Kolokviální přednáška proběhne 2.11.2022 od 17:00 v posluchárně M1.

Nikolay Scherbina

Title: On the integrability of nonsmooth distributions

Abstract:
TThe existence and uniqueness of integral curves for distributions of lines in ${\mathbb R}^n$ is a classical result which was proved in 1894 by Picard and Lindel{\"o}f for the case of distributions with Lip - regularity (in the modern literature this result is usually called "the existence and uniqueness theorem for ODE"). If the distribution is not Lip - regular, but just continuous, then the existence of integral curves was proved in 1890 by Peano, but these curves are not unique in general.

A similar problem of the existence and uniqueness of integral surfaces for distributions of dimension $k > 1$ is more involved and the corresponding result is usually attributed to the work of Frobenius from 1877 (some earlier results on this problem were also obtained by  Clebsch in 1866 and Deahna in 1840). The integral surfaces for distributions of dimension larger than 1 do not exist in general even for smooth distributions and the condition which guarantees the existence and uniqueness of such surfaces is given in terms of the Lie brackets of the vector fields generating the given distribution.

In our talk we present a "Peano version" of the Frobenius result, more precisely, we give a geometric necessary and sufficient condition for the existence of integral surfaces for distributions of hyperplanes which are just continuous. The corresponding problem for distributions of higher codimension is still completely open.

Nikolay Scherbina is a Full Professor at the University of Wuppertal (Germany). His main field of research in Complex Analysis in Several Variables, with strong interactions with Potential Theory and Topology. He has several distinguished results in the field, mainly in the framework of the Pluripotential Theory, published in top math journals, including Acta Mathematica and Duke Math Journal.

Next Colloquial Talk at our department - online ZOOM meeting.

Title:  Towards the Digital City: methods and applications of urban network analysis and AI
Speaker: Stanislav Sobolevsky

Time: Wednesday, 12 May, 2021, 4 pm.

Dr. Stanislav Sobolevsky will present his prior research as well as the project to start at our department under a recent MASH award, see the invitation link.

Abstract:
The growing scale, complexity, and dynamics of urban systems pose tremendous challenges within urban planning and operations. At the same time, the increasing pervasiveness of digital technology in facilitating urban activities generates a vast amount of data. This big urban data creates fresh opportunities to respond to urban challenges and gain an unparalleled understanding of complex urban systems. And recent network analysis and AI techniques help to address the complexity and interconnectedness of the urban data.
I will introduce the network analysis techniques used by my teams at NYU and MIT to study the spatio-temporal transactional data on human mobility and interactions, as well as their applications to smart urban planning and smart transportation solutions.
I further present the proposed cross-disciplinary research program I  look forward to implementing at MUNI: the Digital City Engine - a  unified, scalable analytic framework for multi-layered urban data and its methodological core - Urban Network AI - a novel fusion of network science and deep learning techniques. We shall discuss the methodological foundations of Urban AI as well as applications to predictive modeling and detection of patterns, impacts, and emergent phenomena in spatio-temporal networks of urban activity. 

Záznam přednášky ZDE