This small event is to celebrate the very successful collaboration between the institutes involved in the Institute of the Mathematical Sciences of the Americas Consortium (IMSAC), solidify our connections and expand even more the prosperous cooperation.
A seminar talk by Marta Pieropan, Utrecht University
Abstract: In joint work with Damaris Schindler we develop a new version of the hyperbola method for counting rational points of bounded height that generalizes the work of Blomer and Brüdern for products of projective spaces. The hyperbola method transforms a counting problem into an optimization problem on certain polytopes. For rational points on subvarieties of toric varieties, the polytopes have a geometric meaning that reflects Manin's conjecture, and the same holds for counts of Campana points of bounded height. I will present our results as well as some general heuristics.
From May 27 to 31, 2024, the International Conference Modern Methods in Nonlinear Elliptic and Parabolic PDE is being held in Plovdiv. Organizers of the event are the International Center for Mathematical Sciences at the Institute of Mathematics and Informatics with the financial support of the Simons Foundation.
The International Center for Mathematical Sciences – Sofia is organizing for a fourth consecutive year a meeting of young Bulgarian mathematicians from around the world. The conference will take place at the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences (IMI-BAS) on June 11 – 12, 2024.
The talk aims to review the progress in the study of higher rank P-and Q-polynomial association schemes, starting with the foundational concepts and recent advancements as outlined in the referenced papers. It will provide explicit examples of multivariate P-and Q-polynomial association schemes and explore the implications of these developments. The discussion will draw from multiple sources, including recent significant contributions by Bernard et al. and Crampé et al., and will conclude with speculations on future directions for research in this area.
This talk is based on the two joint papers with Hirotake Kurihara (Yamaguchi University), Da Zhao (East China University of Science and Technology) and Yan Zhu (Shanghai University for Science and Technology).
Classical amoebas serve as a tool to study algebraic varieties and as one of the entry points to tropical geometry. The original definition involves a logarithmic projection of a subvariety of a complex algebraic torus, which can be interpreted as forgetting the phase, i.e. the arguments of complex numbers. In group-theoretic terms, this projection map may be thought of as passing to the quotient by the maximal compact subgroup. Suppose one replaces the algebraic torus with a complex three-dimensional matrix group $PSL_2 \mathbb{C}$. In that case, the analogous projection naturally has a three-dimensional hyperbolic space as its target, and it still makes sense to consider images, i.e. hyperbolic amoebas, of complex algebraic varieties under this map. I will review some of the basic properties of hyperbolic amoebas, extending a fascinating interplay between complex and hyperbolic geometries.
In order to classify objects in singularity theory and algebraic geometry, we define invariants associated to varieties or germs of singularities and hope to have enough tools to compute them easily. From classical projective duality, for any variety X, we can define a dual variety X∗ and we call the class of X the degree of X∗. Plücker’s formula allows one to compute this class for plane curves with a certain set of nodes, cusps and tacnodes.
In the last few decades, Berkovich’s theory of k -analytic space has extended classic rigid geometry. Since k -analytic space has good topological properties, and the Berkovich analytification of algebraic varieties could be also dealt with via the geometry of the model and it has a strong connection with tropical geometry, this allows us to use the combinatorial techniques to study algebraic varieties. Our lectures will mainly discuss the contents of Berkovich space and its application in other aspects of mathematics, we plan to go over the basic properties of Berkovich space in both algebraic and topological ways for the first lecture.
A seminar talk by Daniela Martinez Correa, Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”
In the last few decades, Berkovich’s theory of k -analytic space has extended classic rigid geometry. Since k -analytic space has good topological properties, and the Berkovich analytification of algebraic varieties could be also dealt with via the geometry of the model and it has a strong connection with tropical geometry, this allows us to use the combinatorial techniques to study algebraic varieties. Our lectures will mainly discuss the contents of Berkovich space and its application in other aspects of mathematics, we plan to go over the basic properties of Berkovich space in both algebraic and topological ways for the first lecture.
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