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)


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