Dates: August 14 - 18, 2023
Location: Sunny Beach, Bulgaria
In recent months o-minimal structures have become a powerful tool in Hodge theory. The conference “Generalized and Symplectic Geometry” will serve as a step consolidating the use of o-minimal structure in non Kaehler geometry. Many different approaches to generalized geometries – from logic to Homological Mirror Symmetry will be discussed. The conference will be an excellent opportunity for researchers and students to learn about the latest developments in these fields and to meet and interact with leading experts from around the world. Proceedings will be published.
The event is jointly organized by the Institute for the Mathematical Sciences of the Americas (IMSA) at the University of Miami, and the International Centre for Mathematical Sciences (ICMS-Sofia) at the Institute of Mathematics and Informatics in Sofia.
For the event poster, click here.
9:15am |
Opening |
9:30am |
Johann Davidov, Institute of Mathematics and Informatics, Sofia: Generalized Metrics and Generalized Twistor Spaces In the first part of this talk, basic facts about the generalized complex geometry will be recalled. Then the twistor construction for Riemannian manifolds will be extended to the case of manifolds endowed with generalized metrics. The generalized twistor space associated to such a manifold is defined as the bundle of generalized complex structures on the tangent spaces of the manifold compatible with the given generalized metric. This space admits natural generalized almost complex structures whose integrability conditions will be discussed. An interesting feature of the generalized twistor spaces, which usual twistor spaces do not admit, is the existence of intrinsic isomorphisms. |
10:30am |
Phillip Griffiths, Institute for Advanced Study, USA: Atypical Hodge Loci* Recent work by a number of people has shown that for a general family of smooth varieties of dimension at least 3 the non-trivial Hodge loci have strictly bigger dimension than expected. This talk will sketch the proof of their result and explain why it is true. *Talk based on the paper [BKU] and related works given in the references in that work, and on extensive discussions with Mark Green and Colleen Robles. |
4:00pm |
Lino Grama, University of Campinas – UNICAMP: 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=2𝑠𝐶, where 𝑠 is the Riemannian scalar curvature and 𝑠𝐶 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. |
5:10pm |
Eder Correa, University of Campinas - UNICAMP: t-Gauduchon Ricci-flat metrics on non-Kähler Calabi-Yau manifolds In this talk, we will present recent results related to the construction of t-Gauduchon Ricci-flat metrics on non-Kähler Calabi-Yau manifolds defined by certain suspensions of Sasaki-Einstein manifolds. These include every compact Hermitian Weyl-Einstein manifold, every compact locally conformal hyperKähler manifold, certain suspensions of Brieskorn manifolds, and every generalized Hopf manifold provided by suspensions of exotic spheres. These examples generalize previous constructions known for Hopf manifolds. |
6:10pm |
Mina Teicher, University of Miami, USA: Numbers in the Brain- Can I read your thoughts? Unlocking the mystery of the brain is one of the most intriguing challenges. Trying to reveal the real model of brain activity is a target of many mathematicians and physicists, who try to do it via one of the brain controls - either Motiric, cognitive or emotional. I took as a toy one of the highest brain functions - Numeration. With my students we proved using MEG recording that there is a theoretical concept or control of a number in the brain. More over we succeeded to build a classification metric and identify different numbers in recording of brain activity, and then guess what number one is thinking on. This was never done before! Our next target is to differentiate between math gifted students and regular students, using recordings of brain activity. Later, to differentiate from brain recordings if one is more talented to algebra or to geometry. |
9:00am |
Vestislav Apostolov, Université du Québec à Montréal, Canada and Institute of Mathematics and Informatics, Sofia: The generalized Calabi problem The notion of a generalized Kähler (GK) structure was introduced in the early 2000’s by Hitchin and Gualtieri in order to provide a mathematically rigorous framework of certain nonlinear sigma model theories in physics. Since then, the subject has developed rapidly. It is now realized, thanks to more recent works of Hitchin, Goto, Gualtieri, Bischoff and Zabzine, that GK structures are naturally attached to Kähler manifolds endowed with a holomorphic Poisson structure. Inspired by Calabi’s program in Kähler geometry, which aims at finding a “canonical” Kähler metric in a fixed deRham class, I will present in this talk an approach towards a “generalized Kähler” version of Calabi’s problem motivated by an infinite dimensional moment map formalism. As an application, we give an essentially complete resolution of the problem in the case of a toric complex Poisson variety. |
10:00am |
Geou Grantcharov, Florida International University, USA: Hermitian metrics on compact complex homogeneous spaces Complex manifolds which admit a transitive action by a compact group of biholomorphisms were first studied by Wang and are known to admit a canonical fibration over a rational homogeneous space. Such spaces in general are non-Kähler unless the fibration is trivial. In this talk I'll consider the existence of special Hermitian metrics on them like balanced, pluriclosed, and others. The results are partly based on joint works with A. Fino nd L. Vezzoni. |
11:00am |
Christopher Brav, HSE University, Moscow, Russia: The cyclic Deligne conjecture and Calabi-Yau structures The Deligne conjecture, many times a theorem, states that for a dg category C, the dg endomorphisms End(Id_C) of the identity functor - that is, the Hochschild cochains - carries a natural structure of 2-algebra. When C is endowed with a Calabi-Yau structure, then Hochschild cochains and Hochschild chains are identified up to a shift, and we may transport the circle action from Hochschild chains onto Hochschild cochains. The cyclic Deligne conjecture states that the 2-algebra structure and the circle action together give a framed 2-algebra structure on Hochschild cochains. We establish the cyclic Deligne conjecture, as well as a variation that works for relative Calabi-Yau structures on dg functors D --> C, more generally for functors between stable infinity categories. We discuss examples coming from oriented manifolds with boundary, Fano varieties with anticanonical divisor, and doubled quivers with preprojective relation. This is joint work with Nick Rozenblyum. |
4:00pm |
Yingdi Qin, Institute of Mathematics and Informatics, Sofia: Polydifferential operators, shifted symplectic geometry and Generalized complex branes Generalized complex geometry was introduced by N. Hitchin partly motivated by Mirror Symmetry. It is an interpolation between complex geometry and symplectic geometry. A new approach to generalized complex geometry was introduced based on Tony Pantev’s idea from (shifted) Poisson geometry and motivated by my thesis on Coisotropic Branes. This approach utilizes shifted symplectic geometry and a new theory of homotopy complex structures on derived differential stacks. The theory of homotopy complex structure is built on a strict model of differential forms on formal differential stacks based on polydifferential operators. I will also explain the corresponding results for generalized complex branes which are geared towards a construction of a category of generalized complex branes. |
5:10pm |
Morgan Brown, University of Miami, USA: Birational, Berkovich, and Tropical Geometry I will give an overview of the relationship between constructions of dual complexes in birational geometry, skeletons in Berkovich geometry, and the tropicalization map. |
6:10pm |
Mancho Manev, University of Plovdiv Paisii Hilendarski and Medical University – Plovdiv: Some Almost Riemann Solitons on Conformal Cosymplectic Contact Complex Riemannian Manifolds Almost Riemann solitons are introduced and studied on an almost contact complex Riemannian manifold, i.e. an almost contact B-metric manifold, obtained from a cosymplectic manifold of the considered type by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. The potential of the studied soliton is assumed to be in the vertical distribution, i.e. it is collinear to the Reeb vector field. In this way, manifolds from the four main classes of the studied manifolds are obtained. Curvature properties of the resulting manifolds are derived. An explicit example of dimension five is constructed. The Bochner curvature tensor is used (for dimension at least seven) as a conformal invariant to get properties and construct an explicit example in relation to the obtained results. |
9:00am |
Rodolgo Aguilar, University of Miami, USA: Around homology planes: old and new We will present a survey about old and new results on affine acyclic complex algebraic surfaces, sometimes called homology planes. Relations with topology, symplectic and algebraic geometry will be discussed. |
10:00am |
Sebastian Torres, University of Miami, USA: Windows and moduli of bundles on a curve We survey some results regarding semi-orthogonal decompositions on moduli spaces of vector bundles on a curve, with an emphasis on those that use the machinery of windows. |
11:00am |
Dennis Borisov, University of Windsor: Derived Quot-schemes as dg manifolds A construction of derived Quot-schemes will be presented realizing them as dg manifolds of finite type. This is used to prove that moduli stacks of sheaves on Calabi-Yau 4-folds are critical loci of globally defined shifted potentials. Joint work with L. Katzarkov and A. Sheshmani. |
4:00pm |
Jiachang Xu, Institute of Mathematics and Informatics, Sofia: Aspects of Mirror Symmetry for Abelian Varieties In this talk, I will introduce the mirror symmetry for abelian varieties in terms of moduli spaces associated with abelian varieties. To understand the Hodge theoretical and non-Archimedean aspects of moduli spaces associated with the mirror pairs is an ongoing project joint with Muyuan Zhang. |
5:10pm |
Enrique Becerra, University of Miami, USA: The stringy spectrum of orbifolds In this talk I will introduce the stringy spectrum of a complex algebraic orbifolds as an invariant of motivic nature and state its basic properties. |
6:10pm |
Erik Paemurru, University of Miami, USA: Log canonical thresholds of high multiplicity plane curves We classify log canonical thresholds at points of multiplicity 𝑑−1 for plane curves of degree 𝑑. As a consequence, we describe all possible values of log canonical threshold that are less than 2𝑑/1 for plane curves of degree 𝑑. In addition, we compute log canonical thresholds for all plane curves of degree less than 6. |
Aleksandar Petkov, Sofia University and Institute of Mathematics and Informatics, Sofia, Bulgaria | Ivan Minchev, Sofia University, Bulgaria | Richard Paul Horja,University of Miami, USA | Morgan Brown, University of Miami, USA |
Andras Nemethi, Alfréd Rényi Institute of Mathematics, Hungary | Jaqueline Mesquita, Department of Mathematics at University of Brasília, Brazil | Robert Stephen Cantrell, University of Miami, USA | Christopher Brav, HSE University, Moscow, Russia |
Artan Sheshmani, Harvard University, USA | Jiachang Xu, Institute of Mathematics and Informatics, Sofia, Bulgaria | Rodolfo Aguilar, University of Miami, USA | Tony Pantev, University of Pennsylvania, USA |
Bernardo Uribe, Universidad del Norte, Barranquilla, Colombia | Johann Davidov, Institute of Mathematics and Informatics, Sofia, Bulgaria | Sebastian Torres, University of Miami, USA | |
Dennis Borisov, University of Windsor, Canada | Josef Svoboda, Caltech, USA | Stefan Ivanov, Sofia University and Institute of Mathematics and Informatics, Sofia, Bulgaria | |
Eder de Moraes Correa, University of Campinas, Brazil | Lino Grama, University of Campinas – UNICAMP, Brazil | Velichka Milousheva, Institute of Mathematics and Informatics, Sofia, Bulgaria | |
Enrique Ruby Becerra, University of Miami, USA | Ludmil Katzarkov, University of Miami and Institute of Mathematics and Informatics, Sofia | Vestislav Apostolov, Université du Québec à Montréal and Institute of Mathematics and Informatics, Sofia | |
Erik Paemurru, University of Miami, USA | Mancho Manev, Plovdiv University, Bulgaria | Umut Varolgunes, Department of Mathematics, Boğaziçi University, Istanbul, Turkey | |
Ernesto Lupercio, Cinvestav-IPN, Mexico | Mina Teicher, University of Miami, USA | Daniel Pomerleano, University of Massachusetts, Boston, USA | |
Gueo Grantcharov, Florida International University, USA | Phillip Griifiths, Institute for Advanced Study, USA | Dimitar Kodjabachev, Institute of Mathematics and Informatics, Sofia, Bulgaria |