Knot concordance: Fifty years since Fox and Milnor, June 2008

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Fifty years ago, in a 1957 research announcement for the summer AMS meeting, R. H. Fox and J. Milnor, began their influential collaboration with the title, `Singularities of 2-spheres in 4-space and equivalence of knots.' Here they introduced the seminal idea that the concordance class of the link of a singularity obstructs its removal. Both concordance of knots, and the motivating goal of understanding singularities remain central to topology and algebraic geometry. A conference at Brandeis University will be held on June 2-5, 2008, to bring together a variety of researchers and students in geometric topology whose work connects to this fundamental idea. The conference will fertilize new research directions by encouraging mathematical interaction and collaboration. A substantial number of young investigators will be invited to give them exposure, broaden their perspective, and allow them to get to know each other and the more senior members of these fields. There will be approximately twenty invited addresses, and ample time will be set aside for interaction among the attendees. The conference will also honor the memory of Jerome Levine, a pioneer and key contributor to the field.A knot is a non-intersecting closed curve in three dimensional space. By closed, one means that if one travels along the curve, originating from any point on the curve, one eventually returns to the point of origin. A foundational subject to the field of topology, knot theory interacts with important areas in geometry, biology, and physics as well. The Knot concordance group measures wrinkles, or singularities, of surfaces in four dimensional space, and through this connection, enlightens our understanding of four dimensional shapes. Despite intense effort over the last 50 years, these groups have not been fully computed. Recently, new techniques from low-dimensional topology, including gauge theory, non-commutative algebra, higher- order linking theory, and quantum topology, have resulted in great advances in the study of knot concordance, revealing new structures in the concordance group. The conference will promote interactions between researchers working on these aspects of knot concordance and engender future research on this fundamental problem.