press, Cambridge, 2016.

 Advanced courses

 Course 1 (CANCELLED)

Title: Mahler's method and finite automata

Duration: 6h    
Lecturer’s NAME: ADAMCZEWSKI
Lecturer’s given name: Boris
Lecturer’s gender: Male
Lecturer’s institution: CNRS and Institut Camille Jordan, Université Claude Bernard Lyon 1
Lecturer's country: France

Abstract of the course: An M-function is a power series satisfying a linear difference equation with polynomial coefficients associated with the Mahler operator that maps z to z^q, for some natural number q > 1.  In 1929, Mahler developed a powerful method for proving transcendence and algebraic independence of values at algebraic points of some particular M-functions. From the perspective of transcendence and algebraic independence, recent results show that M-functions lead to a theory that mirrors the more classical one of Siegel E-functions. However, these two theories also have some fundamental differences. On the one hand, the theory of E-functions takes its roots in classical results such as the transcendence of the numbers   and  , and more generally the Lindemann-Weierstrass theorem, while, on the other hand, the theory of M-functions has a very different raison d'être, which comes from its connection with finite automata. The aim of this lectures is to discuss the links between the theory of M-functions and finite automata.

Detailed plan:

  • Introduction to finite automata (definition, classical examples, characterization)
  • Automatic sequences and automatic numbers (main problems and conjectures)
  • Introduction to M-functions (definition, main properties)
  • Links between M-functions and finite automata (proof of Christol's theorem and that automatic series are M-function)
  • Transcendence and algebraic independence of values of M-functions (statements of the main results, ideas of proofs, applications, perspectives)

References:

[AS 03] J.-P. Alllouche and J. Shallit, Automatic sequences. Theory, applications, generalizations, Cambridge University Press, Cambridge, 2003.
[Nis 97] Ku. Nishioka, Mahler functions and transcendence, Lecture Notes in Math. 1631, Springer-Verlag, Berlin, 1997.

Course 2

Title: Ax-Lindemann-Weierstrass theorems and Differential Galois Theory

Duration: 6h    
Lecturer’s NAME: CASALE    
Lecturer’s given name: Guy
Lecturer’s gender: Male
Lecturer’s institution: Institut de Math. de Rennes, Université de Rennes 1
Lecturer's country: France

Abstract of the course: The aim of the course is to introduce the Differential Group of various forms of second order differential equations and to explain how this tool can be used to obtain results of functional transcendency such as generalizations of Ax-Lindemann-Weiertrass theorems.

Detailed plan:

  • Second order linear differential equations / Schwarzian equations / Uniformization equation: Differential Galois group and strong minimality.
  • Uniformization of Riemann surfaces, their differential equations and "gauge" / modular correspondences.
  • Principal connection/ connection form /Lie foliation.
  • Proof of Ax-Lindemann-Weierstrass (with derivatives) for solutions of (generalized) uniformization equations.

References:

[CFN 18] G. Casale, J. Freitag and J. Nagloo, Ax-Lindemann-Weierstrass with derivatives for genus zero Fuchsian groups, arXiv:1811.06583, 2018.
[Sha 96] R.W. Sharpe, Differential Geometry, Graduate Texts in Math. 166, Springer, 1996.
[CH 11] T. Crespo and Z. Hajto, Algebraic Groups and Differential Galois Theory, Graduate Studies in Math. 122 AMS, 2011.
[StG 11] H.P. de Saint Gervais, Uniformisation des surfaces de Riemann, retour sur un théorème centenaire, Lyon, ENS Éditions, 2011.

Course 3

Title: o-minimality and diophantine applications

Duration: 6h
Lecturer’s NAME: YAFAEV    
Lecturer’s given name: Andrei
Lecturer’s gender: Male
Lecturer’s institution: University College London
Lecturer's country: UK

Abstract of the course: o-minimality is a branch of mathematical logic (model theory) which proved very useful for applications to some diophantine problems, namely the Zilber-Pink type conjectures on unlikely intersections in certain types of varieties naturally appearing in diophantine geometry (tori, abelian, Shimura, mixed Shimura). We will present an overview of o-minimality and proofs of some cases of the Zilber-Pink conjectures.

Detailed plan:

  • Introduction to Zilber-Pink type problems.
  • Notions of o-minimality, examples, Pila-Wilkie counting theorem.
  • The o-minimal proof of the Manin-Mumford conjecture.
  • Sketch of the proofs of some cases of the Andre-Oort conjecture.
  • (if time permits) Some cases of the general Zilber-Pink conjectures.

References:

[HRSUY 17] P. Habegger, G. Remond, T. Scanlon, E. Ullmo, A. Yafaev, Around the Zilber-Pink conjecture, Panoramas et Syntheses, Vol 52, 2017.
[JW 15] G. Jones and A. Wilkie (edited), o-minimality and diophantine geometry, LMS Lecture Note Series 421, 2015.
[KUY 18] B. Klingler, E. Ullmo, A. Yafaev, ``Bi-algebraic geometry and the Andre-Oort conjecture'' Proceedings of 2015 AMS Summer Institute in Algebraic Geometry, PSPMS 97-2, AMS, 319-360, 2018.