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\centerline{\Smcps First International Meeting AMS--UMI}

\centerline{\Smcps Pisa, June 12--16, 2002}

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\centerline{\biggtype Program of the special session on}


\centerline{\biggtype Quantum Cohomology and Moduli Spaces}

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\centerline{\ssmcps Organizers:}

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\centerline{\ssmcps Aaron Bertram (University of Utah)}

\centerline{\ssmcps  Angelo Vistoli (Universit\`a di Bologna)}

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\noindent {\smcps Saturday, June 15, morning}
\bigskip

\w 9:00--9:45; Tom Graber (Harvard University): Generalizations of Tsen's
Theorem

\w 9:50--10:20; Paul M.N. Feehan (Rutgers University): Seiberg--Witten
invariants, Donaldson invariants, and blow-up formulae

\w 10:20--10:45; Coffee Break

\w 11:00--11:45;  Barbara Fantechi (Universit\`a di Udine): On the moduli
spaces of $(A,B)$ pointed curves

\w 12:00--12:30; Domenico Fiorenza (Universit\`a di Roma ``Tor
Vergata''): Feynman diagrams and the KdV hierarchy


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\noindent {\smcps Saturday, June 15, afternoon}
\bigskip

\w 15:00--15:45; Dan Abramovich (Boston University): Flips, moduli, and
derived categories

\w 15:50--16:20;  Alessandro Arsie (Universit\`a di Bologna): Stacks of
hyperelliptic curves

\w 16:20--16:45; Coffee Break

\w 16:50--17:35;  Lothar Goettsche (International Center for Theoretical
Physics): Orbifold cohomology and Hilbert schemes

\w 17:45--18:15; Marco Maggesi (Universit\`a di Firenze): On the quantum
cohomology ring of some Fano threefolds





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\centerline{\biggtype Abstracts of the talks}

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\speaker Dan Abramovich (Boston University)

\talk Flips, moduli, and derived categories

\abstract Joint work with J. Chen and T. Bridgeland. I will discuss some
extensions of work of Bridgeland  and of Chen on the construction of flops
as moduli spaces of perverse point sheaves. In these cases the varieties in
question have {\bf Q}-Gorenstein singularities, and the approach involves
using a natural stack structure on these varieties (which also features in
work of Kawamata).


\speaker Alessandro Arsie (Universit\`a di Bologna)

\talk Stacks of hyperelliptic curves

\abstract This is joint work with Angelo Vistoli. Given a integer $g$ at
least equal to 2, we express the stack ${\cal H}_g$ of smooth
hyperelliptic curves of genus $g$ as the quotient of an open subset of a
representation of the group
$G$, where $G$ is ${\rm GL}_2$ for $g$ even, and ${\bf G}_{\rm m}
\times {\rm PGL}_2$ for $g$ odd. As a corollary, we show that the Picard
group of ${\cal H}_g$ is cyclic of order $2(2g+1)$, and we exhibit
generators for its integral Chow ring.


\speaker Barbara Fantechi (Universit\`a di Udine)

\talk On the moduli spaces of $(A,B)$ pointed curves

\abstract $(A,B)$ pointed curves (stable curves where some of the marked
points are allowed to coincide) were defined by Losev and Manin. We
present some results related to the definition of stability and to some
properties of their moduli spaces.


\speaker Paul M.N. Feehan (Rutgers University)

\talk Seiberg--Witten invariants, Donaldson invariants, and blow-up formulae

\abstract Compact smooth 4-manifolds have two kinds of gauge-theoretic
invariants, due to Seiberg--Witten and to Donaldson, which can be used to
distinguish between different smooth structures on the same topological
4-manifold. The invariants are conjecturally related by a formula of Edward
Witten (1994). In this lecture we shall explain how input from an overarching
gauge theory (${\rm SO}(3)$ monopoles), the blow-up formulae for Donaldson
and Seiberg--Witten invariants, and properties of the Donaldson and
Seiberg--Witten theories can be used to prove Witten's formula.


\speaker Domenico Fiorenza (Universit\`a di Roma ``Tor Vergata'')

\talk Feynman diagrams and the KdV hierarchy

\abstract
A classical result of Kontsevich--Witten states that the
generating functional of the intersection numbers of the
stable cohomology classes on moduli spaces of curves
satisfies the string equation and the KdV
hierarchy. Kontsevich's original proof uses a matrix model,
now called the 't Hooft--Kontsevich model, and the matrix
Airy equation. Witten
recasted Kontsevich's results in terms of Virasoro algebras.
His proof is an ingenious mixture of Feynman diagrams
techniques and integrations by parts with some ``rather
formidable choice'' of the integrands.

We show how the
language of Feynman algebras allows to rewrite Witten's proof
entirely in terms of Feynman diagrams. In this approach,
Witten's formulas are found to be completely natural.


\speaker Lothar Goettsche (International Center for Theoretical Physics)

\talk Orbifold cohomology and Hilbert schemes

\abstract Orbifold cohomology rings have recently been introduced for
smooth orbifolds by Chen and Ruan (and Abramovich, Graber and Vistoli in
algebraic geometry). They are conjecturally related to the
cohomology rings of good (crepant or even complex symplectic) resolutions
of the orbifolds. One of the best known example of such a resolution
is the resolution of the symmetric power of a surface by its Hilbert scheme
of points.  We study the orbifold cohomology of symmetric powers of algebraic
surfaces and relate it to the cohomology of Hilbert schemes of points on the
surface.



\speaker Tom Graber (Harvard University)

\talk Generalizations of Tsen's Theorem

\abstract Tsen's theorem states that polynomials of low degree in many
variables over the function field of a curve have nontrivial solutions.  I
will discuss two generalizations of this theorem---one true and one
false---which show that Tsen's theorem is best understood in connection with
the geometry of rationally connected varieties.  This talk is based on joint
work with Joe Harris, Barry Mazur, and Jason Starr.


\speaker Marco Maggesi (Universit\`a di Firenze)

\talk On the quantum cohomology ring of some Fano threefolds

\abstract The aim of the talk is to give an explicit description of the
quantum cohomology ring of all Fano threefolds which are ${\bf P}^1$-bundles
on smooth projective surfaces, with the aid of the classification given
by Szurek and Wi\'sniewski of such varieties.  This involves the
computation of some Gromov--Witten invariants whose moduli spaces of
maps do not have the expected dimension.



\bye


