Prochains séminaires
Jeudi 5 Juin 2014, à 16h15, Villa Battelle
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
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
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
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.
Lundi 25 Novembre 2013, à 15h30, Villa Battelle
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
Séminaire itinérant
More informations
here.
Lundi 24 juin 2013, à 14h30, Villa Battelle
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
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
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
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
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
Séminaire itinérant
More informations
here.
Jeudi 18 Octobre 2012, 14h30 à 16h00, Villa Battelle
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
Séminaire itinérant
More informations
here.
Vendredi 28 Septembre 2012, 14h30 à 16h00, Villa Battelle
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.
Mardi 25 Septembre 2012, 14h30 à 16h30, Villa Battelle
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
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
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
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
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
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
Sergei Galkin (IPMU Tokyo)
Continuation.
Mercredi 21 mars 2012, 14h15, Villa Battelle
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
Sergei Galkin (IPMU Tokyo)
Continuation.
Lundi 19 mars 2012, 14h30, Villa Battelle
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
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
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
Vladimir Fock (Strasbourg)
Continuation
Mardi 22 novembre 2011, 15h30, Villa Battelle
Vladimir Fock (Strasbourg)
Continuation
Lundi 21 novembre 2011, 16h00, Villa Battelle
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
Séminaire itinérant
Toutes les informations
ici.
Lundi 7 novembre 2011, 16h00, Villa Battelle
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
Michel Brion (Grenoble)
Continuation
Jeudi 20 octobre 2011, Toulouse
Séminaire itinérant
More informations
here.
Mercredi le 18 mai 15h30 Villa Battelle
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
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.
Mardi le 10 mai 15h30 Villa Battelle
Barbara Fantechi (Trieste)
Lundi le 9 mai 10h Villa Battelle
Barbara Fantechi (Trieste)
Jeudi le 5 mai 15h45 Villa Battelle
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
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.
Mardi le 29 mars 15h30 Villa Battelle
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
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
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
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
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
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
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
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
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
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
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
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.
Mercredi le 3 novembre Institut de Recherche Mathématiques Avancée, Strasbourg.
Mercredi le 27 octobre 15h30 Villa Battelle
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
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
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
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
Kentaro Hori (IPMU, Tokyo)
Jeudi le 27 mai Institut de mathématiques de Jussieu, Paris.
Lundi le 17 mai 14h15-15h30 Battelle
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
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
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
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
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
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
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:
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
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:
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:
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:
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:
Ernesto Lupercio (Cinvestav)
Mercredi le 24 mars à 15h15-17h Battelle:
Albrecht Klemm (Uni Bonn)
Mercredi le 24 mars à 11h-12h Battelle:
Jacob Mostovoy (Cinvestav)
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.
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.
The geometry of auctions and competitive equilibrium
with indivisible goods.
All manifolds are contact except those which are obviously not.
Cylinders in del Pezzo surfaces.
Lagrangian non-intersection theory.
Séminaires passés
Special joint seminar : Fables Geometriques / Mathematique Physique
Real normalized differentials and their applications
.
Tropical Geometry in Europe
Mahler measure in geometry and topology.
Tropical Schubert varieties.
(Affine) ADE bundles over del Pezzo surfaces.
Degeneration and mirror symmetry for minuscule varieties.
Topological classification of real rational curves in the plane .
Tropical Geometry in Europe
A method for comparing curve counting invariants.
Tropical Geometry in Europe
Topological strings and knot contact homology.
"Special Workshop"
composed of 3 lectures by Andrey Losev (Moscow)
A-I-B mirror symmetry on toric variety
(chiral algebras approach, based on the joint work with E.Frenkel).
Tropical mirror symmetry for toric varieties and conjecture for varieties of general type.
Beta function in conformal field theories as obstruction in homotopical Maurer-Cartan equation (approach based on the configuration space of points).
Symplectic sum formulas in real enumerative geometry.
Mirror Symmetry and Cluster Varieties.
From configuration spaces to stable rational curves: Kohno-Drinfeld way.
Reflecting Fano.
Betti numbers of random real hypersurfaces.
Reflecting Fano.
Reflecting Fano.
Monsky theorem and tropical geometry.
Knots, dimers and clusters.
Variétés amassées, courbes planaires et systèmes intégrables
Variétés amassées, courbes planaires et systèmes intégrables
Variétés amassées, courbes planaires et systèmes intégrables
Perspectives in Tropical Geometry 2011
Tropical Geometry in Europe
Counting points of homogeneous spaces over finite fields
Counting points of homogeneous spaces over finite fields
Tropical Geometry in Europe
Geometry and topology of curves on surfaces II
Geometry and topology of curves on surfaces I
Séminaire GPS - Géometrie tropicale
Virtual class for the moduli of sheaves on a curve.
Algebraic stacks and virtual classes.
Some enumerative results for real curves of fixed cogenus.
Finite subgroups of Cremona groups
Séminaire GPS - Géometrie tropicale
Non-commutative toric varieties
Formal Mirror Symmetry
Middle-dimensional squeezing and non-squeezing.
On plane real algebraic curves, I
On plane real algebraic curves, II
Invariants of 3-manifolds via counting surfaces
Matrix models and tropical geometry I
Matrix models and tropical geometry II
Minicourse - Hodge theory of character varietes I
Minicourse - Hodge theory of character varietes II
Minicourse - Hodge theory of character varietes III
Toric mirror symmetry
Séminaire GPS - Géometrie tropicale
Séminaire GPS - Géometrie tropicale
SYZ mirror symmetry for toric manifolds I
SYZ mirror symmetry for toric manifolds II
Tropical avatar of the Gelfand-Zeitlin integrable system I.
Tropical avatar of the Gelfand-Zeitlin integrable system II.
Mirror Symmetry and Reality.
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)
Minicourse - Geometry of special holonomy I.
Minicourse - Applications of Category Theory to Geometry: Derived Categories of Coherent Sheaves I .
Minicourse - Geometry of special holonomy II.
Minicourse - Applications of Category Theory to Geometry: Derived Categories of Coherent Sheaves II.
Minicourse - Applications of Category Theory to Geometry: Derived Categories of Coherent Sheaves III.
Computing the alpha-invariant.
Dolgachev's surface.
Non-commutative cluster mutations.
Topology of the spaces of rational maps.
Lagrangian topology and disk counting.
Calcul tropical de nombres caractéristiques du plan projectif.
Real enumerative geometry and String theory.
Introduction to motivic integration.
Introduction to direct integration.
Lawson Homology.
Microcourse in Hodge Theory I
Microcourse in Hodge Theory II