p-adic representations, modularity, and beyond

This workshop, sponsored by AIM and the NSF, will be devoted to interactions between p-adic Hodge theory, p-adic Langlands correspondences, and the modularity of Galois representations.

The current progress on modularity of Galois representations originates with Wiles's work on Fermat's Last Theorem. New ideas from p-adic Hodge theory have enabled a number of authors to improve Wiles's results, but significant technical challenges stand in the way of obtaining stronger results. These difficulties now seem to be related to questions about p-adic Langlands correspondences; thanks to the work of many people, understanding of these correspondences has improved rapidly in recent years.

The main goals of the workshop are to clarify the connections between the aforementioned fields, and to identify some target results for both the short and long term. Some specific topics to be discussed include:

• new methods in p-adic Hodge theory, including links to p-adic differential equations;
• the Taylor-Wiles method and modular lifting theorems in higher weight;
• concrete steps towards a p-adic Langlands correspondence.

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 working sessions.

Invited participants include L. Berger, C. Breuil, K. Buzzard, F. Calegari, R. Crew, F. Diamond, M. Emerton, A. Iovita, K. Kedlaya, C. Khare, M. Kisin, A. Mezard, W. Niziol, R. Ramakrishna, K. Ribet, D. Savitt, P. Schneider, C. Skinner, R. Taylor, J. Teitelbaum, M.-F. Vigneras, and H. Zhu.

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

For more information email workshops@aimath.org

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