Abstracts

Hans-Christian von Bothmer		Janko Böhm
Rationality of Hypersurfaces		Tropical mirror symmetry for elliptic curves
tba		Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with certain integrals over Feynman graphs. We prove a tropical generalization of mirror symmetry for elliptic curves, that is, a statement relating certain labeled higher genus Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This implies the tropical analogue of the mirror symmetry statement, and, by proving a correspondence theorem, the mirror symmetry statement itself. Furthermore we use the techniques for computing Feynman integrals to prove that the generating functions are quasimodular forms. The theoretical results are complemented by a Singular package for computing Hurwitz numbers of the elliptic curve as integrals over Feynman graphs. This is joint work with Arne Buchholz, Kathrin Bringmann, and Hannah Markwig.
Ragnar Buchweitz		Wolfram Decker
Ulrich Bundles of Rank 2 on a Cubic Curve		Monads and Syzygies
Ulrich line bundles on, equivalently, linear determinantal presentations of smooth projective curves have been parametrized by various authors. For smooth cubics in Hesse form such parametrization becomes particularly nice when pulled back along a 3-isogeny, in that then such matrix factorizations are given by Moore matrices. In this joint work with Sasha Pavlov we describe in a similar vein all matrix factorizations of a smooth cubic that belong to Ulrich bundles of rank two. The result reveals some, still mysterious identities for Moore matrices and we were lead to it through computer experiments with Singular.		In this talk I will review my collaboration with Frank-Olaf Schreyer.
David Eisenbud		Daniel Erman
Tor as a module over an exterior algebra		Supernatural bundles and resolutions of the diagonal
The Bernstein-Gel'fand-Gel'fand correspondence relates sheaves on projective space and free resolutions over an exterior algebra. I will describe the basic idea of the correspondence - which is a lovely and elementary bit of linear algebra - and a new application of it related to high syzygy modules over complete intersections. This is part of ongoing joint work with Jesse Burke, Irena Peeva and Frank-Olaf Schreyer.		By combining pure resolutions and supernatural vector bundles, we produce resolutions of Ulrich bundles that are supported on the diagonal in P^n x P^n. We then apply these resolutions in several directions, including to categorifications of Boij-Soederberg decompositions. This work is joint with Steven Sam.
Gavril Farkas		Giorgio Ottaviani
The Prym-Green Conjecture		The Waring decomposition of a polynomial and its uniqueness
Green's Conjecture for generic curves (proven by Claire Voisin) asserts that the resolution of a generic canonical curve is natural. The Prym-Green Conjecture predicts analogously that the resolution of a generic Prym-canonical curve is natural. I will discuss a solution of this conjecture in odd genus and a geometric explanation for the surprising failure of the Prym-Green Conjecture in genus 8, which had been previously observed computationally jointly with Eisenbud and Schreyer. This is joint work with Kemeny and with Colombo, Verra and Voisin respectively.		A Waring decomposition of a homogeneous polynomial (over complex numbers) is a minimal expression as a sum of powers of linear forms. The Alexander-Hirschowitz Theorem settles the number of summands needed for a general polynomial, while for any polynomial the problem is still open, for example it is not known the maximum number of summands needed, unless in two variables. In the talk we give a overview on this topic and we address the uniqueness of the Waring decomposition. When uniqueness holds, it gives a canonical form for that family of polynomials.
Kristian Ranestad		Claire Voisin
From powersum varieties to cactus varieties; old and new results		IVHS of Jacobians and syzygies of curves
Powersum varieties have been studied since Sylvester, as part of the study of higher secant varieties. Cactus varieties are generalisations of higher secant varieties. In an attempted overview, I shall discuss the role of syzygies, and present both results and open problems.		The theory of infinitesimal variations of Hodge structures is a very powerful tool in the study of algebraic cycles. In the case of families of Jacobians, the study of these infinitesimal invariants is closely related to the study of syzygies of curves, and suggests natural generalizations of them.