Discrete methods in combinatorial geometry and geometric topology

A group of researchers has a sound tradition in the study of algebraic and geometric structures, which are either discrete or can suitably be treated through discretization techniques: The study of such topics embraces themes of great interest for their applications to science and technology. For instance, graphs are employed nowadays within the most common data structures and computing algorithms, and can even be encountered in the study of social and financial phenomena; knot theory is connected with the study of biological structures (comparison of genetic data) and physical structures (string theory); computational topology has become a basic tool for the computer guided description and comparison of shapes, with a subsequent fallout on graphic manipulation, on the comparison of models and on the acquisition of visual information. A good amount of experience was gained in enumerative problems from various contexts, developed on different platforms, which is susceptible of further growth.

The three main areas of investigation are the following:

Algebraic and differential topology

  • Topology and geometry of manifolds
  • Combinatorial group theory
  • Algebraic topology, homological algebra and L-theory

Applications of crystallization theory to geometric topology and quantum gravity

  • Relations between crystallization theory and colored tensor models in quantum gravity
  • Trisections of PL 4-manifolds with boundary
  • Generation of catalogs of PL 4-manifolds using colored graphs and their classification

Discrete mathematics

  • Graph coloring, decomposition and characterization
  • Matching search, perfect matchings (1-factors), 2-factors
  • Graph theory techniques applied to DNA assembly problems

 

Staff researchers:
Arrigo Bonisoli, Simona Bonvicini, Maria Rita Casali, Alberto Cavicchioli, Paola Cristofori, Carla Fiori, Carlo Gagliardi, Fulvia Spaggiari