Assistant Professor Antoni Rangachev from the Institute of Mathematics and Informatics and Institut de Mathématiques de Jussieu – Paris Rive Gauche receives a 5-year PROMYS grant from the Swiss National Science Foundation (SNSF). PROMYS is a funding programme by the SNSF as part of the second Swiss contribution to selected EU member states. It is intended to create promising and interesting career opportunities for outstanding researchers and thus counteract a “brain drain” from these states.

The “Topological, Metric and Algebraic Equisingularity Theory” project will be carried out at IMI-BAS. Its aim is to help Dr. Rangachev establish a research group in singularities. Dr. Rangachev will be assisted in this endeavor by Prof. Norbert A’Campo (University of Basel) and Prof. Zsolt Patakfalvi (EPFL). The project will fund several postdoctoral positions, graduate students, a singularities seminar, research trips for the team members, and conferences in singularity theory that will be held in Bulgaria.

Dr. Rangachev specializes in singularity theory, algebraic geometry and commutative algebra. He focuses on studying how singular varieties that depend on parameters change under parameter perturbations. Singular varieties are geometric objects that are defined as the solutions of systems of polynomial or analytic equations in many variables and that locally differ from the appearance of Euclidean spaces. When some of the variables in the defining equations of a variety are independent parameters, the variety can be viewed as a family composed of subvarieties called fibers each of which corresponds to a particular choice of parameters. The changes that can take place in a family are of topological, metric or algebrо-geometric nature. There are two extreme cases that can occur. The first one is if there is no change: the fibers of the family exhibit the same nature. Such families are called equisingular. The other extreme case is that of maximum change: the nearby fibers around a singular fiber are smooth (locally resembling Euclidean spaces). Such families are called smoothings.

The main research goals of the project are as follows. In the case of equisingular families the goal is to understand the interplay among various equisingularity notions. In the case of smoothing, the goal is to develop methods to compute topological invariants of the smooth fibers directly from the defining algebraic equations of the singular fiber.

Antoni Rangachev is a Massachusetts Institute of Technology graduate. He obtained his PhD from Northeastern University in 2017 under the supervision of Prof. Terence Gaffney and Prof. Steven Kleiman. He was a Dickson Instructor at the University of Chicago from 2017 to 2021 and a Peter Beron fellow at IMI-BAS from 2021 to 2023. Currently, Antoni is a Marie Skłodowska Curie fellow at Institut de Mathématiques de Jussieu in Paris and the CNRS where he works with Prof. Bernard Teissier.