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Geometric and Mathematical Analysis, and Weak Geometric Structures

23 March 2021 to 23 December 2021
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Thematic program "Geometric and Mathematical Analysis, and Weak Geometric Structures"

March 23 – December 23, 2021

This thematic program will be focused upon two directions of current mathematical research that encompass several interconnected areas related to geometric measure theory (GMT) and function spaces theory:

  • Shape optimization problems and geometric analysis: minimal surfaces (with respect to various notions of surface area/perimeter including nonlocal ones) and related problems, clusters of soap bubbles, problems with prescribed curvature, the Steiner problem and transportation networks, as well as applications of shape optimization in image analysis (e.g. Mumford-Shah problem and similar) and mechanics (e.g. eigenvalue or compliance optimization or ground states of Schrödinger equation).
  • Weak geometric structures related to GMT, such as currents, as well as those arising in optimal mass transportation and the study of geometry of highly irregular metric measure spaces, in particular those without any differentiable structure, with applications to the analysis of PDEs with very low regularity (e.g. weak and/or or probabilistic notions of flows, or “differential equations” with “purely nondifferentiable”, for instance just Hölder continuous unknowns, like those in Rough paths theory) and to some modern harmonic analysis problems.
  • Purely geometric problems involving weak geometric structures, like extensions of Frobenius theorem either to irregular differential forms or distributions of planes or to weaker notions of surfaces like De Rham currents; extensions of Chow-Rachevsky theorem to irregular vector fields or flows; applications to sub-Riemannian geometry, geometric control theory and dynamical systems.
  • Function spaces, especially Banach and Hilbert spaces of holomorphic and/or harmonic functions on the unit disc and polydisc: problems related to the geometry of such spaces and their invariant subspaces (e.g. cyclicity and hypercyclicity, shift operators), fine boundary behavior of their elements (e.g. harmonic measure and  integral spectra estimates), related discrete models on trees and products of trees, time-frequency analysis and Gabor systems.
  • Problems of potential-theoretic nature, such as condensers, their capacities and their asymptotical behavior; fine properties of Cantor-type sets on the plane e.g. Buffon needle with connections to combinatorics and Fourier analysis, Bessel capacity estimates for self-similar sets; properties of potentials on directed graphs; interpolation and sampling in function spaces on the unit disc; approximation problems in several complex variables.

The idea is to explore the deep connections between the above mentioned areas of research and to make them more comprehensible and attractive to both international and local mathematical community.

Minicourses

If you plan to attend minicourses please register here.

Conferences and schools

Organizers

  • Roberta Musina, Università di Udine
  • Matteo Novaga, Università di Pisa
  • Eugene Stepanov, PDMI RAS
  • Alexander Borichev, Université Aix-Marseille
  • Dario Trevisan, Università di Pisa
  • Pavel Mozolyako, Saint-Petersburg University

Institutions participating in the organization of the event

The program is supported by a grant from the Government of the Russian Federation, agreements 075-15-2019-1619 and 075-15-2019-1620, and by a grant from Simons Foundation.

Starts
Ends
Europe/Moscow
Zoom
443 726 1792
"Geometric flows of networks", Lectures 1, 2, 3,4
Loops and Bubbles, Lectures 1,2,3,4
One-dimensional optimization, Lectures 1, 2, 3
Registration
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