%Plain TeX file \magnification\magstep 1 \font\bigtype=cmr12 \font\biggtype=cmbx12 scaled \magstephalf \font\smcps = cmcsc10 \font\ssmcps = cmcsc10 scaled \magstephalf \font\Smcps = cmcsc10 scaled \magstep 1 \def\speaker#1\par{\bigbreak\noindent{\smcps #1}\smallskip} \def\talk#1\par{\noindent{\bf #1}\medskip} \def\abstract{\noindent} \def\w#1; {\smallbreak\noindent \hangindent2.5cm\hangafter1\leavevmode\hbox to 2.5cm{#1\hfill}} %%%%%%%%%%%%%%%%%%%%% \nopagenumbers \bgroup\baselineskip .5cm \centerline{\Smcps First International Meeting AMS--UMI} \centerline{\Smcps Pisa, June 12--16, 2002} \vskip.4cm \centerline{\biggtype Program of the special session on} \centerline{\biggtype Quantum Cohomology and Moduli Spaces} \vskip.4cm \centerline{\ssmcps Organizers:} \smallskip \centerline{\ssmcps Aaron Bertram (University of Utah)} \centerline{\ssmcps Angelo Vistoli (Universit\`a di Bologna)} \egroup \vskip 1cm \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 \vskip.6cm \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 \vfill \break \centerline{\biggtype Abstracts of the talks} \vskip .6cm \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