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## Geometric flows of networks

### Matteo Novaga (Università di Pisa) and Alessandra Pluda (Università di Pisa)

23.03, 25.03, 30.03, 01.04 18.00-19.30 MSK

The study of geometric flows is a very flourishing mathematical field and geometric evolution equations have been applied to a variety of topological, analytical and physical problems. In particular, in the last decades a great attention has been devoted to the analysis of the mean curvature flow. In this series of lectures we will introduce the notion of curvature flow of curves and networks, describing the possible approaches to this geometric evolution. We will present the state of the art of the problem, from the short time existence and uniqueness to a description of the long time behavior and singularity formation. We will conclude the course with some recent extensions to higher order flows and a list of open problems.

Program:

- Lecture 1: Formulation of the curvature flow of curves and networks and preliminary results.
- Lecture 2: Short time existence and uniqueness of the motion by curvature of networks.
- Lecture 3: Monotonicity formula and analysis of singularities.
- Lecture 4: Elastic flow of curves and networks: State of the art and open problems.

## Gabor analysis for rational functions

### Yurii Belov (St. Petersburg State University)

14.04, 21.04, 28.04, 5.05, 12.05, 19.05, 26.05. 19:00 (MSK)

This series of lectures is aimed to present a recent progress in Gabor analysis for rational

functions and is based on the works by A. Kulikov, Yu. Lyubarskii and the author.

Program:

- Gabor systems: what they are, how do they appear and why we need them.
- Main criterion for rational functions. Finite-diagonal matrices.
- Hunting for positivity. Herglotz functions.
- Irrational densities. Daubechies conjecture.
- Near the criticial hyperbola. Large densities.
- Irregular sampling. Toeplitz approach.
- Sum of two Cauchy kernels. Estimates of frame bounds.

This course is accessible to second year students.

The prerequisites are: basic complex analysis and linear algebra.

## One-dimensional optimization problems

### Emanuele Paolini (Università di Pisa)

11.05, 12.05, 18.05, 19.05, 18.00-19.30 MSK

We will consider optimization problems where the unknown is a one-dimensional object (such as a network) constrained by length and optimizing some geometric functional. We will introduce the tools used in the study of the Steiner problem (minimal length network spanning some set of points), the irrigation problem (minimal network which minimizes its distance from a given set of points) and minimal cluster (the minimal interface required to enclose and separate regions of the plane with given area).

Program:

- The Steiner Problem. Melzak's construction. Hausdorff one-dimensional measure, Hausdorff distance, Golab's theorem.
- Irrigation problems, duality formulation. Generalized Steiner Problem.
- Minimal clusters: existence & Regularity.
- The double bubble, weak formulation. Moebius transformation, monotonicity. The honeycomb problem.

## Loops and Bubbles

### Roberta Musina (Università di Udine)

13.04, 14.04, 20.04, 21.04, 18.00-19.30 MSK

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.

## Mathematics and applications of manifold learning: reconstructing hidden geometric structures in the data

### Serguei Barannikov (Skoltech and CNRS), Sergey Nechaev (CNRS), Vladimir Spokoiny (WIAS) and Dario Trevisan (Università di Pisa)

dates TBA

We will give an overview of the basic problems (both solved and still open) and approaches in manifold learning and topological data analysis aimed to reconstruct hidden geometric structures in the data. They receive growing attention due to the constantly increasing need for statistical analysis of big data in various applications. Such problems happen to have deep connections to several areas of pure mathematics, from metric geometry to harmonic analysis and calculus of variations, which we will discuss together with their algorithmic and applicative aspects.

Program:

- Reconstructing manifolds from (intrinsic) distances. Euclidean distance geometry. Multidimensional scaling. Hilbert space embedding problems. Spectral methods. Laplacian eigenmaps. Variational approach for reconstruction and semidefinite programming. (2-4h, Dario Trevisan)
- Reconstructing manifolds from noisy data and dimensionality reduction. (2h, Vladimir Spokoiny)
- Topological data analysis. Persistent homology. (2h, Sergey Barannikov)
- Applications. (2h, Sergey Nechaev)

## Sobolev vector fields and their flows

### Elia Brué (IAS) and Dario Trevisan (Università di Pisa)

Irregular vector fields and associated flows are ubiquitous in the mathematical description of many natural phenomena, such as fluids and moving particles. The theory of DiPerna and Lions largely extends the classical (Lipschitz) one, providing well-posedness for less regular fields (e.g., Sobolev) at the price of a generalized notion of flow, nowadays called regular Lagrangian flow. Aim of this course is to describe the main results of the theory and some recent breakthroughs on failure of uniqueness and regularity properties.

Program:

- Lecture 1: Lipschitz theory and the superposition principle.
- Lecture 2: Well-posedness for Sobolev and BV fields.
- Lecture 3: Uniqueness of Lagrangian flows vs classical uniqueness.
- Lecture 4: A convex integration approach to non-uniqueness of transport equations.