# Geometric and Mathematical Analysis, and Weak Geometric Structures

Europe/Moscow
443 726 1792 (Zoom)

Description

## 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.

### 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.

Registration
Registration for Minicourses
• Tuesday, 23 March
• 18:00 19:30
Geometric flows of networks (1/4) 1h 30m
Speakers: Alessandra Pluda (Università di Pisa), Matteo Novaga (Università di Pisa)
• Thursday, 25 March
• 18:00 19:30
Geometric flows of networks (2/4) 1h 30m
Speakers: Alessandra Pluda (Università di Pisa), Matteo Novaga (Università di Pisa)
• Tuesday, 30 March
• 18:00 19:30
Geometric flows of networks (3/4) 1h 30m
Speakers: Alessandra Pluda (Università di Pisa), Matteo Novaga (Università di Pisa)
• Thursday, 1 April
• 18:00 19:30
Geometric flows of networks (4/4) 1h 30m
Speakers: Alessandra Pluda (Università di Pisa), Matteo Novaga (Università di Pisa)
• Tuesday, 13 April
• 18:00 19:30
Loops and Bubbles (1/4) 1h 30m

An important class of problems in Riemannian geometry can be stated as follows: given a smooth and orientable Riemanninan manifold M, find an hypersphere U in M having prescribed mean curvature K at each point. We will be mainly focused on the case when the target M is the Euclidean plane and the unknown U is a planar loop. Besides its geometrical interpretation, this (apparently) simple problem naturally arises in the study of the planar motion of an electrified particle that experiences a Lorentz force produced by a magnetostatic field. It can be regarded as a model for a more general question raised by V.I. Arnold in [Uspekhi Mat. Nauk 1986]. We will first discuss some Alexandrov-type uniqueness results in case the prescribed curvature is a positive constant or, more generally, a positive and radially non increasing function. To obtain existence results we will choose a parametric point of view, which will lead us to study certain variational, noncompact systems of second order ODE's for functions on the circle. This will give us the opportunity to briefly introduce some variational (mountain pass lemma) and nonvariational (Lyapunov-Schmidt dimension reduction)basic techniques. In the last part of the course will overview some recent results and open problems in case the target space M is the hyperbolic plane, or the Euclidean/hyperbolic 3-dimensional space.

Program:

The curvature of planar curves. Planar loops and physical interpretation: a related ODE system and Arnol'd problem. Planar loops of positive curvature. Homework: the curvature of circles and ellipses, radially symmetric prescribed curvatures.
The four vertex theorem (Osserman's proof).
Uniqueness results: Alexandrov (1956), Aeppli (1960) and more (2011).
The (pseudo)-length functional and the weighted, signed area functional.
The variational approach. A quick introduction to variational methods.
Palais-Smale condition, the Mountain Pass Lemma, saddle points.
A non-variational approach: the Lyapunov-Schmidt dimension reduction.
Loops of prescribed curvature in the hyperbolic plane. Bubbles in the Euclidean and in hyperbolic spaces: recent results and open problems.

Speaker: Roberta Musina
• Wednesday, 14 April
• 18:00 19:30
Loops and Bubbles (2/4) 1h 30m

An important class of problems in Riemannian geometry can be stated as follows: given a smooth and orientable Riemanninan manifold M, find an hypersphere U in M having prescribed mean curvature K at each point. We will be mainly focused on the case when the target M is the Euclidean plane and the unknown U is a planar loop. Besides its geometrical interpretation, this (apparently) simple problem naturally arises in the study of the planar motion of an electrified particle that experiences a Lorentz force produced by a magnetostatic field. It can be regarded as a model for a more general question raised by V.I. Arnold in [Uspekhi Mat. Nauk 1986]. We will first discuss some Alexandrov-type uniqueness results in case the prescribed curvature is a positive constant or, more generally, a positive and radially non increasing function. To obtain existence results we will choose a parametric point of view, which will lead us to study certain variational, noncompact systems of second order ODE's for functions on the circle. This will give us the opportunity to briefly introduce some variational (mountain pass lemma) and nonvariational (Lyapunov-Schmidt dimension reduction)basic techniques. In the last part of the course will overview some recent results and open problems in case the target space M is the hyperbolic plane, or the Euclidean/hyperbolic 3-dimensional space.

Program:

The curvature of planar curves. Planar loops and physical interpretation: a related ODE system and Arnol'd problem. Planar loops of positive curvature. Homework: the curvature of circles and ellipses, radially symmetric prescribed curvatures.
The four vertex theorem (Osserman's proof).
Uniqueness results: Alexandrov (1956), Aeppli (1960) and more (2011).
The (pseudo)-length functional and the weighted, signed area functional.
The variational approach. A quick introduction to variational methods.
Palais-Smale condition, the Mountain Pass Lemma, saddle points.
A non-variational approach: the Lyapunov-Schmidt dimension reduction.
Loops of prescribed curvature in the hyperbolic plane. Bubbles in the Euclidean and in hyperbolic spaces: recent results and open problems.

Speaker: Roberta Musina
• Tuesday, 20 April
• 18:00 19:30
Loops and Bubbles (3/4) 1h 30m

An important class of problems in Riemannian geometry can be stated as follows: given a smooth and orientable Riemanninan manifold M, find an hypersphere U in M having prescribed mean curvature K at each point. We will be mainly focused on the case when the target M is the Euclidean plane and the unknown U is a planar loop. Besides its geometrical interpretation, this (apparently) simple problem naturally arises in the study of the planar motion of an electrified particle that experiences a Lorentz force produced by a magnetostatic field. It can be regarded as a model for a more general question raised by V.I. Arnold in [Uspekhi Mat. Nauk 1986]. We will first discuss some Alexandrov-type uniqueness results in case the prescribed curvature is a positive constant or, more generally, a positive and radially non increasing function. To obtain existence results we will choose a parametric point of view, which will lead us to study certain variational, noncompact systems of second order ODE's for functions on the circle. This will give us the opportunity to briefly introduce some variational (mountain pass lemma) and nonvariational (Lyapunov-Schmidt dimension reduction)basic techniques. In the last part of the course will overview some recent results and open problems in case the target space M is the hyperbolic plane, or the Euclidean/hyperbolic 3-dimensional space.

Program:

The curvature of planar curves. Planar loops and physical interpretation: a related ODE system and Arnol'd problem. Planar loops of positive curvature. Homework: the curvature of circles and ellipses, radially symmetric prescribed curvatures.
The four vertex theorem (Osserman's proof).
Uniqueness results: Alexandrov (1956), Aeppli (1960) and more (2011).
The (pseudo)-length functional and the weighted, signed area functional.
The variational approach. A quick introduction to variational methods.
Palais-Smale condition, the Mountain Pass Lemma, saddle points.
A non-variational approach: the Lyapunov-Schmidt dimension reduction.
Loops of prescribed curvature in the hyperbolic plane. Bubbles in the Euclidean and in hyperbolic spaces: recent results and open problems.

Speaker: Roberta Musina
• Wednesday, 21 April
• 18:00 19:30
Loops and Bubbles (4/4) 1h 30m

An important class of problems in Riemannian geometry can be stated as follows: given a smooth and orientable Riemanninan manifold M, find an hypersphere U in M having prescribed mean curvature K at each point. We will be mainly focused on the case when the target M is the Euclidean plane and the unknown U is a planar loop. Besides its geometrical interpretation, this (apparently) simple problem naturally arises in the study of the planar motion of an electrified particle that experiences a Lorentz force produced by a magnetostatic field. It can be regarded as a model for a more general question raised by V.I. Arnold in [Uspekhi Mat. Nauk 1986]. We will first discuss some Alexandrov-type uniqueness results in case the prescribed curvature is a positive constant or, more generally, a positive and radially non increasing function. To obtain existence results we will choose a parametric point of view, which will lead us to study certain variational, noncompact systems of second order ODE's for functions on the circle. This will give us the opportunity to briefly introduce some variational (mountain pass lemma) and nonvariational (Lyapunov-Schmidt dimension reduction)basic techniques. In the last part of the course will overview some recent results and open problems in case the target space M is the hyperbolic plane, or the Euclidean/hyperbolic 3-dimensional space.

Program:

The curvature of planar curves. Planar loops and physical interpretation: a related ODE system and Arnol'd problem. Planar loops of positive curvature. Homework: the curvature of circles and ellipses, radially symmetric prescribed curvatures.
The four vertex theorem (Osserman's proof).
Uniqueness results: Alexandrov (1956), Aeppli (1960) and more (2011).
The (pseudo)-length functional and the weighted, signed area functional.
The variational approach. A quick introduction to variational methods.
Palais-Smale condition, the Mountain Pass Lemma, saddle points.
A non-variational approach: the Lyapunov-Schmidt dimension reduction.
Loops of prescribed curvature in the hyperbolic plane. Bubbles in the Euclidean and in hyperbolic spaces: recent results and open problems.

Speaker: Roberta Musina