**Algebraic Geometry** (03.10.07-20.12.07)

Abstract algebraic varieties as function spaces. Schemes as ringed spaces. Properties of schemes. Coherent sheaves. We will cover most of the material in Hartshorne, Algebraic Geometry Chapter II.

Notes from a previous course; diary of the current course.

**Closed Model Categories of Differential Graded Algebras and
Schemes** (20.10.07-19.12.07)

This is a seminar, run jointly with Dr. Donatella Iacono. Starting with the definition of category, we will introduce the notion of closed model category structure on a category, and analyze in detail differential graded algebras. We will then introduce differential graded schemes, state without proof the main results about them (due to Ciocan-Fontanine and Kapranov) and illustrate the relationship between dg schemes and the cotangent complex of a scheme. The seminar is a preliminary to research work towards interpreting dg-Hilb and dg-Quot as moduli spaces in Toen and Vezzosi's derived/homotopical algebraic geometry language; the methods proposed are replacing each dg scheme by its formal completion along the support scheme, and use the results of Manetti on extended deformation functors.

**Virtual Fundamental Class** (May-July, dates to be fixed)

We briefly outline the notion of algebraic stack in the sense of Artin.
We then introduce the notion of absolute and relative obstruction theory and
define the virtual class. We prove a deformation-theoretic criterion for
obstruction theories, and then discuss computational techniques relating
different virtual classes: in particular pullback formula, deformation
invariance, Costello's pushforward formula, and
Graber-Pandharipande's virtual localization.

**Homological Methods in Algebraic Geometry**
(20.04.08-05.06.08)

Derived categories and functors. Cohomology.
Pushforward and pullback. Ext and Tor. Spectral sequences. Dualizing
complex. Cotangent complex.

**Dalide Pontoni**, *Quantum cohomology of
$Hilb^2(P^1\times P^1)$*, Università di Padova, January
2004.

**Fabio Perroni**, *Orbifold cohomology of ADE
singularities*, Sissa, September 2005 (joint with L.
Göttsche).

**Elena Andreini**, *Quantum cohomology of gerbes*, Sissa,
expected October 2007.

**Francesco Noseda**, *Virtual fundamental classes for Artin
stacks*, Sissa, expected November 2007.

**Fabio Nironi**, *Moduli spaces of sheaves on an algebraic
stack*, Sissa, expected fall 2008.

**Cristina Manolache**, Sissa, expected fall 2009 (joint with I.
Ciocan-Fontanine).

**Nicola Tito Pagani**, *Orbifold cohomology of $\bar M_{1,n}$*,
Sissa, expected fall 2010.

** Algebraic Geometry at Kinosaki** Kinosaki,
Japan, 22.10.07-26.10.07

** Moduli Spaces** Bonn, 02.01.08-11.01.08

**
Representation Theory, Algebraic Geometry and Physics** Princeton,
USA, 14.01.08-11.04.08

**
Classical Algebraic Geometry** Oberwolfach, Germany,
09.06.08-14.06.08

** Aspects of Moduli** Pisa, Italy, 16.06.08-28.06.08

**60 Miles**
London, UK, 16.07.07-18.07.08

**Integrable Structures and Topological Strings**, Trieste, Italy,
24.08.08-29.08.08

Levico Terme, Italy, 01.09.08-06.09.08

**
Algebraic Geometry at MSRI** Berkeley, USA, sometime within
12.01.09-22.05.09

Via Beirut 4

34014 Trieste

email:

Last updated on October 12, 2007.

¶ttsche).

**Elena Andreini**, *Quantum cohomology of gerbes*, Sissa,
expected October 2007.

**Francesco Noseda**, *Virtual fundamental classes for Artin
stacks*, Sissa, expected November 2007.

**Fabio Nironi**, *Moduli spaces of sheaves on an algebraic
stack*, Sissa, expected fall 2008.

**Cristina Manolache**, Sissa, expected fall 2009 (joint with I.
Ciocan-Fontanine).

**Nicola Tito Pagani**, *Orbifold cohomology of $\bar M_{1,n}$*,
Sissa, expected fall 2010.