### Equipe Analyse Algébrique

Responsable d’équipe : François Loeser
Responsables adjoints : Penka GEORGIEVA, Maxime ZAVIDOVIQUE
Gestionnaire : Julienne PASSAVE

### L’après-midi de l’équipe, le 10/03/2022

Programme de l’après-midi de l’équipe AA, le 10/03/2022

En salle 15.16.413

16h-17h30 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)
16h00 – 16h40 Erman Cineli16h50 – 17h30 Marvin Hahn
17h40 – 18h : Réunion de l’équipe

À partir de 18h15 : pot dans la salle de convivialité

Résumés des exposés :
Erman Cineli
Title: Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Abstract: In this talk I will introduce barcode entropy and discuss its connections to topological entropy. The barcode entropy is a Floer-theoretic invariant of a compactly supported Hamiltonian diffeomorphism, measuring, roughly speaking, the exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. The topological entropy bounds from above the barcode entropy and, conversely, the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set. As a consequence, the two quantities are equal for Hamiltonian diffeomorphisms of closed surfaces. The talk is based on a joint work with Viktor Ginzburg and Basak Gurel.

Marvin Hahn
Title: Quasimodularity of weighted Hurwitz numbers
Abstract: Hurwitz numbers enumerate branched mophisms between Riemann surfaces with fixed numerical data. When the target surface is an elliptic curve, these enumerative invariants are intimately related to mirror symmetry, which e.g. predicts a quasimodular structure of the generating series of elliptic Hurwitz numbers. This prediction was confirmed in seminal work of Dijkgraaf in 1995. In the past years, several variants of Hurwitz numbers were introduced that arise in various different contexts, e.g. monotone Hurwitz numbers in random matrix theory or strictly monotone Hurwitz numbers in the theory of Grothendieck dessins d’enfants. Recently, in work of Guay-Paquet and Harnard a unified framework for these different variants was introduced under the name of weighted Hurwitz numbers. Here the idea is to consider a Hurwitz numbers-like enumeration that depends on a weight function. For different choices of this weight function, weighted Hurwitz numbers specialise to essentially all known variants of Hurwitz numbers. In this talk, we present results regarding the structure of elliptic Hurwitz numbers. In particular, we generalise Dijkgraaf’s work to this case and derive a quasimodular structure of generating series. Our methods heavily rely on tropical geometry. This talk is based on a joint work in progress with Danilo Lewanski and Jonas Wahl.

### Journée de rentrée, le 21/10/2021

Programme de la journée de rentrée de l’équipe AA, le 21/10/2021

En salle 15.16.413

15h-17h20 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)

15h-15h40 : Dusan Joksimovic

15h50-16h30 : Bernhard Reinke

16h40-17h20 : Jinhe Ye

17h30-18h : Réunion de l’équipe

À partir de 18h15 : pot sous la barre 24-25 niveau Jussieu

Résumés des exposés :

Dusan Joksimovic
Title : No symplectic-Lipschitz structures on $S^{2n \geq 4}$

Abstract: One of the central questions in $C^0$-symplectic geometry is whether spheres (of dimension at least 4) admit symplectic topological atlas (i.e. atlas whose transition functions are symplectic homeomorphisms). In this talk, we will prove that the answer is « no » if we replace the word « topological » with « Lipschitz ». More precisely, we will prove that every closed symplectic-Lipschitz manifold has non-vanishing even degree cohomology groups with real coefficients. The proof is based on the fact that one can define analogs of differential forms and de Rham complex on Lipschitz manifolds which share similar properties as in the smooth setting.

Bernhard Reinke
Title : Connections between complex dynamics and algebra

Abstract: Complex dynamics and algebra are deeply connected. I will present two examples of their connection: the dynamics of root-finding methods, and iterated monodromy groups of transcendental maps.

Finding roots of a polynomial is a fundamental numerical problem. Many root-finding methods, such as the Newton’s method or the Weierstrass/Durand-Kerner method can be understood as complex dynamical systems. I will sketch how computer algebraic tools were used to show that the Weierstrass method is not generally convergent.

Iterated monodromy groups are self-similar groups associated to partial self-coverings. I will give an overview of iterated monodromy groups of post-singularly finite entire transcendental functions. These groups act self-similarly on a regular rooted tree, but in contrast to IMGs of rational functions, every vertex of the tree has countably infinite degree.

I will discuss the similarities and differences of IMGs of entire transcendental functions and of polynomials, in particular in the direction of amenability.

Jinhe Ye

Title : Curve-excluding fields

Abstract : Consider the class of fields with Char(K)=0 and x^4+y^4=1 has only 4 solutions in K, we show that this class has a model companion, which we denote by CXF, curve-excluding fields. Curve-excluding fields provides examples to various questions. Model theoretically, they are model complete and algebraically bounded. Field theoretically, they are not large. This answers a question of Junker and Macintyre negatively. Joint work with Will Johnson and Erik Walsberg.