Topics in Mathematics 2014/2015

Mechanical engineers use submanifolds of the special Euclidean group SE(3) to characterize the motion pattern of a mechanism or robot manipulator. So far, Lie subgroups of SE(3) are well known to the mechanism community, and their extended application in mechanism analysis and synthesis are well documented in the literature. General submanifolds, on the other hand, are hard to classify and not all of them are useful for real applications. Recently, we discovered that there is a special class of submanifolds of SE(3) that have potentially broad application in mechanism analysis and synthesis, yet are largely overlooked for forty years. These are the symmetric subspaces (or totally geodesic submanifolds) of SE(3), with SE(3) treated as a sym- metric space. In this seminar, I will give an informal introduction of these submanifolds from an mechanical engineering point of view. First, I will briefly review the state-of-the-art application of Lie subgroups of SE(3) in mechanism analysis and synthesis. The limitation of Lie subgroups is illustrated by some motivating engineering examples. Second, I will give a brief introduction to symmetric subspaces of SE(3) using Ottmar Loos’ elementary approach. I will study properties of these submanifolds and show how they can be used to design new mechanisms.

We shall discuss the geometric problem of reconstructing the topology of a three-dimensional (not necessarily connected) shape, starting from its apparent contour. To face this problem, some techniques from singularity theory will be needed. We shall also briefly discuss some algorithmic applications, obtained using the program appcontour. The motivation for this study came from computer vision and image segmentation.

Ultimo appuntamento del minicorso su moto per curvatura media.

Sesto appuntamento (lezioni 11 e 12) del minicorso di dottorato: Moto per curvatura media

Quinto appuntamento (lezioni 9 e 10) del minicorso di dottorato: Moto per curvatura media

Quarto appuntamento (lezioni 7-8/8) del minicorso di dottorato: Moto per curvatura media

Terzo appuntamento (lezioni 5 e 6 su 8) del minicorso per il dottorato sul moto per curvatura media.

Lectures on Mean Curvature Flow - Carlo Mantegazza In this series of lectures I will introduce the mean curvature flow of a compact hypersurface in the Euclidean space with particular attention to the cases of curves and surfaces. The basic properties and the main analytic and geometric techniques used in the analysis of this flow will be discussed, for instance maximum and comparison principles. Moreover, I will present the fundamental Huisken.s monotonicity formula and the Harnack inequality by Hamilton, which are the key tools in the study of the singularity formation. Up to now, the classification of possible asymptotic shape of a singularity is almost complete for some classes of evolving hypersurfaces. For others it seems difficult and quite far off. Time allowing, in the last lectures I will try to outline an up.to.date scenario of the "state of the art".

Lectures on Mean Curvature Flow - Carlo Mantegazza In this series of lectures I will introduce the mean curvature flow of a compact hypersurface in the Euclidean space with particular attention to the cases of curves and surfaces. The basic properties and the main analytic and geometric techniques used in the analysis of this flow will be discussed, for instance maximum and comparison principles. Moreover, I will present the fundamental Huisken.s monotonicity formula and the Harnack inequality by Hamilton, which are the key tools in the study of the singularity formation. Up to now, the classification of possible asymptotic shape of a singularity is almost complete for some classes of evolving hypersurfaces. For others it seems difficult and quite far off. Time allowing, in the last lectures I will try to outline an up.to.date scenario of the "state of the art".

We show different notions of convexity for function defined on sub-Riemannian ge- ometries: geodesic convexity, H-convexity, horizontal convexity, viscosity convexity and convexity along vector fields. We first investigate the relations between them and then state some properties for functions convex in these various settings. Joint work with Martino Bardi (Universit`a di Padova).

Lectures on Mean Curvature Flow - Carlo Mantegazza In this series of lectures I will introduce the mean curvature flow of a compact hypersurface in the Euclidean space with particular attention to the cases of curves and surfaces. The basic properties and the main analytic and geometric techniques used in the analysis of this flow will be discussed, for instance maximum and comparison principles. Moreover, I will present the fundamental Huisken.s monotonicity formula and the Harnack inequality by Hamilton, which are the key tools in the study of the singularity formation. Up to now, the classification of possible asymptotic shape of a singularity is almost complete for some classes of evolving hypersurfaces. For others it seems difficult and quite far off. Time allowing, in the last ectures I will try to outline an up.to.date scenario of the "state of the art".

Lectures on Mean Curvature Flow - Carlo Mantegazza In this series of lectures I will introduce the mean curvature flow of a compact hypersurface in the Euclidean space with particular attention to the cases of curves and surfaces. The basic properties and the main analytic and geometric techniques used in the analysis of this flow will be discussed, for instance maximum and comparison principles. Moreover, I will present the fundamental Huisken.s monotonicity formula and the Harnack inequality by Hamilton, which are the key tools in the study of the singularity formation. Up to now, the classification of possible asymptotic shape of a singularity is almost complete for some classes of evolving hypersurfaces. For others it seems difficult and quite far off. Time allowing, in the last lectures I will try to outline an up.to.date scenario of the "state of the art.

The goal of these lectures is to discuss how methods of coarse geometry can be applied to estimate analytic quantities such as the eigenvalues of the Laplacian. As applications I will for instance discuss (1) the recurrence properties of the simple random walk on planar graphs, or (2) prove that every Riemannian manifold whose fundamental group surjects onto a free group admits a tower of covers which forms an expander. To a large extent the methods are rather elementary, although some familiarity with Riemannian geometry should be helpful.

The goal of these lectures is to discuss how methods of coarse geometry can be applied to estimate analytic quantities such as the eigenvalues of the Laplacian. As applications I will for instance discuss (1) the recurrence properties of the simple random walk on planar graphs, or (2) prove that every Riemannian manifold whose fundamental group surjects onto a free group admits a tower of covers which forms an expander. To a large extent the methods are rather elementary, although some familiarity with Riemannian geometry should be helpful

The goal of these lectures is to discuss how methods of coarse geometry can be applied to estimate analytic quantities such as the eigenvalues of the Laplacian. As applications I will for instance discuss (1) the recurrence properties of the simple random walk on planar graphs, or (2) prove that every Riemannian manifold whose fundamental group surjects onto a free group admits a tower of covers which forms an expander. To a large extent the methods are rather elementary, although some familiarity with Riemannian geometry should be helpful

The goal of these lectures is to discuss how methods of coarse geometry can be applied to estimate analytic quantities such as the eigenvalues of the Laplacian. As applications I will for instance discuss (1) the recurrence properties of the simple random walk on planar graphs, or (2) prove that every Riemannian manifold whose fundamental group surjects onto a free group admits a tower of covers which forms an expander. To a large extent the methods are rather elementary, although some familiarity with Riemannian geometry should be helpful

The goal of these lectures is to discuss hoe methods of coarse geometry can be applied to estimate analytic quantities such as the eigenvalues of the Laplacian. As applications I will for instance discuss (1) the recurrence properties of the simple random walk on planar graphs, or (2) prove that every Riemannian manifold whose fundamental group surjects onto a free group admits a tower of covers which forms an expander. To a large extent the methods are rather elementary, although some familiarity with Riemannian geometry should be helpful

The goal of these lectures is to discuss hoe methods of coarse geometry can be applied to estimate analytic quantities such as the eigenvalues of the Laplacian. As applications I will for instance discuss (1) the recurrence properties of the simple random walk on planar graphs, or (2) prove that every Riemannian manifold whose fundamental group surjects onto a free group admits a tower of covers which forms an expander. To a large extent the methods are rather elementary, although some familiarity with Riemannian geometry should be helpful.