We study the conjectured Gorenstein property for the tautological rings of the moduli spaces of smooth curves, of curves of compact type, of stable curves, and of their analogues in the pointed case, and we will try to establish it.
Zagier (Max-Planck Institut für Mathematik) and I continue our work on obtaining an explicit description of the Gorenstein quotient of the tautological ring of the moduli space of curves of genus g.
Pandharipande (Princeton University) and I study the question whether the standard localization sequence is exact for the tautological algebras, this in relation with the problem of finding explicit relations between tautological classes.
Van der Geer (Universiteit van Amsterdam) and I try to obtain an explicit formula for the motivic Euler characteristic of the moduli space of n-pointed smooth curves of genus 2. Together with known results, this would lead to formulas for the Hodge numbers of the moduli spaces of stable n-pointed curves of genus 2. We will explore the relation with vector valued Siegel cusp forms. It is also natural to study the motives that appear. Jonas Bergström studies similar questions for higher genus. Lars Halvard Halle studies stable reduction in positive characteristic.