TROPICAL AND TORIC GEOMETRY

Lecturers.

S. Di Rocco (KTH Stockholm – Se)

G. Mikhalkin  (Université de Genève – CH)

Supporting lecturer and tutor.

Organizers.
The School/Workshop is organized by G. Casnati, C. Fontanari, F. Galluzzi, R. Notari, F. Vaccarino. For contacting the organizers send a mail to

The School/Workshop is supported by CIRM-Fondazione Bruno Kessler (formerly CIRM-ITC), by GNSAGA-INdAM, by Dipartimento di Matematica-Politecnico di Torino and by XTERAFOOD project cofinanced by Regione Piemonte.
The School and the Workshop will take place at

Aim of the School.
The School is mainly aimed to Phd students and young researchers in Algebraic Geometry, introducing the participants to research, beginning from a basic level with a view towards the applications and to the most recent results.
S. Di Rocco. Toric Geometry.
Toric varieties have provided a remarkably fertile testing ground for general theories in Algebraic Geometry. The action of the algebraic torus  gives a  rich combinatorial structure which not only serves as a useful computational tool, but, equally importantly, as a bridge to neighboring areas such as Combinatorics, Statistics, Biology and more. The emphasis of this series of lectures is on the study of the geometry of projective toric varieties and its relation  to convex geometry. In the first lecture normal projective toric varieties and the correspondence with lattice convex polytopes will be introduced. Basic notions and basic examples will be presented. The remaining lectures will each introduce research areas in toric geometry where problems are still open, typically in the singular case.
G. Mikhalkin. Tropical Geometry.
The lectures will cover basic notions of tropical geometry with a focus on their topological treatment. Tentatively the following topics will be treated. Real 1-parametric families of complex manifolds and their tropical limits. Matroids, integer affine structure, formal definition of tropical variety. Projective spaces. Hypersurfaces and complete intersections. Chow group vs. homological cycles in tropical varieties. Intersection theory. Tropical waves and detecting algebraic cycles. Moduli spaces of curves in tropical manifolds. Enumerative problems. Reality questions. Connection to Mirror Symmetry and further topics.

Aim of Workshop.
The Workshop is intended to discuss the state of the art. Up to now the following speakers have confirmed their participation: F. Catanese, J. Draisma, I. Itenberg, D. Maclagan. People interested in delivering a short communication are kindly requested to submit the title and an abstract within August 22^nd to

Schedule.
The School will start on Monday, 12^th at 13.00 and it will end on Friday, 16^th at 12.30. Workshop will start on Friday 16^th at 14.30 and it will end on Saturday 17^th at 14.00. On Friday evening a social dinner open to the participants of the School/Workshop will be organized.

Registration.
Participants who do not require financial support are expected to fill in the on line registration form before August, 22^nd.