Programme :

Lectures :

• Erwan Lanneau (Université de Grenoble): Pseudo-Anosov maps and their expansions factors.
• Maria Cumplido Cabello (Universidad de Sevilla): Parabolic subgroups of Artin-Tits groups.
• Alexandra Kjuchukova (University of Notre Dame, Indiana, USA): Knot invariants from branched covers.

Special talk :

• Erwan Brugallé (Université de Nantes): Enumeration of rational curves in algebraic surfaces or symplectic 4-varieties.

Schedule :

	Monday 3rd	Tuesday 4th	Wednesday 5th	Thursday 6th
9:00 - 9:30	CUMPLIDO I	KJUCHUKOVA II	LANNEAU II	LANNEAU III
9:30 - 10:00
10:00 - 10:30	Coffee Break	Coffee Break	Coffee Break	Coffee Break
10:30 - 11:00	LANNEAU I	CUMPLIDO II	KJUCHUKOVA III	Bagherifard
11:00 - 11:30				Chemin
11:30 - 12:00	Lunch	Lunch	Lunch	Gavazzi
12:00 - 14:00				Farewell Lunch
14:00 - 14:30	KJUCHUKOVA I	BRUGALLÉ	CUMPLIDO III
14:30 - 15:00
15:00 - 15:30	Di Prisa	Malech	Zhang
15:30 - 16:00	Coffee Break	Coffee Break	Coffee Break
16:00 - 16:30	Haïoun	Flash talks	Jouteur
16:30 - 17:00	Haladjian	Poster session	Panda

Abstracts :

Pseudo-Anosov maps and their expansions factors

Erwan Lanneau (Université de Grenoble)

Pseudo-Anosov maps play an important role in the study of the geometry and dynamics of moduli spaces and mapping class groups, such as braid groups. A pseudo-Anosov map has an expansion factor (or dilatation), which records the exponential growth rate of the lengths of the curves under iterations. Thurston's train track theory relates the dynamics of pseudo-Anosov maps to that of Perron-Frobenius matrices: their dilatation is then the Perron-Frobenius eigenvalue. Less well known is the Rauzy-Veech induction, which is a powerful machinery for computing dilatations. In this talk, we will first explain some classical constructions of pseudo-Anosov maps (such as the Thurston-Veech construction). We will then look at the Rauzy-Veech induction and use it to get several applications.

Parabolic Subgroups of Artin-Tits Groups

Maria Cumplido Cabello (Universidad de Sevilla)

Artin-Tits groups are deeply connected to braid groups, which are a fundamental example of Artin-Tits groups. These connections reveal a rich interplay between the combinatorial, geometric, and topological aspects of both group families. In particular, braid groups and their actions on configuration spaces and the curve complex provide insights into broader questions about Artin groups and their representations. Given an Artin-Tits group with its standard presentation, a standard parabolic subgroup is the subgroup generated by a subset of the group's generators. More generally, a parabolic subgroup refers to any conjugate of a standard parabolic subgroup. These subgroups are not only a natural part of the group's structure but also play a crucial role in understanding the algebraic and geometric properties of Artin-Tits groups and the complexes on which they act. In this mini-course, we will explore the importance of parabolic subgroups in studying Artin groups. We will also discuss the key techniques and ideas that have driven recent breakthroughs in the field, including tools from combinatorics, topology, and geometric group theory. The goal is to provide a comprehensive picture of how parabolic subgroups and related constructions have shaped our understanding of Artin groups and their rich mathematical landscape.

Knot invariants from branched covers

Alexandra Kjuchukova (University of Notre Dame, Indiana, USA)

(See PDF file)

Enumeration of rational curves in algebraic surfaces or symplectic 4-varieties

Erwan Brugallé (Université de Nantes)

Over the last 20 years, considerable progress has been made in the study of the enumerative geometry of rational curves in algebraic or symplectic varieties, particularly in the case of real curves. In this talk, I will present methods for degenerating/cutting ambient varieties that are particularly effective in dimension 2 (complex) / 4 (real). The talk will focus on curves in surfaces. Time permitting, I will also discuss recent variations in algebraic geometry, over any field, of Gromov-Witten and Welschinger invariants.