Workshop Geometry and Physics 22 Novembre 2007 Dipartimento di Matematica, Aula C 10:30- 11:20 String Theory and non compact Toric Varieties Sergio Cacciatori, Università dell’Insubria String theories predict for space-time to a Lorentzian manifold with (at least) ten dimensions. Usually one thinks that six of the spatial dimensions are compactified on a very thin compact variety, a Calabi-Yau manifold, to justify physical observations. However, recently some alternatives to compactification have been developed, which make use of non compact complex varieties in place of Calabi-Yau manifolds. After a short introduction to physical motivations, we will show how some results for the compact cases (related to mirror symmetry) seem to extend for the non compact ones, at least for the case of non compact toric varieties. However many of such results are either partial or expressed in form of conjectures, and deserve a deeper mathematical understanding. 11:30-12:20 Black Holes in 3D Gravity and Earthquakes on Hyperbolic Surfaces with Boundary Francesco Bonsante, Università degli Studi di Pavia An interesting feature of 3D-gravity is that many global aspects of the realistic gravity have some realization in this simple model. In particular examples of constant curvature (2+1)-spacetimes containing black holes are well known and studied. Moduli spaces of such spacetimes have been completely described by Barbot in terms of Teichmüller space of the spatial slice. In a recent work with Schlenker, we use this description to get an earthquake theorem for hyperbolic surfaces with geodesic boundary. In the talk I will describe 3d-spacetimes containing black holes and the relation with Teichmüller theory. 14:00-14:50 Vertex Algebras avant Borcherds Domenico Fiorenza, Università “La Sapienza” di Roma We present a geometric setting for basic CFT/string theory constructions. It is not a rigorous mathematical treatment of string theory, rather it is the “rules of the game”: once one believes (or imposes) that these rules apply to path integrals for the Polyakov action, one is immediately led to the whole machinery of vertex algebras: algebras of currents, OPEs, Ward identities, Virasoro operators, the state-field correspondence. 15:00-16:00 Enumerative Invariants of Calabi-Yau Spaces Balazs Szendroi, University of Oxford I’ll discuss two seemingly very different enumerative problems for Calabi-Yau threefolds, that of counting curves and sheaves, and their conjectural relation. I will also show how to compute some invariants in examples. I then discuss how at least one of the enumerative problems also makes sense in a non-commutative setting, and make further computations, leading to some pretty infinite products.