Conference: Quantum Toric Geometry, Generalized Geometries and O-Minimality

Tony Pantev, University of Pennsylvania: Generalized complex branes, doubling, and shifted symplectic geometry, part I

I will describe a symplectic approach to generalized complex geometry which is geared towards a construction of a natural category of  generalized complex branes (at the large volume limit) and allows us to define and work with generalized complex  branes with substrates which are not submanifolds. The approach utilizes shifted symplectic geometry and a new theory of homotopy complex structures on derived differentiable stacks. I will discuss the general  theory and will explain its implications for generalized complex and generalized Kaehler  geometry. This is a report of a joint work in progress with P. Safronov and B. Pym and on recent work of Y. Qin.  Video

Tony Pantev, University of Pennsylvania: Generalized complex branes, doubling, and shifted symplectic geometry, part II

I will describe a symplectic approach to generalized complex geometry which is geared towards a construction of a natural category of  generalized complex branes (at the large volume limit) and allows us to define and work with generalized complex  branes with substrates which are not submanifolds. The approach utilizes shifted symplectic geometry and a new theory of homotopy complex structures on derived differentiable stacks. I will discuss the general  theory and will explain its implications for generalized complex and generalized Kaehler  geometry. This is a report of a joint work in progress with P. Safronov and B. Pym and on recent work of Y. Qin.  Video

Ernesto Lupercio, CINVESTAV: Quantum Toric Geometry and Tame Geometry via Motivic Rings I

In these talks, we will investigate the fascinating connection between the non-commutative geometry of quantum toric manifolds and tame geometry. Through explicit computations of Hodge numbers of LVM manifolds, we aim to provide a conjectural roadmap for further explorations of this relationship. This is joint work with L. Katzarkov, K.S: Lee, y L. Meersseman.  Video

Ernesto Lupercio, CINVESTAV: Quantum Toric Geometry and Tame Geometry via Motivic Rings II

In these talks, we will investigate the fascinating connection between the non-commutative geometry of quantum toric manifolds and tame geometry. Through explicit computations of Hodge numbers of LVM manifolds, we aim to provide a conjectural roadmap for further explorations of this relationship. This is joint work with L. Katzarkov, K.S: Lee, y L. Meersseman.  Video  

Discussion

Kyoung Seog-Lee, University of Miami: Logarithmic Transformations and vector bundles on elliptic surfaces via o-minimal geometry I

Logarithmic transformation is an important operation introduced by Kodaira in the 1960s. One can obtain an elliptic surface with multiple fibers by performing logarithmic transformations of an elliptic surface without multiple fibers. On the other hand, vector bundles on elliptic surfaces are important objects in many branches of mathematics, e.g., algebraic geometry, gauge theory, mathematical physics, etc. In this talk, I will discuss how certain vector bundles on elliptic surfaces are changed under logarithmic transformations from the perspective of o-minimal geometry. This talk is based on several joint works (some in progress) with L. Katzarkov and E. Lupercio.  Video

Kyoung Seog-Lee, University of Miami: Logarithmic Transformations and vector bundles on elliptic surfaces via o-minimal geometry II

Logarithmic transformation is an important operation introduced by Kodaira in the 1960s. One can obtain an elliptic surface with multiple fibers by performing logarithmic transformations of an elliptic surface without multiple fibers. On the other hand, vector bundles on elliptic surfaces are important objects in many branches of mathematics, e.g., algebraic geometry, gauge theory, mathematical physics, etc. In this talk, I will discuss how certain vector bundles on elliptic surfaces are changed under logarithmic transformations from the perspective of o-minimal geometry. This talk is based on several joint works (some in progress) with L. Katzarkov and E. Lupercio.  Video

C. Ruiz  Video

C. Ruiz  Video

Discussion

Lino Grama, CAMPINAS: T-duality and generalized G_2 structures

In the first part of the talk we will describe the construction of the (infinitesimal) T-duality on nilmanifolds and explore some consequences. On the second part,  given a classical G_2-structure on certain seven dimensional manifolds, either closed or co-closed, we use T-duality to obtain integrable generalized G_2-structures which are no longer a usual one, and with non-zero three form in general. In particular we obtain manifolds admitting closed generalized G_2-structures not admitting closed (usual) G_2-structures.This is a joint work with V. del Barco and L.Soriani  Video

Lino Grama, CAMPINAS: Kähler-like scalar curvature on homogeneous spaces

In this talk, we will discuss the curvature properties of invariant almost Hermitian geometry on generalized flag manifolds. Specifically, we will focus on the "Kähler-like scalar curvature metric" - that is, almost Hermitian structures (g,J) satisfying s=2s_C, where s is the Riemannian scalar curvature and s_C is the Chern scalar curvature. We will provide a classification of such metrics on generalized flag manifolds whose isotropy representation decomposes into two or three irreducible components. This is a joint work with A. Oliveira.  Video

Gueo Grantcharov, Florida International University: Introduction to generalized K\"ahler geometry I

We'll start with description of the equivalence between generalized K\"ahler and bihermitian structures and first focus on the main features in the dimension four. Then the results will be extended to higher dimensions. I'll explain the relations with holomorphic Poisson geometry and the analog of the standard K\"ahler properties like existence of local K\"ahler potential and stability under small deformations in the generalized setting.  Video

Gueo Grantcharov, Florida International University: Introduction to generalized K\"ahler geometry II

We'll start with description of the equivalence between generalized K\"ahler and bihermitian structures and first focus on the main features in the dimension four. Then the results will be extended to higher dimensions. I'll explain the relations with holomorphic Poisson geometry and the analog of the standard K\"ahler properties like existence of local K\"ahler potential and stability under small deformations in the generalized setting.  Video

Carolina Araujo, IMPA: Frontiers Lecture  Video

Aldo Witte, UTRECHT: T-duality with fixed points

T-duality provides a powerful framework for describing symmetries between quantum field theories with very different behaviour. Mathematically it can be described as a symmetry between principal torus bundles, and can be used to transport geometric structures (e.g. generalized complex structures) between these. However the existing theory cannot incorporate torus actions which have fixed points. In this talk we will establish such a framework, making use of a class of Lie algebroids called elliptic tangent bundles. This will provide us with interesting new examples of T-dual generalized complex structures. Joint work with Gil Cavalcanti.  Video

Y. Ustinovskii, Leigh University: Generalized Kähler constant scalar curvature problem

In this talk we consider the space of symplectic generalized Kähler (GK) structures on a given holomorphic Poisson manifold $(M,J,\pi)$. Fixing the cohomology class of the underlying symplectic form $[F]$ we denote this space by $GK_{[F]}(M,J,\pi)$. Recently Goto and Boulanger arrived at the moment map framework for the action of the group of hamiltonian diffeomorphisms $Ham(F)$ on the space of generalized almost Kähler structures, and used it to define the notion of the GK scalar curvature. Following the classical Kähler setup, we formulate the (generalized Kähler) Calabi Problem of finding a constant GK scalar curvature (cscGK) in $GK_{[F]}(M,J,\pi)$. We use the GIT interpretation of the moment map to develop the GK analogues of the notions of  Futaki invariant, Mabuchi energy and Mabuchi metric familiar from the Kähler setting. This allows us to recast Calabi-Lichnerowicz-Matsushima obstruction for the cscGK metrics in terms of the automorphism group of (J,\pi) and prove conditional uniqueness of cscGK metrics in $GK_{[F]}(M,J,\pi)$. In the special case when $(M,J,\pi)$ is a \emph{toric} Poisson manifold, we prove an existence result for the cscGK problem, constructing new examples of the cscGK metrics on P^2. (Based on a joint work with V.Apostolov and J.Streets)  Video

Velichka Milousheva, IMI, Sofia: Click here for Title & Abstract  Video

Anna Fino, Florida International University: Canonical metrics in Complex Geometry

An important tool to study  complex non Kaehler manifolds is to look for "canonical metrics",  where the word canonical is referred to some special properties of the associated fundamental form. A Hermitian metric on a complex manifold is called  pluriclosed if the torsion of the associated Bismut connection is closed and it is called balanced if its fundamental form is co-closed. In the talk I will focus on pluriclosed and balanced metrics, showing some  general results and new  constructions of compact
non-Kaehler manifolds. In particular, we will see how pluriclosed metrics lie halfway between generalized Hermitian and generalized Kaehler structures.  Video

Carolina Araujo, IMPA: Frontiers Lecture  Video

Leonardo Cavenaghi, CAMPINAS: A symmetric approach to General Relativity: Applications to exotic spheres

A procedure known as Cheeger deformation, developed initially in the 70s by Jeff Cheeger to produce manifolds with non-negative sectional curvature, consists of shrinking the geometry of a manifold with an isometric action in the direction of the orbits. The idea is to start with a Riemannian manifold (M,g) with a group action by a Lie Group G, regarded with a bi-invariant metric Q, and induce a one-parameter of metrics on M, g_t, via the submersion metric (MxG, g + t^{-1}Q) -> (M,g_t). If we take M = R^4 and G = R with the translation action in the last factor, plugging a specific choice of t < 0 induces on R^4 the
Minkowski metric and the Lorentz group can be derived from the
symmetries originally defined on R^4 (given by the translations by R).
If we take S^3 instead of R^4 and G = S^1, repeating the procedure can lead to the "analogous compact model" for the Minkowski space. In this talk, we intend to repeat the analysis in dimension 7. Here, M = S^7 or Sigma^7, i.e., it is either a classical sphere or a manifold homeomorphic but not diffeomorphic to S^7 (it is an exotic sphere). One of our interests in Sigma^7 relies on the fact that this manifold, with a Riemannian metric, can never satisfy the hypothesis of "isotropic" in the physical sense, so it is of interest to see how physical laws work, lacking such a hypothesis. As applications, however, we show that fixed points exist for the S^1-action on S^3 and M. However, the metric can be continued smoothly to define global Lorentzian metrics on these manifolds (which are time-oriented). We show that such metrics enjoy the same global geometric properties, indicating that smooth structures need not distinguish physics for compact models. This is an inaugural work made in collaboration with Prof. Lino Grama.  Video

Enrique Becerra, University of Miami  Video

V. Tsanov, IMI, Sofia: On some fans and toric varieties related to branching laws for reductive groups

Given an embedding H<G of connected complex reductive linear algebraic groups, the problem of describing the H-invariant vectors in finite dimensional representations of G can be translated in terms of Geometric Invariant Theory for the H-action on the complete flag variety G/B. An approach initiated in this generality by Heckman and further developed by Berenstein-Sjamaar, Belkale-Kumar, Ressayre, among others, has lead to a description of the H-ample cone. In this talk, based partly on joint work with Seppaenen, I shall present a combinatorial description of the GIT-classes and exhibit some special properties of the GIT-fan. This construction allows to encode representation theoretic information on H<G into properties of toric varieties associated to this fan and some relevant lattices.  Video

Aleksander Petkov, Sofia University: The Almost Schur Lemma and the Positivity Conditions in Quaternionic Contact Geometry

The main goal of this talk is to present quaternionic contact (qc) versions of the so called Almost Schur Lemma, which give estimations of the qc scalar curvature on a compact qc manifold to be a constant in terms of the norm of the [−1]-component and the norm of the trace-free part of the [3]-component of the horizontal qc Ricci tensor and the torsion endomorphism, under certain positivity conditions. The talk is based on a joint work with Stefan Ivanov.  Video

Eder M. Correa, UNICAMP: Deformed Hermitian Yang-Mills equation on rational homogeneous varieties

The deformed Hermitian Yang-Mills equation is a differential equation on the compact Kähler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry. Motivated by mirror symmetry in string theory, this equation was independently discovered, around 2000, by Mariño-Minasian-Moore Strominger and Leung-Yau-Zaslow. Since then, it has been extensively studied by both physicists and mathematicians because of its relevance to gauge theory, quantum field theory, and algebraic geometry. In this talk, we will focus on studying the dHYM equation on rational homogeneous varieties. Our main goal is to show that the dHYM equation on a rational homogeneous variety, equipped with any invariant Kähler metric, always has a solution. Additionally, we will explore algebraic obstructions to the existence of specific solutions, known as supercritical solutions, using central charges defined by analytic subvarieties.  Video