2024/2025🔗
Wojciech Kryński🔗
Institute of Mathematics of the Polish Academy of Sciences
Date: 29th of April 2025
Title: Path geometries
Abstract: A path geometry on a manifold is a collection of unparameterized curves with the property that through any point, there is exactly one curve passing through this point in each tangent direction. A flat model is given by the set of lines in Euclidean space. More generally, one can consider geodesics of a metric. Locally, path geometries are described by second-order systems of ordinary differential equations. In this talk, I will begin with a review of the general theory of path geometries, in particular describing the notions of torsion and curvature. I will then show how such structures naturally arise in many geometric contexts, enabling the study of various geometric structures in a unified way. I will focus on (para)CR-geometry.
Christophe Eyral🔗
Institute of Mathematics of the Polish Academy of Sciences
Date: 15th of April 2025
Title: Quasi-convenient Newton non-degenerate line singularities and Bekka’s (c)-regularity
Abstract: PDF
Gabriel Pietrzkowski🔗
Warsaw University of Technology
Date: 8th of April 2025
Title: Introduction to the Cartan-Kähler theory
Abstract: After introducing the concept of an external differential system (EDS), and its connection with partial differential equations and differential geometry, I will review the classical theorems of Frobenius, Darboux, and Goursat for Pfaffian systems. This will give basic examples for general theorem of Cartan and Kähler who gave sufficient conditions for the existence of integral manifolds of analytic EDS.
Jarosław Buczyński🔗
University of Warsaw
Date: 1st of April 2025
Title: Singularities of complex projective curves of low degree
Abstract: In order to study projections of smooth curves and the singularities of the image curve, we introduce multifiltrations obtained by combining flags of osculating spaces. In our research we classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree d rational curves in CP^n when d-n <4 and d<2n. During the talk I will focus on an illustrative example. The set of linear spaces giving a fixed singularity type is related to a specific Schubert cycle (which depends on the singularity).
Based on a joint work with Nathan Ilten and Emanuele Ventura, arxiv, International Mathematical Research Notices.
Maksymilian Safarewicz🔗
Warsaw University of Technology
Date: 25th of March 2025
Title: The geometry of the third-order preserving set
Abstract: The presentation aims to introduce and explain the 3rd order preserving sets i.e. sets which support functions preserve only every third coefficient of its Fourier series. We will consider geometric properties of such sets, formulate their isoperimetric type problem, investigate their singularities, and consider possible generalizations.
Tomasz Miller🔗
Jagiellonian University
Date: 15th of October 2024
Title: Causal evolution of probability measures and continuity equation
Abstract: One of the most important concepts of relativistic physics is that of a world line, i.e., the space-time trajectory of a point particle. In Lorentzian geometry, it is modelled by the so-called causal curve, and the questions concerning which spacetime points can be connected by means of causal curves lead to a vast area of study known as causality theory. In the talk, I will present how the basic notions of causality theory can be naturally extended to probability measures on spacetimes (what is motivated by both classical and quantum physics). In particular, I will discuss the notion of a causal evolution of probability measures, its deep connection with the continuity equation (known from elementary physics) and a surprisingly nice topological properties of the space of causal curves.
Maksymilian Safarewicz🔗
Warsaw University of Technology
Date: 8th of October 2024
Title: Isoperimetric inequalities
Abstract: This presentation explores isoperimetric inequalities, which are fundamental results in mathematics linking the area of a shape to its perimeter. We will discuss the historical context and significance of these inequalities, highlighting classical results such as the isoperimetric theorem in the plane, which asserts that among all simple closed curves with a given length, the circle encloses the maximum area. Additionally, we will examine various extensions and generalizations of these inequalities. The presentation aims to provide a comprehensive overview of the principles underlying isoperimetric inequalities.