at the

American Institute of Mathematics, Palo Alto, California

organized by

Jarod Alper, Maksym Fedorchuk, Brendan Hassett, and David Smyth

This workshop, sponsored by AIM and the NSF, will be devoted to applications of the minimal model program (MMP) to the study of geometry of moduli spaces of algebraic varieties.

A fundamental problem in algebraic geometry is the problem of constructing a moduli space for a nice class of varieties (e.g. smooth curves, smooth polarized K3 surfaces) and of finding a geometrically meaningful compactification for this moduli space. Once a compactification is constructed, one applies the methods of deformation theory and projective geometry to study, respectively, the local and the global geometry of the moduli space. Compactifications of moduli spaces of smooth objects can be constructed using different methods, including stack-theoretic methods, Geometric Invariant Theory (GIT), and the MMP. The principal focus of the workshop will be to use the minimal model program as a framework for understanding the relationships between these different compactifications.

More specifically, the main topics for the workshop are:

- The minimal model program for the moduli space of stable curves and stable maps; in particular, the search for a modular interpretation of the log canonical models of the moduli space of stable curves.
- Geometric Invariant Theory of polarized varieties; especially, stability of Hilbert points of pluricanonically embedded curves and surfaces.
- Extending recently developed techniques in moduli theory of curves to construct and study geometrically meaningful compactifications of moduli spaces of surfaces.

An important goal of the workshop will be to consolidate and disseminate the variety of different techniques, heuristics, and approaches that has been applied to these problems in recent years.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email *workshops@aimath.org*

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