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

Current members: Thomas Blomme, Francesca Carocci, Aloïs Demory, Gurvan Mével, 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

Joé Brendel (ETHZ), Friday, Feb 21, 15h15, room 6-13 (Seminaire "Fables Géométriques")

“Split tori in S^2 x S^2, billiards and ball-embeddability”

Abstract: In this talk we will discuss the symplectic classification of Lagrangian tori that split as circles in S^2 x S^2. As it turns out, this classification is equivalent to playing mathematical billiards on a rectangular table. This has many interesting applications, for example to Lagrangian packing and the topological study of the space of Lagrangians. We will focus on one application in particular, asking which Lagrangian tori are contained in the image of a symplectic ball embedding. There are many open questions of more general interest surrounding this property of “ball-embeddability” of Lagrangians, which we will discuss at the end of the talk. This is joint work with Joontae Kim.

Gurvan Mével (UNIGE), Wednesday, Feb 19, 14h00, room 1-07 (Seminaire "Fables Géométriques")

“Floor diagrams and some tropical invariants in positive genus”

Abstract : Göttche-Schroeter invariants are a rational tropical refined invariant, i.e. a polynomial counting genus 0 curves on toric surfaces, that can be computed with a floor diagrams approach. In this talk I will explain that this approach extends in any genus. This gives new invariants, related to ones simultaneously defined by Shustin and Sinichkin. I will then say few words on a quadratically enriched (and not refined !) version of this extension.

Uriel Sinichkin (Tel-Aviv), Wednesday, Feb 5, 14h00, room 1-07 + Zoom (Seminaire "Fables Géométriques")

“Refined Tropical Invariants and Characteristic Numbers”

Abstract: In this talk I will present a generalization of Goettche-Schroeter and Schroeter-Shustin refined counts of tropical curves that splits to a product of terms on small fragments of the curves. This count is invariant in each of the following situations: either genus at most one, or a single contact element, or point conditions in Mikhalkin position. I will compare our results to Mével’s floor diagram approach, and discuss the specialization of the count at q=1, which recovers certain characteristic numbers.

Thomas Blomme (Neuchâtel), Friday, Jan 31, 14h00, room 1-07 (Seminaire "Fables Géométriques")

“Une preuve courte d’une formule de revêtement multiple”

Abstract: Enumérer les courbes de genre g passant par g points dans une surface abélienne est un problème naturel, et d’une difficulté surprenamment inégale en fonction du degré des courbes étudiées. Pour les degrés « primitifs », il est aisé d’obtenir une formule close par une résolution simple et explicite. Pour les classes « divisibles », une telle résolution est en revanche assez fastidieuse et souvent hors de portée. Pour autant, les invariants de ces dernières s’expriment aisément en fonction des invariants primitifs au travers de la formule de revêtement multiple, conjecturée par G. Oberdieck. Dans cet exposé, on va montrer comment la géométrie tropicale permet de prouver cette formule en esquivant toute forme concrète d’énumération.

Ajith Urundolil-Kumaran (Cambridge), Wednesday, Dec 11, 14h00, room 06-13 (Seminaire "Fables Géométriques")

“Tropical correspondence theorems, Scattering diagrams and Quantum Mirrors”

Abstract: The mirror algebras constructed in the Gross-Siebert program come with a natural trace pairing. The Frobenius conjecture gives an enumerative interpretation for this pairing. In the Log Calabi-Yau surface case there exists a deformation quantization of the mirror algebra. We prove a quantum version of the Frobenius conjecture by interpreting it as a refined tropical correspondence theorem. This is joint work with Patrick Kennedy-Hunt and Qaasim Shafi.

Marvin HAHN (Dublin), Wednesday, Dec 4, 14h00, room 06-13 (Seminaire "Fables Géométriques")

“A tropical twist on Hurwitz numbers”

Hurwitz numbers count branched morphisms between Riemann surfaces with fixed numerical data. While a classical invariant, having been introduced in the 19th century, Hurwitz numbers are an active topic of study, among others due to their interplay with Gromov-Witten theory and their role in mirror symmetry. In recent work of Chapuy and Dołęga a non-orientable generalisation of Hurwitz numbers was introduced, so-called b-Hurwitz numbers. These invariants are a weighted enumeration of maps between non-orientable surfaces weighted by a power of a parameter b. This parameter should be viewed as measuring the non-orientability of the involved covers. For b=0, one recovers classical Hurwitz numbers, while b=1 represents a non-weighted count of non-orientable maps yielding so-called twisted Hurwitz numbers. In this talk, we derive a combinatorial model of twisted Hurwitz numbers via tropical geometry and employ it to derive a wide array of new structural properties. This talk is based on joint work with Hannah Markwig.

Aloïs DEMORY (Genève), Wednesday, Nov 20, 14h00, room 06-13 (Seminaire "Fables Géométriques")

“Primitive real algebraic surfaces in 3-dimensional toric varieties”

The study of topology of real algebraic hypersurfaces is classically divided into two complementary directions : on one hand, finding restrictions on the topology of the real part of real algebraic hypersurfaces with given Newton polytope, and on the other hand, constructing real algebraic varieties with interesting topological properties of their real part. Primitive patchworking is a very fruitful combinatorial construction tool introduced by O. Viro that allowed to construct many maximal (with respect to the Smith-Thom inequality) real algebraic hypersurfaces in various smooth ambient spaces.

The very specific topological properties of the hypersurfaces produced using this method are quite well studied in the case of hypersurfaces in smooth toric varieties. We present an ongoing attempt to extend some of these properties to primitive surfaces in arbitrary 3-dimensional toric varieties. As a consequence, new maximal surfaces in certain singular and non-singular toric 3-folds are constructed.

Monday, Nov 11, 14h00, 01-15, Nikon KURNOSOV (Glasgow)

“Bounds on Betti numbers of holomorphically symplectic manifolds and conjectures all around”

Abstract. I will review how to construct holomorphically symplectic manifolds, there are four series of hyperkahler ones, one non-Kahler (BG-manifolds) and some singular ones known. I will talk on ideas how to bound the Betti numbers of holomorphic symplectic manifolds. And explain on the connection to some other conjectures like Nagai’s conjecture and SYZ conjecture.

Friday, Nov 1, 14h, 06-13, and Monday, Nov 4, 14h, 01-15, minicourse

Vladimir Fock (Strasbourg)

“Goncharov-Kenyon integrable systems and plane curves”

Goncharov-Kenyon constructed integrable system starting form any Newton polygon which generalize plenty of known integrable systems. The phase space of such system is the space of plane curves provided with a line bundle. On the other hand the same space admit a description as a cluster variety and thus can be parameterized by algebraic tori. The aim of the talk is to describe these two points of view on the integrable system as well as discuss some other geometric interpretations of them.

Friday, Oct 18, 14h, 06-13, Stepan Orevkov (Toulouse)

“An algebraic curve with small boundary components in the 4-ball”

Abstract. We construct an algebraic curve in a ball in C^2 which passes through the origin, and such that all its boundary components are arbitrarily small.

Wednesday, Oct 16, 14h30, 06-13, Stepan Orevkov (Toulouse)

“On Korchagin's conjectures about M-curves of degree 9 in RP^2”

Abstract: Anatoly Korchagin formulated 4 conjectures about the ovals of an M-curve of degree 9 in RP^2. Now two of them are proven (one by myself and Viro, another by Severine Fiedler-LeTouzé) and two are disproven (by myself). Most of these results required to involve some (more ore less) new technique.

Monday, Sep 23, 14h30, room 01-15 and Wednesday, Sep 25, 14h00, room 06-13, minicourse

Rostislav MATVEEV (Leipzig).

“Corks, light-bulbs and other 4D objects”

Аbstract. I will describe some hands-on 4D-topological constructions and an attempt (after S.Akbulut) to use them to prove 4D-Poincare conjecture.

GeNeSyS Workshop in Belalp, Tuesday September 17th to Thursday 19th, Belalp.

Prof. Ilia Itenberg (Sorbonne AV����Ƶ), 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 (AV����Ƶ 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.

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