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Accueil > Agenda > Colloques et rencontres > Participants > Cluster Algebras and Tropical Geometry.

Titres et résumés.

**Atomic Bases of Cluster Algebras : From Surfaces to Representations.**

**Grégoire Dupont**

*Abstract :*Atomic bases of cluster algebras are linear bases consisting of minimal elements in the positive cone. Viewing (coefficient-free) cluster algebras of types A and \tilde*A* as cluster algebras associated with unpunctured surfaces, we will prove the existence of a unique atomic basis for those cluster algebras. We will then show that the Laurent expansions of the elements in these atomic bases can be interpreted as generating series of certain subvarieties in the quiver grassmannians.

This is based on joint works with Giovanni Cerulli, Francesco Esposito and Hugh Thomas.

**Mirror symmetry and cluster varieties**

*Abstract :* In my three talks I will explain my conjecture, joint with Gross and Hacking, which gives a synthetic construction of the mirror to an affine CY manifold (with maximal boundary) as the spectrum of an explicitly described algebra : The vector space with basis the integer points of the tropicalization, and structure constants for the multiplication rule given by counts of tropical discs. I’ll discuss our partial results, joint with Kontsevich, for the simplest non-trivial class of CYs, cluster varieties, where our conjecture restricts to a generalization of the (corrected) Fock-Goncharov dual basis conjecture. We produce in particular for any cluster algebra A, an algebra A^*\mid*, squeezed between A and the upper cluster algebra, with canonical basis, a subset of the integer tropical points (the lattice where g-vectors live), containing all cluster monomials. The basis elements are universal positive laurant polynomials, and the structure constants for the algebra are non-negative integers. In particular, when A is equal to the upper cluster algebra, e.g. A =k[SL_n], we have a canonical basis of each. In any case our results include the positivity part of the Laurant phenomenon conjecture, and ou construction also proves the conjectured sign coherance of g-vectors. To us most interestingly, our construction of the canonical basis, e.g. for k[SL_n], has nothing a priori to do with representation theory, or cluster algebras — it’s a general, and conceptually quite simple, construction that applies to any variety with the right sort of volume form.

**Quantum cluster algebra structures on quantum nilpotent algebras**

*Abstract :* We will describe a result that proves the existence of quantum cluster algebra structures on a very general, axiomatically defined class of quantum nilpotent algebras. This has a wide range of applications, among which are a proof of the Berenstein-Zelevinsky conjecture for quantum double Bruhat cells and the construction of quantum cluster algebra structures on quantum unipotent groups in full generality. A Poisson version of the main result also settles affirmatively the question of whether the upper cluster algebras of Berenstein, Fomin and Zelevinsky for the coordinate rings of double Bruhat cells are equal to the corresponding cluster algebras.

The proofs are based on techniques from noncommutative and Poisson unique factorization domains. The talk is based on a joint work with Ken Goodearl (Univ California, Santa Barbara).

**Counting curves on Hirzebruch surfaces with tropical methods.**

*Abstract :*We use tropical methods to generalize a formula of Abramovich-Bertram relating enumerative numbers of curves in the Hirzebruch surfaces F_0 and F_2. The method involves the tropical enumerative geometry of a modified plane and different projections of the involved curve,

Joint with Erwan Brugallé.

**On Gromov-Witten invariants of del Pezzo surfaces.**

*Abstract :* The genus zero Gromov-Witten invariants of rational surfaces are well understood via the quantum cohomology theory. We suggest a recursive formula for Gromov-Witten invariants of del Pezzo surfaces in any genus which is a combination of Caporaso-Harris and Abramovich-Bertram-Vakil formulas.Geometry behind the formula reflects certain degenerations of curves which we analyze using some tropical techniques.

**Patchworking by ribbon graphs.**

*Abstract :*We’ll revisit patchworking of real algebraic curves from the tropical perspective. As a bonus we obtain a generalization of the patchworking to the case of planar immersed curves as well as for real algeraic knots and links. This revisiting can be formulated in terms of ribbon graphs as they expliciitly correspond to real tropical curves.

**Balancing conditions in global tropical geometry.**

*Abstract.* We study tropical geometry in the global setting using the deformation retraction of a non-Archimedean analytic space to its skeleton constructed by V. Berkovich. We state and prove the generalized balancing conditions in this setting. Starting with a semi-stable formal scheme, we calculate certain sheaves of vanishing cycles using analytic étale cohomology, then interpret tropical weights by these cycles. We obtain the balancing condition for tropical curves on the skeleton associated to the formal scheme in terms of the intersection theory on the special fibre. Our approach works over any complete discrete valuation field.

**Limiting mixed Hodge structures and tropical homology.**

*Abstract :*Tropical homology is defined for any tropical variety. In case the tropical variety is realizable (i.e. is a limit of a smooth projective complex family) the tropical homology can be used to calculate the Hodge numbers of a general fiber. I will outline the proof of this statement.

This is a joint project with Itenberg, Katzarkov and Mikhalkin.

**Horn problem and planar networks.**

*Abstract :* The Horn problem is the classical linear algebra problem of determining the inequalities on the eigenvalues of the sum of two Hermitian matrices with given spectra. We describe a tropicalization of this problem : the same set of inequalities appears in the context of maximal paths on weighted planar networks. Based on a joint work with A. Alekseev and A. Szenes.

**Frobenius splittings and cluster algebras.**

*Abstract :* We consider reductions of cluster algebras to fi-elds of odd characteristic. Algebras in positive characteristic have a Frobenius endomorphism, which sends every element to its p-th power. We show that the Frobenius map on an upper cluster algebra admits a standard splitting. When the cluster algebra is locally acyclic, this Frobenius splitting is ’regular’ ; that is, it does not -x any non-trivial ideals. This has geometric consequences for locally acyclic cluster algebras over Z ; in particular, they have (at worst) log terminal singularities.

** Cluster structure and categorification for Lusztig’s strata in partial flag varieties (after Nicolas Chevalier’s thesis).**

Let X=G/P be a partial flag variety for a simple algebraic group G of type A,D,E. Lusztig has introduced a stratification of X, which yields a cell decomposition of the non-negative part of X. The big cell of this decomposition is the totally positive part of X. In joint work with Geiss and Schröer, we have introduced a cluster

structure on the multi-homogeneous coordinate ring of X, and Chevalier has proved in his PhD thesis that the clusters give positive charts for the totally positive part of X. In the second part of his thesis, he has taken the first steps towards a generalization of these results to the lower-dimensional cells of the non-negative part of G/P. He has considered appropriate 2-Calabi-Yau subcategories of the module

category of the preprojective algebra with the same Dynkin type as G, and conjectured that they yield cluster algebra structures in the coordinate rings of the unipotent Lusztig strata of X. The conjecture would give new results even in the case of type A Grassmannians.

**Spectral Networks and Moduli of Local Systems.**

*Abstract :* In joint work with Davide Gaiotto and Greg Moore we recently introduced some new geometric/combinatorial objects which we call "spectral networks." A spectral network is a collection of curves on a punctured Riemann surface, carrying some discrete decorations and obeying certain local rules. I will explain what spectral networks are and one of their main applications, the construction of Darboux coordinate systems on moduli spaces of GL(K) local systems. Some of these Darboux coordinate systems coincide (essentially) with the ones introduced by Fock-Goncharov.

**Tropical del Pezzo surfaces and quantum cluster surfaces.**

*Abstract :* Given a tropical del Pezzo surface one may consider a tropical version of Fukaya-Oh-Ohta-Ono obstruction, the generating function for Maslov index 2 tropical discs. This potential is not invariant with respect to tropical modifications, and it depends on the choice of generic point : two such potentials are related by a birational change of coordinates prescribed by Auroux. Conjecturally the similar story happens with non necessary tropical varieties of any dimension, however in case of tropical surfaces one can assign q-coefficients to tropical discs by Block-Goettsche-Itenberg-Mikhalkin’s recipe to deform the Laurent polynomials into functions on two-dimensional quantum torus. Thus mirror partners of del Pezzo surface have structure of (quantum) cluster varieties. For del Pezzo surfaces of degrees 1 and 2 not a single FOOO’s potential have been computed so far, but the method above predicts what should they be equal to.

**Subtraction-free complexity and cluster transformations.**

*Abstract* Subtraction-free computational complexity is the version of algebraic complexity that allows only three arithmetic operations : addition, multiplication, and division. I will present some examples of using cluster transformations to design efficient subtraction-free algorithms.

This is joint work with Dima Grigoriev and Gleb Koshevoy.

**Piecewise polynomiality of double Gromov-Witten invariants of Hirzebruch surfaces.**

*Abstract :* Double Hurwitz numbers count ramified coverings of a fixed genus of the projective line, with arbitrary ramification profiles A and B over two particular points, and simple ramification values elsewhere. These numbers are known to be piecewise polynomials in the entries of A and B, once one has fixed their length. Recently Cavalieri, Jonhson, and Markwig, using tropical methods, were able to give wall crossings formulas for Hurwitz numbers. A bit later Ardila reinterpreted Cavalieri-Jonhson-Markwig’s proof within the framework of De Concini-Procesi-Vergne’s remarkable spaces.

Similarly to the case of Hurwitz numbers, one can define double Gromov-Witten invariants of Hirzebruch surfaces, with two prescribed intersection profiles with two non-intersecting sections. The goal of this talk is to explain how Ardila’s approach extends to this case via floor diagrams.

This is a joint work with Federico Ardila.

Dernière mise à jour le 26-05-2013