Table des mati猫res
Web-page of the Geneva 玉美人传媒 tropical group
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, Alo茂s Demory, Gurvan M茅vel, Grigory Mikhalkin, Antoine Toussaint.
Alumni: Ivan Bazhov, Johan Bjorklund, R茅mi Cr茅tois, 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:
S茅minaire "Fables G茅om茅triques".
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
Jo茅 Brendel (ETHZ), Friday, Feb 21, 15h15, room 6-13 (Seminaire "Fables G茅om茅triques")
鈥淪plit tori in S^2 x S^2, billiards and ball-embeddability鈥
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 鈥渂all-embeddability鈥 of Lagrangians, which we will discuss at the end of the talk. This is joint work with Joontae Kim.
Gurvan M茅vel (UNIGE), Wednesday, Feb 19, 14h00, room 1-07 (Seminaire "Fables G茅om茅triques")
鈥淔loor diagrams and some tropical invariants in positive genus鈥
Abstract : G枚ttche-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 G茅om茅triques")
鈥淩efined Tropical Invariants and Characteristic Numbers鈥
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 M茅vel鈥檚 floor diagram approach, and discuss the specialization of the count at q=1, which recovers certain characteristic numbers.
Thomas Blomme (Neuch芒tel), Friday, Jan 31, 14h00, room 1-07 (Seminaire "Fables G茅om茅triques")
鈥淯ne preuve courte d鈥檜ne formule de rev锚tement multiple鈥
Abstract: Enum茅rer les courbes de genre g passant par g points dans une surface ab茅lienne est un probl猫me naturel, et d鈥檜ne difficult茅 surprenamment in茅gale en fonction du degr茅 des courbes 茅tudi茅es. Pour les degr茅s 芦 primitifs 禄, il est ais茅 d鈥檕btenir une formule close par une r茅solution simple et explicite. Pour les classes 芦 divisibles 禄, une telle r茅solution est en revanche assez fastidieuse et souvent hors de port茅e. Pour autant, les invariants de ces derni猫res s鈥檈xpriment ais茅ment en fonction des invariants primitifs au travers de la formule de rev锚tement multiple, conjectur茅e par G. Oberdieck. Dans cet expos茅, on va montrer comment la g茅om茅trie tropicale permet de prouver cette formule en esquivant toute forme concr猫te d鈥櫭﹏um茅ration.
Ajith Urundolil-Kumaran (Cambridge), Wednesday, Dec 11, 14h00, room 06-13 (Seminaire "Fables G茅om茅triques")
鈥淭ropical correspondence theorems, Scattering diagrams and Quantum Mirrors鈥
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 G茅om茅triques")
鈥淎 tropical twist on Hurwitz numbers鈥
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 Do艂臋ga 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.
Alo茂s DEMORY (Gen猫ve), Wednesday, Nov 20, 14h00, room 06-13 (Seminaire "Fables G茅om茅triques")
鈥淧rimitive real algebraic surfaces in 3-dimensional toric varieties鈥
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)
鈥淏ounds on Betti numbers of holomorphically symplectic manifolds and conjectures all around鈥
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鈥檚 conjecture and SYZ conjecture.
Friday, Nov 1, 14h, 06-13, and Monday, Nov 4, 14h, 01-15, minicourse
Vladimir Fock (Strasbourg)
鈥淕oncharov-Kenyon integrable systems and plane curves鈥
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)
鈥淎n algebraic curve with small boundary components in the 4-ball鈥
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)
鈥淥n Korchagin's conjectures about M-curves of degree 9 in RP^2鈥
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-LeTouz茅) 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).
鈥淐orks, light-bulbs and other 4D objects鈥
袗bstract. 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 玉美人传媒), Friday, March 15, SM 01-05, 15h15-17h
鈥淏asic algebra and algebraic geometry special talk: Real plane sextic curves without real singular points鈥
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
鈥淒imers and M-curves鈥
We develop a general approach to dimer models analogous to Krichever鈥檚 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 (Gen猫ve), Dec 8, 14h30, Salle 06-13
鈥淒egenerations of Limit linear series鈥
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鈥揘oether 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
鈥淭ropical mirror symmetry and real Calabi-Yaus鈥
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, universit茅 de Gen猫ve, Thursday, Nov 9, 16h15, Room 1-15.
鈥淕romov-Witten invariants of bielliptic surfaces鈥
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, universit茅 de Gen猫ve, Monday, Oct 23, at 15h, Salle 06-13
鈥淩eal Structures of Phase Tropical Surfaces鈥
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 (玉美人传媒 of Luxembourg), Monday, Oct 16, at 15h, Salle 06-13
鈥淐omplexities in backpropagation and tropicalization in neural networks鈥
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 M茅vel (Universit茅 de Nantes), Wednesday, Oct 18, at 14h15, Salle 06-13
鈥淯niversal polynomials for coefficients of tropical refined invariant in genus 0鈥
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, G枚ttsche'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, Brugall茅 and Jaramillo-Puentes proved that some coefficients of these polynomials behave polynomially, bringing back a G枚ttsche's conjecture in a dual and refined setting. In this talk we will investigate the existence of universal polynomials for these coefficients.
Geneva-Neuch芒tel Symplectic Geometry Seminar
Schedule and more details: seminar page
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