Groupe d’études et de recherche en analyse des décisions

# Mixed Atiyah determinants for graphs in the Euclidean or hyperbolic space

## Dragutin Svrtan – Department of Mathematics, University of Zagreb, Croatie

In 2001 Sir Michael Atiyah, inspired by physics (Berry Robbins problem related to spin statistics theorem of quantum mechanics) associated a remarkable determinant to any $$n$$ distinct points in Euclidean 3-space (or hyperbolic 3-space), via an elementary construction. Although the problem of nonvanishing of the Atiyah determinants is very intricate (Atiyah’s first conjecture), we shall show how one can associate a mixed Atiyah determinant to any graph with the given points as vertices. For the sum of all mixed determinants we can prove an identity ($$n$$ conjecture) which implies that for any configuration of $$n$$ distinct points in hyperbolic 3-space at least one of the mixed determinants is nonzero. For two stronger Atiyah-Sutcliffe conjectures, in case of four Euclidean points, a direct geometric proof was obtained five years ago by the speaker (and presented first at MATH/CHEM/COMP 2010, Dubrovnik, Croatia, June 7-12, 2010). Recently another proof is obtained, via linear programming, by M.J. Khuzam and M.J. Johnson. Both proofs use the famous Eastwood-Norbury formula for 4-pt Atiyah determinant (hyperbolic analogue of which is not yet completely known!). For more information (https://bib.irb.hr/prikazi-rad?&rad=553790, http://www.emis.de/journals/SLC/wpapers/s73vortrag/svrtan.pdf)

Entrée gratuite.
Bienvenue à tous!