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Eugenii Shustin

Title: Examples of real algebraic and tropical enumerative invariants

Abstract: We present a series of new examples of real enumerative invariants counting real rational curves on real del Pezzo surfaces. Some of them admit tropical analogs defined for arbitrary toric surfaces and for arbitrary genus.

Lionel Lang

Title: On approximation of harmonic tropical curve.

Abstract: The generalization of the amoeba map introduced by I.Krichever suggest the consideration of a wider class of tropical curves, with non rational slopes. To be more precise, it suggest a generalization of the notion of tropical morphism from abstract tropical curves to R^n. Both generalization of amoebas and immersed tropical curve will be called harmonic. After motivating the terminology, we will see how one can approximate any harmonic tropical curve by a family of harmonic amoebas. With a bit of extra work, it gives an “alternative” proof to Mikhalkin's theorem on approximation of complex tropical curves in the plane.

Karim Adiprasito

Title: Tropical Lefschetz theorems and filtered geometric lattices

Abstract: I will review some analogues of the classical Lefschetz Section Theorem for smooth tropical varieties. More precisely, I will present tropical analogues of the section theorems of Lefschetz, Andreotti-Frankel, Bott-Milnor-Thom, Hamm-L{\^e} and Kodaira-Spencer, and the vanishing theorems of Andreotti-Frankel and Akizuki-Kodaira-Nakano. We start the paper by resolving a conjecture of Mikhalkin and Ziegler (2008) concerning the homotopy Cohen-Macaulayness of certain filtrations of geometric lattices, generalizing earlier work on full geometric lattices by Folkman and others.

Ilia Itenberg

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Ludmil Katzarkov

Title: Categorical base loci and applications

Abstract: We introduce the notion of categorical base loci. Possible applications will be considered.

Sergey Galkin

Title: Polynomial relations between holomorphic curves and discs

Abstract: I will tell about polynomial relations between numbers of pseudo-holomorphic curves, passing through a point, and numbers of pseudo-holomorphic discs, bounded on some Lagrangian torus. The relations comes from a comparison of different pictures of mirror symmetry, and can be proved only using tropical geometry (so far).

Hannah Markwig

Title: Tropicalizing rational relative Gromov-Witten theory of P^1

Abstract: We show that the relative stable map compactification of M_0,n (for maps to P^1, relative to two points) is a tropical compactification. Furthermore, the tropicalization of the open part equals the tropical space of relative stable maps to P^1. Consequently, the Chow ring of the relative stable map space can be computed by means of tropical intersection theory in an intuitive way. This is joint work with Renzo Cavalieri and Dhruv Ranganathan.

Luis-Felipe Tabera

Title: Singular tropical hypersurfaces in positive characteristic.

Abstract: In this talk I will consider tropical varieties as non-archimedean amoebas defined over fields in positive characteristic. I will define a tropical singularity and give a method to compute the singular locus of a tropical hypersurface in positive characteristic. As a consequence, I will describe some cells of the discriminant in characteristic p. I will briefly mention the case of the p-adics and we will see that the p-adic tropical discriminant is an “interpolation” of the tropical discriminants in characteristic 0 and p.

Takeo Nishinou

Title: Degeneration and curves on K3 surfaces.

Abstract: There is a well-known conjecture which states that all projective K3 surfaces contain infinitely many rational curves. By calculating obstructions in deformation theory through degeneration, we give a new approach to this problem. In particular, we show that there is a Zariski open subset in the moduli space of projective complex K3 surfaces whose members fulfil the conjecture.

Erwan Brugalle

Title: Some applications of degeneration methods in real enumerative geometry

Tony Yue Yu

Title: Counting holomorphic cylinders and GAGA theorems

Abstract: I will talk about the enumeration of holomorphic cylinders in log Calabi-Yau varieties.

One ingredient is the study of tropicalization via non-archimedean geometry in the sense of Berkovich. I will recall some general results concerning enumerative geometry in this framework. Then I will explain how to apply these general results.

Another important ingredient is the theory of non-archimedean analytic stacks. It is a separate project joint with M. Porta. In fact, we considered more generally higher analytic stacks using the language of infinity categories. We proved the analog of Serre’s GAGA theorem. I will explain the objects and the results by making a lot of analogies with algebraic geometry if the audience is not familiar with non-archimedean geometry.

Ilia Zharkov

Title: Skeleta for affine surfaces

Abstract: I will show how to build skeleta for affine surfaces using their degeneration to tropical surfaces. The skeleta are “kind of Lagrangian”, except there are misterious “holomorphic” pieces which I seem cannot avoid.

Ilya Tyomkin

Title: Applications of tropical geometry to the study of Severi varieties in arbitrary characteristic.

Abstract: In my talk I’ll discuss the geometry of Severi varieties on toric surfaces. I’ll review the progress made during the past 5 years. In some problems the tropical point of view is very helpful, although the proofs and constructions can be made purely algebraically. In others, the tropical tools are the only currently available tools to obtain the results.

Anton Alekseev

Title: Tropicalization of Poisson brackets

Maksim Karev

Title: Arnold's J^- -invariant and Kontsevich integral.

Abstract: In 1994, V. Arnold has defined three order one invariants of smooth immersions S^1 –> R^2 related to three components of the discriminant subset in the space of immersions. One of these invariants, namely the one related to the discriminant component corresponding to the locus of immersions with a point of inverse self-tangency, is called J^- -invariant. In my talk I will remind the definition of the J^- -invariant and explain its relation to configuration space integrals and Kontsevich integral.

Jens Forsgard

Title: Discriminants and Hyperfields

wiki/abstracts.txt · Dernière modification : 2014/11/18 12:40 de serjl