Dates: May 1 - 5, 2023
Location: Lakeside Village Auditorium, University of Miami (May 1, 3-5), Coral Gables, FL & Whitten Learning Center, Room 180 (May 2), University of Miami, Coral Gables, FL
Organizers: Ernesto Lupercio, Ludmil Katzarkov
Please register here
This is a joint event between the International Centre for Mathematical Sciences Bulgarian Academy of Sciences (ICMS), and the Institute for the Mathematical Sciences of the Americas.
The Institute of Mathematical Sciences of the Americas (IMSA) at the University of Miami is pleased to announce an upcoming Quantum Toric Geometry and O-Minimality conference. The conference will be held from May 1 - 5, 2023 at the UM campus. The study of Quantum Toric Geometry has emerged as a new and important area of research in recent years. It is a rich and exciting field with deep connections to a wide range of topics in mathematics, including algebraic geometry, symplectic geometry, representation theory, and mathematical physics. Quantum Toric Geometry provides a new perspective on classical Toric Geometry, which studies the geometry of algebraic varieties that admit an action of a torus.O-Minimality is another area of mathematics that has seen significant recent developments. It is a branch of mathematical logic that studies the properties of definable sets in o-minimal structures. O-Minimality has become an important tool in algebraic geometry, providing a way to study the geometry of varieties using tools from logic .Recent work suggests a deep and fundamental relationship between Quantum Toric Geometry and O-Minimality. This conference aims to bring together experts from both fields to explore these connections and to discuss recent advances in both areas. The conference will feature a series of talks and discussions by leading researchers in the field, covering a wide range of topics in Quantum Toric Geometry and O-Minimality. The conference will be an excellent opportunity for researchers and students to learn about the latest developments in these exciting and important fields and to meet and interact with leading experts from around the world.
9:30am |
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 |
11:00am |
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 |
1:30pm |
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 |
3:00pm |
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 |
4:00pm |
Discussion |
9:30am |
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 |
11:00am |
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 |
1:30pm |
C. Ruiz Video |
3:00pm |
C. Ruiz Video |
4:00pm |
Discussion |
9:30am |
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 |
11:00am |
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 |
1:30pm |
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 |
3:00pm |
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 |
5:00pm |
9:30am |
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 |
11:00am |
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 |
1:30pm |
Velichka Milousheva, IMI, Sofia: Click here for Title & Abstract Video |
3:00pm |
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 |
4:00pm |
9:30am |
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 |
11:00am |
Enrique Becerra, University of Miami Video |
1:30pm |
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 |
2:45pm |
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 |
4:00pm |
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 |
Speakers
C. R. Alonso* | |
Carolina Araujo | IMPA |
V. del Barco* | UNICAMPI |
Enrique Becerra | IMSA |
A. Boivin* | NDSU |
Leonardo Cavenaghi | UNICAMPI |
Anna Fino | FIU |
Lino Grama | UNICAMPI |
Gueo Grantcharov | FIU |
Kyoung Seog-Lee | UMiami |
Ernesto Lupercio | CINVESTAV |
Velichka Milousheva | BAS |
Eder de Moares | UNICAMPI |
Tony Pantev | UPenn |
Aleksander Petkov | Sofia Univeristy |
Josef Svoboda | UMiami |
Y. Ufnarovskii* | Lehigh University |
B. Uribe | Universidad del Norte, Barranquilla, Colombia |
Yuri Ustinovski | Lehigh University |
A. Witte | Utecht |
Mirroslav Yotov | FIU |
*To be confirmed