Première École d'Été Franco-Nordique de Mathématiques, EEFN

Den första fransk-nordiska sommarskolan i matematik

First French-Nordic Summer School in Mathematics

Annonce en français
Announcement in English

Digital geometry. Lectures by Jean Serra

The course on digital geometry will be presented from the lattice point of view. Complete lattices are algebraic structures that emphasize ordering relationships, in contrast to topological or metric ones. They offer a common denominator to continuous and discrete spaces, so that the digital specificity can be introduced relatively late in the discourse.
The course is organized in nine lectures, as follows:
1. Introduction to complete lattices (definitions, examples, basic theorems);
2. Dilation and erosion (Galois adjunction, representation of increasing mappings, translation invariance, Steiner decomposition and digital convexity);
3. Opening and granulometry (inverse operators, idempotence, duality with respect to complement, semi-groups of operators, discrete pyramids);
4. Morphological filtering (alternating filters, center and self duality, activity lattice);
5. Theorem of structure (point of view of the invariants, under which conditions a subset of a lattice is itself a lattice for the same ordering?);
6. Connection (generalization of the concept of connectivity, so that it encompasses digital spaces and clusters of disjoint sets, partition and connections, connected filters);
7. Segmentation (generation of partitions with connected classes, watersheds);
8. Application to 3D structures (3D wave fronts and digital geodesy, Euler-Poincaré constant, trees and branching);
9. Circular data (morphological approach for data on the unit circle, e.g., hue or direction values, increment based operators, application to multidimensional data and to color processing).

The course will be significantly illustrated by examples, drawn from still and moving images, and from micro- and macroscopic scales.

The text of the course is available here in PDF format (2.5 MB).

Projective geometry. Lectures by Anders Heyden

The course in projective geometry will cover both the basic theory and some applications. We will start with the definition and properties of projective spaces. Special emphasis will be on the connection between projective, affine and Euclidean spaces; the so called stratified approach. Another aspect of special importance to applications is projections from a higher-dimensional projective space to another of lower dimension. We will also introduce some classical construction problems and invariant theory.

One application that we will look into in a little bit more detail is multiple view geometry, a topic arising from trying to reconstruct a rigid three-dimensional world from a number of its two- dimensional images. We will introduce some tensor formalism to describe the relations between corresponding point in different images. In this case, the projections are from 3D to 2D or sometimes from 2D to 1D. We shall also look into the problem of reconstructing independently moving objects, which will require the analysis of projections from higher-dimensional projective spaces.

The text of the transparencies is available at the author's homepage.

Complex geometry. Lectures by Mikael Passare

In this course we shall focus on a few concrete objects occurring in modern complex geometry. Unlike most of the theory of several complex variables, which takes place in at least four real dimensions, many of our objects are possible to actually draw in a quite accurate way, and we are going to look at several such pictures. The names of some of these objects are: amoebas, Newton polytopes, Monge--Ampère measures, and toric manifolds.

The amoeba of a complex analytic function f is by definition the image in Rⁿ of its zero locus under the simple mapping (z₁,...,z_n) |---> (log|z₁|,..., log|z_n|). The terminology was introduced in the nineties by the famous (biologist and) mathematician Israel Gelfand and his co-authors. Associated with f is also a natural convex potential function N_f, defined by
x |---> mean value of log|f| over the torus |z_j| = exp(x_j).
The Hessian of N_f gives rise to a positive measure with support on the amoeba having total mass equal to the volume of the Newton polytope of f. By approximating N_f with a piece-wise linear function one gets striking combinatorial information regarding the amoeba and the Newton polytope of f, and by actually computing the Monge--Ampère measure we find sharp bounds for the area of amoebas in R². This has unexpected connections to the geometry of real polynomial curves.

The URL of this page is: http://www.math.uu.se/~kiselman/eefncourses.html
Christer Kiselman. 2001 03 16. Last change 2001 09 22.