# Prochains séminaires

Jeudi 5 Juin 2014, à 16h15, Villa Battelle

### The geometry of auctions and competitive equilibrium with indivisible goods.

Elizabeth Baldwin (Oxford) Auctioneers may wish to sell related but different indivisible goods in a single process. To develop such techniques, we study the geometry of how an agent's demanded bundle changes as prices change. This object is the convex-geometric object known as a `tropical hypersurface'. Moreover, simple geometric properties translate directly to economic properties, providing a new taxonomy for economic valuations. When considering multiple agents, we study the unions and intersections of the corresponding tropical hypersurfaces; in particular, properties of the intersection are deeply related to whether competitive equilibrium exists or fails. This leads us to new results and generalisations of existing results on equilibrium existence. The talk will provide an introductory tour to relevant economics to show the context of these applications of tropical geometry. This is joint work with Paul Klemperer, Oxford.

Vendredi 13 Juin 2014, à 14h30, Villa Battelle

### All manifolds are contact except those which are obviously not.

Yakov Eliashberg (Stanford) We prove that all closed almost contact manifolds admit contact structures, and moreover, in the space of contact structures there exists an open-closed subspace of the so-called “overtwisted” contact structures whose classification up to isotopy coincides wth the homotopy classification as almost contact structures. This is a joint work with Strom Borman and Emmy Murphy

Jeudi 26 Juin 2014, à 14h30, Villa Battelle

### Cylinders in del Pezzo surfaces.

Ivan Cheltsov (Edinburgh) For a projective variety X and an ample divisor H on it, an H-polar cylinder in X is an open ruled affine subset whose complement is a support of an effective Q-divisor Q-rationally equivalent to H. This notion links together affine, birational and Kahler geometries. I will show how to prove existence and non-existence of cylinders in smooth and mildly singular del Pezzo surfaces. This will answer an old question of Zaidenberg and Flenner about additive group actions on the cubic Fermat affine threefold cone. This is a joint work with Park and Won. See http://arxiv.org/abs/1303.2648 and http://arxiv.org/abs/1311.5257.

Jeudi 26 Juin 2014, à 16h15, Villa Battelle

### Lagrangian non-intersection theory.

Yakov Eliashberg (Stanford) It turns out that the Lagrangian self-intersection problem is controlled in a general case by differential and not symplectic topology. For instance, for any orientable $3$-manifold there exists a Lagrangian immersion to the standard symplectic R^6 with exactly one transverse self-intersection point. This is a joint work with Emmy Murphy, Tobias Ekholm and Ivan Smith.

# Séminaires passés

## Special joint seminar : Fables Geometriques / Mathematique Physique

Lundi 25 Novembre 2013, à 15h30, Villa Battelle

### Real normalized differentials and their applications .

Igor Krichever (Columbia) A general concept of real normalized meromorphic differentials has arisen in the framework of the Whitham perturbation theory of integrable systems. In the talk we will discuss some constructions associated motivated by the Whitham theory and their applications to a study of geometry of moduli spaces of curves with punctures.

Lundi 14 octobre 2013, Toulouse, France

### Tropical Geometry in Europe

Lundi 24 juin 2013, à 14h30, Villa Battelle

### Mahler measure in geometry and topology.

Eriko Hironaka (Florida State) Lehmer's problem on Mahler measures relates two measures of complexity for an algebraic integer, the house and the degree. Many analogous questions can be asked in the realm of dynamical systems in geometry and topology, for example in studying the growth rate of groups, the entropy of surface automorphisms, and the volumes of hyperbolic polyhedra. This talk will give a survey of how Lehmer's conjectural minimum Mahler measure appears in each of these contexts.

Mercredi 5 juin 2013, 14h30 à 16h00, Villa Battelle

### Tropical Schubert varieties.

Kristin Shaw (Toronto) In this talk I will describe a method for identifying Schubert varieties on the tropical Grassmannian, as well as some examples of intersection products. Also we will explain how to identify Schubert varieties with faces of Gelfand-Zetlin like polytopes coming from integrable systems given by faces of the tropical Grassmannian.

Lundi 13 mai 2013, 11h00 à 12h30, Villa Battelle

### (Affine) ADE bundles over del Pezzo surfaces.

Conan Leung (Chinese University of Hong Kong) I will explain intimate relationships between the geometry of (almost) del Pezzo surfaces and (affine) ADE bundles over these surfaces. This project was originally motivated from the duality between heterotic string and F-theory.

Jeudi 21 mars 2013, à 16h20, Villa Battelle

### Degeneration and mirror symmetry for minuscule varieties.

Alexey Bondal (Steklov Institute (Moscow) / Institute for Physics and Mathematics of the Universe (Tokyo) ) Minuscule varieties is a particular nice class of homogeneous spaces. We discuss geometry of toric degenerations of minuscule varieties and mirror symmetry for them.

Mercredi 27 février 2013, 14h30 à 16h00, Villa Battelle

### Topological classification of real rational curves in the plane .

Johannes Rau (Saarbrücken) The study of the topological shape of smooth real algebraic curves in the plane has a long history. In this talk, however, we focus on singular curves, namely (irreducible) rational curves of degree 5 (and 4) with ordinary double points. We will study their topological classification, showing that all the possible types can be easily constructed using tropical curves (joint work with Ilia Itenberg and Grigory Mikhalkin).

Jeudi 21 février 2013, Saarbrücken, Allemagne

### Tropical Geometry in Europe

Jeudi 18 Octobre 2012, 14h30 à 16h00, Villa Battelle

### A method for comparing curve counting invariants.

Cristina Manolache (Imperial College) In the past twenty years several curve counting invariants of a space X have been constructed: Gromov-Witten invariants, stable quotient invariants, quasi-maps invariants. These invariants are numbers which ideally count fixed genus and degree curves on X with certain properties (e.g. intersect some given subvarieties of X). One is then tempted to expect all these invariants to be closely related. There are by now several theorems and conjectures in this direction. In this seminar I will discuss a few examples in which we can prove the expected relationships between various Gromov-Witten-type invariants by geometric methods. In particular, one can understand geometrically that Gromov-Witten invariants are equal to stable quotient invariants.

Jeudi 4 octobre 2012, Toulouse

### Tropical Geometry in Europe

Vendredi 28 Septembre 2012, 14h30 à 16h00, Villa Battelle

### Topological strings and knot contact homology.

Tobias Ekholm (Uppsala) The talk discusses an emerging picture relating knot contact homology to Chern-Simons theory of knots and topological strings in the resolved conifold via large N duality. In particular, the so called augmentation polynomial in knot contact homology plays a central role in constructing Calabi-Yau mirrors of the resolved conifold, and is related Lagrangian fillings of the Legendrian conormal torus of a knot in the resolved conifold. We introduce knot contact homology and topological strings, describe the proposed relation, and discuss it from the perspective of symplectic geometry and open Gromov-Witten theory. The talk reports on joint work with Aganagic, Ng, and Vafa.

## "Special Workshop"

composed of 3 lectures by Andrey Losev (Moscow)

Mardi 25 Septembre 2012, 14h30 à 16h30, Villa Battelle

### A-I-B mirror symmetry on toric variety (chiral algebras approach, based on the joint work with E.Frenkel).

Andrey Losev (Moscow) We consider holomorphic maps from the Riemann surface $\Sigma$ into a toric manifold X (say, $CP_1$) as maps from $\Sigma-{set of points}$ to $(C^*)^{\otimes N}$ with prescribed behavior at deleted points. Such maps are realized in the theory of chiral fields. Prescribe behavior may be realized by special operators called holomortexes. The T-duality in the compact direction on the target space makes holomortexes local operators in the mirror fields. They form superpotential of the mirror LG theory.

Mercredi 26 Septembre 2012, 14h30 à 16h30, Villa Battelle

### Tropical mirror symmetry for toric varieties and conjecture for varieties of general type.

Andrey Losev (Moscow) We consider tropical curve (graph embedded in $R^n$) as a trajectory of a particle. Hamiltonian of such particle is just a Lie derivative along the vector field depending on the integer vector in $Z^n$. Different integer vectors correspond to different sectors of the Hilbert space of the same particle. At the vertexes of the graph sectors are changed and corresponding operator is the tropical holomortex. Considering integer vector as a Fourier component of the mirror field we obtain BCOV formulation of the counting of tropical curves.

Jeudi 27 Septembre 2012, 11h00 à 12h00, Villa Battelle

### Beta function in conformal field theories as obstruction in homotopical Maurer-Cartan equation (approach based on the configuration space of points).

Andrey Losev (Moscow) I will consider Segal axioms for QFT, so called "figure" observables and local observables among them. I will describe deformation of the QFT due to all types of observables, and particular case of deformations due to local observables. I will show that singularities due to collision of points need regularization, and such regularization causes anomalous dependence of the regularized theory on metric on the worldsheet. In this way we discover the zero beta-function condition as the homotopical Maurer-Cartan equation. In application we explain how Einstein equations of bosonic string theory can be written in this form and propose their generalizations to Q-manifolds.

Jeudi 28 juin 2012, 14h30 à 16h30, Villa Battelle

### Symplectic sum formulas in real enumerative geometry.

Erwan Brugallé (Paris) Ionel and Parker's symplectic sum formulas relate the (relative) Gromov-Witten invariants of a symplectic sum X#Y with the ones of X and Y. In the presence of a real structure on X and Y, these formulas have a natural real version. In particular, to any recursion among Gromov-Witten invariants obtained via a symplectic sum decomposition correspond one or several analogous recursions involving Welschinger invariants, e.g. Caporaso-Harris formula, Abramovich-Bertram-Vakil formula, ... In this talk I will explain how to use this approach to relate Welschinger invariants for different real structures on the same symplectic 4-manifold (up to deformation). This is a joint work with Nicolas Puignau (UFRJ, Rio de Janeiro).

Mercredi 27 juin 2012, 14h30 à 16h30, Villa Battelle

### Mirror Symmetry and Cluster Varieties.

Mark Gross (San Diego) I will talk about the intersection of the theory of cluster varieties with recent work of myself with Paul Hacking and Sean Keel. The cluster X- and A-varieties as defined by Fock and Goncharov have natural interpretations in the world of mirror symmetry, and canonical bases of the associated cluster algebras can then be constructed via tropical methods.

Vendredi 1er juin 2012, 14h30 à 16h30, Villa Battelle

### From configuration spaces to stable rational curves: Kohno-Drinfeld way.

Alexander Veselov (University of Loughborough) I am going to explain that Gaudin subalgebras of Kohno-Drinfeld Lie algebra and stable rational curves have the same moduli space. No special knowledge will be required.

Jeudi 22 mars 2012, 14h30, Villa Battelle

### Reflecting Fano.

Sergei Galkin (IPMU Tokyo) Continuation.

Mercredi 21 mars 2012, 14h15, Villa Battelle

### Betti numbers of random real hypersurfaces.

Jean-Yves Welschinger (Lyon) What are the expected Betti numbers of a real projective hypersurface taken at random? I will explain how to estimate them from above asymptotically thanks to the theory of peak sections of Hörmander. This is a joint work with Damien Gayet.

Mardi 20 mars 2012, 16h30, Villa Battelle

### Reflecting Fano.

Sergei Galkin (IPMU Tokyo) Continuation.

Lundi 19 mars 2012, 14h30, Villa Battelle

### Reflecting Fano.

Sergei Galkin (IPMU Tokyo) I'll describe what is known (and not known) about mirror symmetry phenomenon for positively curved manifolds and try to envisage the development of this theory in next decade.

Vendredi 16 mars 2012, 14h30, Villa Battelle

### Monsky theorem and tropical geometry.

Daniil Rudenko (St. Petersbourg) Monsky theorem claims that a square can not be cut into an odd number of triangles of equal areas. We will discuss its consequences and generalizations, as well as connections with tropical geometry. We will also discuss the 3-colorings of the projective plane with each line containing only points of two colors. This colorings in some sense correspond to non-archimedean valuations.

Vendredi 2 mars 2012, 14h30, Villa Battelle

### Knots, dimers and clusters.

Michael Polyak (Technion, Haifa) I will discuss various relations of knots and curves on surfaces to dimers, cluster variables and Poisson brackets.

Mercredi 23 novembre 2011, 15h30, Villa Battelle

### Variétés amassées, courbes planaires et systèmes intégrables

Mardi 22 novembre 2011, 15h30, Villa Battelle

### Variétés amassées, courbes planaires et systèmes intégrables

Lundi 21 novembre 2011, 16h00, Villa Battelle

### Variétés amassées, courbes planaires et systèmes intégrables

Vladimir Fock (Strasbourg) Les variétés amassées sont des variétés construites à partir de données combinatoires - dont la principale est une matrice antisymétrique à valeurs entières. Ces variétés sont recollées des cartes par une classe très restreinte des transformations birationnelles. Ces variétés possèdent une structure de Poisson, une quantification, une action d'un groupe discret et une base canonique des fonctions régulières. Elles sont définies sur un corps ou un semi-corps, en particulier sur le semi-corps tropical. Dans le mini-cours nous donnerons la définition formelle des variétés amassées et discuterons ces propriétés sur deux exemples principales - espaces de Teichmueller et les groupes de Lie simples. Puis nous considérons la construction récente de A.Goncharov et R.Kenyon d'une classe des variétés amassées munies d'un système intégrable - application de Poisson sur l'espace des courbes planaires dans une surface torique. En particulier nous discuterons la relation entre l'espace des courbes planaires tropicaux et les variétés amassées sur le semi-corps tropical, explicitée par cette construction.

Du dimanche 18 au mercredi 21 décembre 2011, Arolla, Suisse

### Tropical Geometry in Europe

Séminaire itinérant Toutes les informations ici.

Lundi 7 novembre 2011, 16h00, Villa Battelle

### Counting points of homogeneous spaces over finite fields

Michel Brion (Grenoble) Given a system of polynomial equations with (say) complex coefficients, one may first reduce it to obtain a system over a finite field, and then count the solutions over all larger finite fields. The resulting function of the cardinality of the field satisfies deep regularity properties due to Weil, Dwork, Grothendieck, and Deligne. The talk, based on joint work with E. Peyre, will address a very symmetric situation, where the solutions of the system admit a transitive action of an algebraic group. Then the counting function turns out to be a periodic polynomial in the cardinality of the field.

Mercredi 9 novembre 2011, 15h30, Villa Battelle

### Counting points of homogeneous spaces over finite fields

Michel Brion (Grenoble) Continuation

Jeudi 20 octobre 2011, Toulouse

### Tropical Geometry in Europe

Mercredi le 18 mai 15h30 Villa Battelle

### Geometry and topology of curves on surfaces II

Oleg Viro (Stony Brook) There are two closely related classes of objects: generic differentiable maps of the circle to a smooth closed surface and generic real algebraic curves on a smooth real algebraic surface. Both can be studied from the viewpoint of the global singularity theory. In the talks, the first steps of such study will be outlined. The notions of generic curve, discriminant hypersurface, its natural stratification, finite type invariants will be discussed. We will consider Arnold's invariants of generic immersed curves, splittings and generalizations of these invariants, similar invariants of real algebraic curves. Arnold's invariants and other finite type invariants will be used in solution of enumerative problems.

Mardi le 17 mai 15h30 Villa Battelle

### Geometry and topology of curves on surfaces I

Oleg Viro (Stony Brook) There are two closely related classes of objects: generic differentiable maps of the circle to a smooth closed surface and generic real algebraic curves on a smooth real algebraic surface. Both can be studied from the viewpoint of the global singularity theory. In the talks, the first steps of such study will be outlined. The notions of generic curve, discriminant hypersurface, its natural stratification, finite type invariants will be discussed. We will consider Arnold's invariants of generic immersed curves, splittings and generalizations of these invariants, similar invariants of real algebraic curves. Arnold's invariants and other finite type invariants will be used in solution of enumerative problems.

Jeudi le 12 mai Strasbourg.

## Séminaire GPS - Géometrie tropicale

Mardi le 10 mai 15h30 Villa Battelle

### Virtual class for the moduli of sheaves on a curve.

Barbara Fantechi (Trieste)

Lundi le 9 mai 10h Villa Battelle

### Algebraic stacks and virtual classes.

Barbara Fantechi (Trieste)

Jeudi le 5 mai 15h45 Villa Battelle

### Some enumerative results for real curves of fixed cogenus.

Benoît Bertrand (Toulouse)

Unlike its complex counterpart, the number of real curves of fixed cogenus c and passing through d(d+3)/2 -c real points in generic position depends on the configuration of points. It is obviously bounded by the number of such complex curves. One says that the problem is maximal if there is a configuration such that the real and complex count coincide (i.e. all solutions are real). I will introduce a signed version of floor diagrams and use them to prove that, in cogenus 1, the above problem is maximal and that for a fixed cogenus c, it is asymptotically maximal when d tends towards infinity and that it is maximal in cogenus 1.

Vendredi le 6 mai 15h30 Villa Battelle

### Finite subgroups of Cremona groups

Yuri Prokhorov (Moscow and Grenoble)

The Cremona group Cr_n(k) over a field k is the group of k-automorphisms of the field of rational functions k(x_1,,x_n) in n independent variables. The group Cr_1(k) is isomorphic to the projective linear group PGL_2. Already in the case n=2 the group Cr_2(k) is not well understood. Very little is known about the Cremona groups for n>2. In this talk, I will discuss general method of describing finite subgroups in Cr_2(k) and Cr_3(k). The method will be applied to the case of simple groups and the symmetric group S_6.

Jeudi le 31 mars Paris.

## Séminaire GPS - Géometrie tropicale

Mardi le 29 mars 15h30 Villa Battelle

### Non-commutative toric varieties

Ernesto Lupercio (CINVESTAV, Mexico City)

This is joint with Katzarkov, Meersseman and Verjovski. We propose a category of NC torics, that suggests the existence of non-commutative tropical geometry.

Lundi le 28 mars 16h30 Villa Battelle

### Formal Mirror Symmetry

Ernesto Lupercio (CINVESTAV, Mexico City)

Several aspects of mirror symmetry are only formal and algebraic, and in this talk I want to illustrate how much one can recover this way.

Vendredi le 25 mars 15h30 Villa Battelle

### Middle-dimensional squeezing and non-squeezing.

Rostislav Matveyev (Leipzig)

Theorem of Gromov states that no symplectic diffeomorphism can map 2n-dimensional ball of radius r into the 2D-cylinder of radius s, that is a product of (2n-2)-dimensional vector-space and a 2-dimensional disk of radius s, unless s \geq r. This result is known as 2D non-squeesing. Liouville's theorem, on the other hand, asserts that volume is preserved by symplectic diffeomorphisms. This can be thought of as top-dimensional non-squeezing. We investigate what could be said about (non-)squeezing in the intermediate dimensions. Joint work with A. Abondandolo.

Lundi le 21 mars 14h30 Villa Battelle

### On plane real algebraic curves, I

Stephan Orevkov (Toulouse)

We study which configurations of ovals in RP2 are realizable by algebraic curves. For that purpoese we use topological and symplectic properties of algebraic curves in CP2.

Mardi le 22 mars 15h30 Villa Battelle

### On plane real algebraic curves, II

Stephan Orevkov (Toulouse)

We study which configurations of ovals in RP2 are realizable by algebraic curves. For that purpoese we use topological and symplectic properties of algebraic curves in CP2.

Samedi le 19 mars 14h30 Villa Battelle

### Invariants of 3-manifolds via counting surfaces

Michael Polyak (Technion, Haifa)

We discuss a combinatorial construction of 3-manifold invariants by counting certain subdiagrams in a diagram of a surgery link. This construction may be interpreted as counting maps of surfaces of a fixed genus.

Jeudi le 10 mars 15h30 Villa Battelle

### Matrix models and tropical geometry I

Marcos Marino (UNIGE)

Integrals over matrices, in the limit of very large rank, are often described in terms of algebraic curves, and many interesting quantities are computed in terms of periods of meromorphic forms on these curves. The tropical limit of these curves turns out to describe in an elegant way an interesting limit of the matrix integral, which has important physical applications. I will describe some concrete matrix models with an interesting tropical limit and sketch the physics behind them.

Vendredi le 11 mars 15h30 Villa Battelle

### Matrix models and tropical geometry II

Marcos Marino (UNIGE)

Integrals over matrices, in the limit of very large rank, are often described in terms of algebraic curves, and many interesting quantities are computed in terms of periods of meromorphic forms on these curves. The tropical limit of these curves turns out to describe in an elegant way an interesting limit of the matrix integral, which has important physical applications. I will describe some concrete matrix models with an interesting tropical limit and sketch the physics behind them.

Mercredi le 2 mars 16h00 Villa Battelle

### Minicourse - Hodge theory of character varietes I

Luca Migliorini (Università di Bologna)

In the first lecture I will review some basic facts about mixed Hodge structure mostly focusing on: MHS associated to singular and non compact curves MHS associated to degenerating families MHS on the moduli space of character varieties

Jeudi le 3 mars 15h30 Villa Battelle

### Minicourse - Hodge theory of character varietes II

Luca Migliorini (Università di Bologna)

In lecture 2 I will describe, in the spirit of Simpson's non abelian Hodge theory, 3 ways of parametrizing the representations of the fundamental group of a Riemann surface, leading to 3 different algebraic varieties, one of which is the character variety of the first lecture while the other is the famous moduli space of Higgs bundles.

Vendredi le 4 mars 15h30 Villa Battelle

### Minicourse - Hodge theory of character varietes III

Luca Migliorini (Università di Bologna)

In lecture 3 I will discuss a recent result due to de Cataldo Hausel and myself which describes the MHS on the character variety in terms of the topology of a map on the moduli space of Higgs bundles, and speculate on possible generalization to algebraic completely integrable systems.

Lundi le 13 décembre 15h30 Villa Battelle

### Toric mirror symmetry

Victor Batyrev (Tuebingen)

The aim of the talk is to give an overview of some ideas and results concerning the inverstigation of mirror symmetry by methods of toric geometry.

le 6 au 8 décembre Autour des surfaces tropicales, Arolla, Valais.

## Séminaire GPS - Géometrie tropicale

Mercredi le 3 novembre Institut de Recherche Mathématiques Avancée, Strasbourg.

## Séminaire GPS - Géometrie tropicale

Mercredi le 27 octobre 15h30 Villa Battelle

### SYZ mirror symmetry for toric manifolds I

Kwokwai Chan (IHES)

I will discuss mirror symmetry for toric manifolds from the SYZ point of view. There is significant difference between Fano and non-Fano cases. The first talk will be a review of the Fano case, which is easier. In the second talk, I will describe recent progress in the non-Fano case, based on joint work with S.-C. Lau and N.-C. Leung and work of Fukaya, Oh, Ohta and Ono.

Jeudi le 28 octobre 15h30 Villa Battelle

### SYZ mirror symmetry for toric manifolds II

Kwokwai Chan (IHES)

I will discuss mirror symmetry for toric manifolds from the SYZ point of view. There is significant difference between Fano and non-Fano cases. The first talk will be a review of the Fano case, which is easier. In the second talk, I will describe recent progress in the non-Fano case, based on joint work with S.-C. Lau and N.-C. Leung and work of Fukaya, Oh, Ohta and Ono.

Mercredi le 6 octobre 15h30 Villa Battelle

### Tropical avatar of the Gelfand-Zeitlin integrable system I.

Anton Alekseev (UNIGE)

I'll recall the definition of the Gelfand- Zeitlin (extended eigenvalue) map $\gamma$ for Hermitian and upper- triangular n by n matrices. Using the coordinate system defined by a certain planar network $N$, we define a "tropical" analogue $\gamma_trop$ of the Gelfand-Zeiltin map. This is a piece-wise linear transformation of $\mathbb{R}^{n(n+1)/2}$ with interesting combinatorial properties described in terms of multiple paths on $N$. Finally, we establish a relation between fibers of $\gamma$ and $\gamma_trop$. This is a joint work with I. Davydenkova, M. Podkopaeva and A. Szenes.

Jeudi le 7 octobre 15h30 Villa Battelle

### Tropical avatar of the Gelfand-Zeitlin integrable system II.

Anton Alekseev (UNIGE)

I'll recall the standard construction of the Poisson structure on the spaces of Hermitian matrices . The Gelfand- Zeitlin map $\gamma$ defines a completely integrable Hamiltonian system with respect to this Poisson structure. And the "tropical" map $\gamma_trop$ provides an open dense Darboux chart. This is a joint work with I. Davydenkova, M. Podkopaeva and A. Szenes.

Lundi le 24 mai 14h15 Villa Battelle

### Mirror Symmetry and Reality.

Kentaro Hori (IPMU, Tokyo)

Jeudi le 27 mai Institut de mathématiques de Jussieu, Paris.

## Séminaire GPS - Géometrie tropicale

### 10h00 Clusters versus tropical clusters

Bernhard Keller (Université de Paris VII)

### 11h30 Configuration spaces of circles and spheres

Igor Dolgachev (University of Michigan)

### 14h00 3200 rational plane sections of a generic tropical quartic in P3

Grigory Mikhalkin (Université de Genève)

Lundi le 17 mai 14h15-15h30 Battelle

### Minicourse - Geometry of special holonomy I.

Conan Leung (CUHK Hong Kong)

In this lecture series, I will explained a unified approach to describe various geometries for Riemannian manifolds. They include Kahler geometry, Calabi-Yau geometry, hyperkahler geometry, G_2 geometry, geometry of Riemannian symmetric spaces and so on. All these geometries play important roles in modern mathematics, as well as in physical theories of string, M-theory and so on. We will unveil the underlying mathematical structures of them.

Mardi le 18 mai 16h15-17h30 Battelle

### Minicourse - Applications of Category Theory to Geometry: Derived Categories of Coherent Sheaves I .

Dmitri Orlov (Steklov Institute Moscow)

I am going to give a few lectures on derived categories of coherent sheaves and triangulated categories and their applications to geometry. We especially will discuss applications to homological mirror symmetry. I plan to concentrate an attention on categories of D-branes of type B in sigma-models and Landau-Ginzburg models. We will discuss different properties of these categories some of which are also coming from physics. Useful notions of exceptional collections, classical and strong generators will be introduced. I am also going to describe a procedure of constructing of mirror symmetric models that is known as Batyrev-Givental-Hori-Vafa procedure and to describe mirror symmetry for some varieties and their noncommutative deformations. Generalization of strange Arnold duality will be described and some results and conjectures will be given in the last lecture.

Mercredi le 19 mai 14h15-15h30 Battelle

### Minicourse - Geometry of special holonomy II.

Conan Leung (CUHK Hong Kong)

In this lecture series, I will explained a unified approach to describe various geometries for Riemannian manifolds. They include Kahler geometry, Calabi-Yau geometry, hyperkahler geometry, G_2 geometry, geometry of Riemannian symmetric spaces and so on. All these geometries play important roles in modern mathematics, as well as in physical theories of string, M-theory and so on. We will unveil the underlying mathematical structures of them.

Mercredi le 19 mai 16h15-17h30 Battelle

### Minicourse - Applications of Category Theory to Geometry: Derived Categories of Coherent Sheaves II.

Dmitri Orlov (Steklov Institute Moscow)

I am going to give a few lectures on derived categories of coherent sheaves and triangulated categories and their applications to geometry. We especially will discuss applications to homological mirror symmetry. I plan to concentrate an attention on categories of D-branes of type B in sigma-models and Landau-Ginzburg models. We will discuss different properties of these categories some of which are also coming from physics. Useful notions of exceptional collections, classical and strong generators will be introduced. I am also going to describe a procedure of constructing of mirror symmetric models that is known as Batyrev-Givental-Hori-Vafa procedure and to describe mirror symmetry for some varieties and their noncommutative deformations. Generalization of strange Arnold duality will be described and some results and conjectures will be given in the last lecture.

Vendredi le 21 mai 14h15-15h30 Battelle

### Minicourse - Applications of Category Theory to Geometry: Derived Categories of Coherent Sheaves III.

Dmitri Orlov (Steklov Institute Moscow)

I am going to give a few lectures on derived categories of coherent sheaves and triangulated categories and their applications to geometry. We especially will discuss applications to homological mirror symmetry. I plan to concentrate an attention on categories of D-branes of type B in sigma-models and Landau-Ginzburg models. We will discuss different properties of these categories some of which are also coming from physics. Useful notions of exceptional collections, classical and strong generators will be introduced. I am also going to describe a procedure of constructing of mirror symmetric models that is known as Batyrev-Givental-Hori-Vafa procedure and to describe mirror symmetry for some varieties and their noncommutative deformations. Generalization of strange Arnold duality will be described and some results and conjectures will be given in the last lecture.

Mercredi le 12 mai à 15h15 Battelle

### Computing the alpha-invariant.

Ivan Cheltsov (University of Edinburgh)

We describe various algebraic methods how to compute an algebraic counterpart of the so-called alpha-invariant of Tian of Fano orbifolds. As an application we show the existence of an orbifold Kahler-Einstein metrics on many Fano orbifolds. We also show some application in birational geometry.

Lundi le 10 mai à 15h15-17h Battelle

### Dolgachev's surface.

Selman Akbulut (Michigan State)

Are there exotic copies of S^4 or CP^2 ?. It is known that if they exist they must contain 1- or 3- handles. About 24 years ago Donaldson gave the first example of an exotic closed orientable smooth 4-manifold, i.e. he proved that Dolgachev's complex surface E(1)_{2,3} is an exotic copy of CP^2 # 9(- CP^2); right about the same time Harer Kas and Kirby wrote a book about E(1)_{2,3} where they conjectured that it must contain 1- or 3- handles. We will discuss the recent solution of this conjecture (in the negative). In this context we will relate the proof to "corks" and "plugs", which are roughly freely floating objects in 4- manifolds determining their exotic structures.

Mercredi le 28 avril à 15h15 Battelle:

### Non-commutative cluster mutations.

Alexander Usnich

We will define non-commutative analog of cluster mutations of two variables. The non-commutative Laurent phenomen is conjectured to hold for them as well. We will reinterpret the Laurent phenomenon in terms of the derived category of coherent sheaves on a projective plane. An important ingredient of this interpretation would be the tropicalization of mutations.

Vendredi le 23 avril à 14h15 Battelle

### Topology of the spaces of rational maps.

Yakov Mostovoy (CINVESTAV, Mexico City)

It often happens in geometry that spaces of continuous objects (such as maps, cycles etc) are approximated by the spaces of corresponding algebraic objects. A typical example of this phenomenon is the well-known theorem of Segal which says that the space of rational functions of degree d on a Riemann sphere is homotopy equivalent, up to dimension d, to the space of all continuous maps of degree d form S^2 to S^2. I will discuss the generalizations of this result and their proof via the Vassiliev's theory of complements of discriminants.

Mercredi le 21 avril à 15h15 Battelle:

### Lagrangian topology and disk counting.

Paul Biran (ETH)

Abstract: In this talk we will explain a new approach to for constructing invariants of Lagrangian submanifolds in symplectic manifolds. We will explain how to construct the so called Lagrangian quantum homology, its basic properties and how to extract from it new invariants. We will then show how these are related to questions on enumeration of holomorphic disks. If time permits we will show how this theory is related to questions on Lagrangian cobordisms. The talk is based on a series of joint works with Octav Cornea.

Vendredi le 16 avril à Battelle:

### Calcul tropical de nombres caractéristiques du plan projectif.

Benoit Bertrand (Institut de Mathématiques de Toulouse)

Les nombres caractéristiques Z(d,g,t), aussi appelés nombres de Zeuthen, sont les nombres de courbes de degré d et de genre g passant pas 3d-1+g-t points et tangentes à t courbes. J'expliquerai comment tropicaliser le problème et utiliser les diagrammes en étages pour calculer ces nombre en genre 0 et 1. Travail en commun avec E. brugalle et G. Mikhalkin

Lundi le 29 mars à 15h15 Battelle:

### Real enumerative geometry and String theory.

Johannes Walcher (CERN)

My plan would be threefold: 1. A few real enumerative predictions that can (probably) be understood using "classical" methods. 2. Real enumeration in the context of toric varieties, since we have a good idea of what a complete theory would look like there. 3. Some physics background about stringy intuition, and a few ideas from the B-model.

Jeudi le 25 mars à 14h30-15h30 Battelle:

### Introduction to motivic integration.

Ernesto Lupercio (Cinvestav)

Mercredi le 24 mars à 15h15-17h Battelle:

### Introduction to direct integration.

Albrecht Klemm (Uni Bonn)

Mercredi le 24 mars à 11h-12h Battelle:

### Lawson Homology.

Jacob Mostovoy (Cinvestav)

### Microcourse in Hodge Theory I

Luca Migliorini (University of Bologna)

Tuesday, April 28, 16:15, Room 17

In this lecture I will give the definition of a pure polarized Hodge structure(PHS), show the link of PHS of weight -1 with abelian varieties and curves and illustrate the first general results about variations of Hodge structures (VHS) focusing on the important property of transversality.

### Microcourse in Hodge Theory II

Luca Migliorini (University of Bologna)

Wenesday, April 29, 15:00

After the definition of a Mixed Hodge structure (MHS) and an informal discussion of the theorem of Deligne on the MHS associated with an algebraic variety, I will illustrate the single most important result in the theory of variation of Hodge structures: the orbit theorem of Schmid, and discuss the MHS arising this way and its geometric significance in the case of VHS arising from degenerations of algebraic varieties.