Extremal Combinatorics is one of the most active topics in Discrete Mathematics, dealing with problems that are often motivated by questions in other areas, including Theoretical Computer Science, Geometry and Game Theory. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers. The topics c...

We discuss the use of the Newton method in the computation of max(p 7→ Ep [f ]), where p belongs to a statistical exponential family on a finite state space. In a number of papers, the authors have applied first order search methods based on information geometry. Second order methods have been widely used in optimization on manifolds, e.g., matrix manifolds, but appear to be new in statistical ...

We describe some combinatorial problems in finite projective planes and indicate how Rédei’s theory of lacunary polynomials can be applied to them. 2000 Mathematics Subject Classification: 05.

My research is at the intersection of algebra, geometry, and combinatorics. My dissertation work under Pramod Achar involved a deep study of the singularities of certain topological spaces through the lens of perverse sheaves. Concurrently, I began working with Greg Muller on cluster algebras, a subject that is related to a large number of areas in mathematics, but which has a somewhat easier p...

I have published during my very long life many papers with similar titles . In this note, I report on progress made on some of the old problems and discuss also some new ones, and I repeat some old problems which seem to have been forgotten by everybody (sometimes including myself) . First, a very incomplete list is given of some papers dealing with these and related questions . I have publishe...

The Marquis du Condorcet recognized 200 years ago that majority rule can produce intransitive group preferences if the domain of possible (transitive) individual preference orders is unrestricted. We present results on the cardinality and structure of those maximal sets of permutations for which majority rule produces transitive results (consistent sets). Consistent sets that contain a maximal ...

OF HONORS THESIS GEOMETRIC OBJECTS WITH A MORE COMBINATORIAL FLAVOR We are studying geometric objects, defined over finite fields, with a more combinatorial flavor and present the results of the investigation of classification problems in geometry and combinatorics. Objects called BLT-sets, living in a vector space over a finite field, are of great interest to finite geometry, as they provide a...

Final version, layout for two-sided printing using A4 paper (December 2004). Preface These notes are based upon a series of lectures given at the Summer School on Combina-torial Geometry and Optimisation 2004 " Giuseppe Tallini ". It took place at the Catholic University of Brescia, Italy, from June 4th until June 10th, 2004, within the MIUR COFIN 2004 research project Strutture Geometriche, Co...

This paper defines the basis of a new hierarchical framework for segmentation algorithms based on energy minimization schemes. This new framework is based on two formal tools. First, a combinatorial pyramid encode efficiently a hierarchy of partitions. Secondly, discrete geometric estimators measure precisely some important geometric parameters of the regions. These measures combined with photo...