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Noncommutative Algebraic Geometry and its Applications to Physics

Scientific report Noncommutative algebraic geometry and its applications to physics" 19
– 23 March 2012 The
main purpose of the workshop was to create a unified view of the landscape of different
noncommutative geometries, and their applications in
theoretical physics, notably quantum field theory and string theory. As
indicated by the title of the workshop, it was our intention to stress in
particular the algebraic approaches to noncommutative
geometry. Moreover, we wanted to highlight the connections of this theory with
geometric invariants, enumerative geometry, string theory, and integrable systems. The
algebraic approach to noncommutative geometry was
illustrated by Lieven Le Bruyn,
with his algebraic theory of Dbranes, Paul Smith (noncommutative curves and Penrose tilings),
Gonçalo Tabuada, who gave
an introduction to the theory of noncommutative motives, and Jan Jitse Venselaar (Spin structures
on noncommutative tori and their Morita
`equivalences'). Related to this aspect of the theory were also the talks by Dimitri Kaledin (about the HochschildWitt complex), Yuri Berest
(Derived representation schemes), Alexander Kuznetsov
(Categorical resolution of singularities), Lucio Cirio (categorification of the KniznhikZamolodchikov connection), Sebastian Klein (Chow
groups for tensortriangulated categories). Links
with the physics of quantum fields were established by the talks by Walter van Suijlekom (Renormalizability conditions
for almost commutative manifolds) and Alexander
Gorsky (Supersymmetric QCD, integrability
and cyclic RG flows). Relations with integrable
systems were discussed by Vladimir Sokolov (Integrable nonabelian ODE's: a biHamiltonian
approach). Interesting applications to the geometry of moduli spaces were proposed
by Tom Sutherland (Stability conditions for Painlev
quivers), Richard Szabo (Instantons
and noncommutative toric
varieties), Ludmil Katzarkov
(From Higgs bundles to stability conditions), and Simon Brain (Gaugetheoretic
Invariants of toric noncommutative
manifolds). Interesting
connections with other areas of mathematics were explored by Ralph Kaufmann (Noncommutative geometry of wire network graphs) and Matilde Marcolli (Quantum statistical
mechanics, Kolmogorov complexity, and the asymptotic bound of codes). It is the
opinion of the organizers that the workshop fulfilled its scopes in a
satisfactory way. The talks were interesting, and the structure of the workshop
has left space for personal discussion. The workshop has allowed many of the
participants to get in touch with the most recent advances in the field. The
organization of the Lorentz Center, and the work of its personnel, have been impeccable,
and have given a fundamental contribution to the success of the workshop. [Back] 