# Mini-courses

Location: Auditório 3, IMPA

## Mini-course 1

SEPTEMBER 26, 28, 30, 2011

1:30 to 3:00 PM

**Title**: Topology of discrete manifolds (and non-manifolds) through cell complexes

Thomas Lewiner (PUC-Rio)

**Description**: Geometric objects are modeled in the computer through several representations, depending on the applications. In particular, the discrete surfaces are generally defined as deformations of triangular or polygonal meshes for surfaces, and similarly tetrahedral or polyhedral for solids or simplicial polytopes

in higher dimensions. This formalization of mesh deformation is precisely defined in mathematics through cell complexes. The first step in analyzing a geometric object is to characterize its topology (how many connected components, with or without boundary, which sets are open) in particular, to define continuous (before differentiable) functions on this object. However, the topology of cell complexes, and in particular discrete manifolds, presents subtle properties that support robust and efficient algorithms to manipulate them. In addition, cell complexes may also represent in a simple manner non-manifolds, providing a basis to extend several results to those yet little-studied objects.

This course will start with the concept of simplicial complex, which is a simpler concept as cell complex. Then, it will present the definitions and properties of cell complexes, alternating between the necessary and constructive approaches. These properties will be translated into algorithmic terms. It will then introduce the notions of induced and CW topology, and their relationship to homotopy. It will conclude on the characterizations of regular, pure, and manifold complexes.

**References**:

A.T. Lundell, S. Weingram. *The Topology of CW Complexes*. Springer, 1969.

L. de Floriani, A.Hui. *Shape Representations based on Cell and Simplicial Complexes*, EG STAR 2007.

## Mini-course 2

OCTOBER 10, 12, 14, 2011

1:30 to 3:00 PM

**Title**: Topological Simplification of Discrete Functions on Surfaces

Carsten Lange Freie Universitat Berlin

**Description**:

1. Discrete Morse Theory

Introduction to Robin Forman's discrete Morse theory (vector fields,
Morse inequalities, cancelation of critical points, ...)

**Literature:**
R. Forman, Morse theory for cell complexes, Adv Math, 134:90-145 (1998)

2.Persistence Theory

Introduction to Edelsbrunner's Persistence Theory (persitence pairings of
critical points, persistence diagrams, ...)

**Literature:**
Edelsbrunner, Letscher, Zomorodian,
Topological persistence and simplification
Discrete and Computational Geometry 30:87-107 (2003)

3. Use discrete Morse theory and persistence theory for an optimal topological
simplification of discrete functions on surfaces

**Literature:**
Bauer, Lange, Wardetzky, as above