### Table des matières

# Web-page of the Geneva University tropical group

PhD graduated: Kristin Shaw (December 2011), Lionel Lang (December 2014), Nikita Kalinin (December 2015), Mikhail Shkolnikov (June 2017), Johannes Josi (February 2018).

Current members: Thomas Blomme, Francesca Carocci, Slava Goncharov, Grigory Mikhalkin, Antoine Toussaint.

Alumni: Ivan Bazhov, Johan Bjorklund, Rémi Crétois, Weronika Czerniawska, Yi-Ning Hsiao, Jens Forsgard, Maxim Karev, Ilya Karzhemanov, Sergei Lanzat, Michele Nesci, Alina Pavlikova, Mikhail Pirogov, Johannes Rau, Arthur Renaudineau.

We organize several seminars:

Séminaire "Fables Géométriques".

pre-2017 (historical) Battelle Seminar and

Tropical working group Seminar.

Also, you can check how tropical curves (and hypersurfaces, in general) emerge from abelian sandpile models: tropicalsand

# Seminars and conferences

Prof. Ilia Itenberg (Sorbonne University), Friday, March 15, SM 01-05, 15h15-17h

“Basic algebra and algebraic geometry special talk: Real plane sextic curves without real singular points”

We will start with a brief introduction to topology of real algebraic curves, and then will discuss in more details the case of curves of degree 6 in the real projective plane. We will prove that the equisingular deformation type of a simple real plane sextic curve with smooth real part is determined by its real homological type, that is, the polarization, exceptional divisors, and real structure recorded in the homology of the covering K3-surface (this is a joint work with Alex Degtyarev).

Alexander Bobenko (TU Berlin), Feb 16, 2024, at 14h30, Salle 01-05

“Dimers and M-curves”

We develop a general approach to dimer models analogous to Krichever’s scheme in the theory of integrable systems. This leads to dimer models on doubly periodic bipartite graphs with quasiperiodic positive weights. This generalization from Harnack curves to general M-curves leads to transparent algebro-geometric structures. In particular explicit formulas for the Ronkin function and surface tension as integrals of meromorphic differentials on M-curves are obtained. Based on Schottky uniformizations of Riemann surfaces we compute the weights and dimer configurations. The computational results are in complete agreement with the theoretical predictions. Also relation to discrete conformal mappings and to hyperbolic polyhedra is explained. This is a joint work with N. Bobenko and Yu. Suris.

Francesca Carocci (Genève), Dec 8, 14h30, Salle 06-13

“Degenerations of Limit linear series”

Maps to projective space are given by basepoint-free linear series, thus these are key to understanding the extrinsic geometry of algebraic curves. How does a linear series degenerate when the underlying curve degenerates and becomes nodal? Eisenbud and Harris gave a satisfactory answer to this question when the nodal curve is of compact type. Eisenbud-Harris's theory of limit linear series gives proofs via degenerations of many foundational results in Brill–Noether theory, and it is powerful enough to answer several birational geometry questions on the moduli space of curves. I will report on a joint work in progress with Lucaq Battistella and Jonathan Wise, in which we review this question from a moduli-theoretic and logarithmic perspective. The logarithmic prospective helps understanding the rich polyhedral and combinatorial structures underlying degenerations of linear series. These are linked with matroids and Bruhat-Titts buildings.

Diego MATESSI (Milano), Dec 4, 15h, Salle 06-13

“Tropical mirror symmetry and real Calabi-Yaus”

I will present some work in progress jont with Arthur Renaudineau. The goal is to understand the topology of real Calabi-Yaus by combining the Renaudineau-Shaw spectral sequence with mirror symmetry. We will consider mirror pairs of Calabi-Yau hypersurfaces X and X' in toric varieties associated to dual reflexive polytopes. The first step is to prove an isomorphism between tropical homology groups of X and X', reproducing the famous mirror symmetry exchange in hodge numbers. We then expect that the boundary maps in the Renaudineau-Shaw spectral sequence, computing the homology of the real Calabi-Yaus, can be interpreted, on the mirror side, using classical operations on homology.

Thomas Blomme, université de Genève, Thursday, Nov 9, 16h15, Room 1-15.

“Gromov-Witten invariants of bielliptic surfaces”

Bielliptic surfaces were classified by Bagnera & de Francis more than a century ago. They form a family spread into seven subfamilies of the Kodaira-Enriques surface classification which have nearly trivial canonical class in the sense that it is non-zero, but torsion. Thus, the virtual dimension of the moduli space of curves only depends on the genus, and contrarily to abelian and K3 surfaces, it yields non-zero invariants. In this talk we'll focus on some techniques to compute GW invariants of these surfaces along with some regularity properties.

Antoine Toussaint, université de Genève, Monday, Oct 23, at 15h, Salle 06-13

“Real Structures of Phase Tropical Surfaces”

Phase tropical surfaces can appear as a limit of a 1-parameter family of smooth complex algebraic surfaces. A phase tropical surface admits a stratified fibration over a smooth tropical surface. We study the real structures compatible with this fibration and give a description in terms of tropical cohomology. As an application, we deduce combinatorial criteria for the type of a real structure of a phase tropical surface. Time permitting, we will also discuss the connection with Renaudineau and Shaw's spectral sequence and Kalinin's spectral sequence.

Ozgur CEYHAN (University of Luxembourg), Monday, Oct 16, at 15h, Salle 06-13

“Complexities in backpropagation and tropicalization in neural networks”

The backpropagation algorithm and its variations are the primary training method of multi-layered neural networks. The backpropagation is a recursive gradient descent technique that works on large matrices. This talk explores backpropagation via tropical linear algebra and introduces multi-layered tropical neural networks as universal approximators. After giving a tropical reformulation of the backpropagation algorithm, we verify the algorithmic complexity is substantially lower than the usual backpropagation as the tropical arithmetic is free of the complexity of usual multiplication.

Gurvan Mével (Université de Nantes), Wednesday, Oct 18, at 14h15, Salle 06-13

“Universal polynomials for coefficients of tropical refined invariant in genus 0”

In enumerative geometry, some numbers of curves on surfaces are known to behave polynomially when the cogenus is fixed and the linear system varies, whereas it grows more than exponentially fast when the genus is fixed. In the first case, Göttsche's conjecture expresses the generating series of these numbers in terms of universal polynomials.

Tropical refined invariants are polynomials resulting of a weird way of counting curves, but linked with the previous enumerations. When the genus is fixed, Brugallé and Jaramillo-Puentes proved that some coefficients of these polynomials behave polynomially, bringing back a Göttsche's conjecture in a dual and refined setting. In this talk we will investigate the existence of universal polynomials for these coefficients.

# Geneva-Neuchâtel Symplectic Geometry Seminar

Schedule and more details: seminar page

We had our page http://www.unige.ch/math/folks/langl/battelle/

Also, there is **Séminaire de Géométrie Tropicale** in Paris:http://erwan.brugalle.perso.math.cnrs.fr/Seminaires/Geotrop/Geotrop.html

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