Topic: The 2016 Talbot workshop, aimed toward graduate students and other young researchers, was about Hill-Hopkins-Ravenel's proof of the Kervaire invariant one problem.

The Kervaire invariant one problem, formulated in the 1960s and solved in 2009 by Hill, Hopkins, and Ravenel, has far-reaching implications in manifold theory and in stable homotopy theory. Their unexpected solution to the question of existence of elements of Kervaire invariant one answered questions about framed bordism and provided a construction of triangulated manifolds without smooth structure in all but finitely many dimensions, as well as topological spheres that are not diffeomorphic to the standard sphere, but also indicated that the behavior of the Adams spectral sequence and the behavior of the EHP spectral sequence are more mysterious than previously thought. Hill, Hopkins, and Ravenel's solution involved major developments in chromatic homotopy theory, the theory of structured ring spectra, and equivariant stable homotopy theory, and has initiated growing interest in these areas, particularly equivariant stable homotopy theory and equivariant derived algebraic geometry.

The workshop started with an introduction to equivariant stable homotopy theory, and then delved into the proof in detail. A syllabus for the talks, with suggested references, can be found here.

Mentors: The 2016 Talbot workshop was mentored by Prof. Douglas Ravenel of University of Rochester and Prof. Mike Hill of UCLA.

Format: The workshop discussions had an expository character and most of the talks were given by participants. The afternoon schedule was kept clear for informal discussions and collaborations. The workshop took place in a communal setting, with participants sharing living space and cooking and cleaning responsibilities.

Timeline: Applications for the 2016 Talbot Workshop are now closed. Check back in late Fall 2016 for the topic and application form for the next Talbot workshop.