Biology, Analysis, Geometry, Energies, Links [bagel19]: A Program on Low-dimensional Topology, Geometry, and Applications

The [bagel19] program is divided into two events. The first is a week-long summer school for graduate students and early career researchers that emphasizes both the interplay of low-dimensional  topology, geometry and knot theory, and their applications to other disciplines. This is immediately followed by a workshop that builds on  the topics covered in the prior week and features speakers from a broad set of disciplines with connections to low-dimensional topology, geometry and knots.

Summer School (June 17 - 22, 2019)

The week-long program will strongly emphasize interactivity with one third of the time dedicated to small groups of participants actively working on sample problems. It combines training and presentation of recent research results to an interdisciplinary audience. The study of (knotted) curves has always been a place of interdisciplinarity. Its roots trace back to Gauss’s studies on electromagnetism which led to the investigation of wires under electric current. While knots as objects in their own right have been studied for centuries, topologically knotted or linked structures recently gained much attention in particular with respect to applications, for example, molecular biology and topological fluid dynamics. In fact, these applications demand elaborate techniques and in turn ask new challenging questions to the theorists.

The summer school sheds some light on these aspects by presenting a state-of-the-art introduction to several mathematical fields concerned not only with curves but with higher-dimensional analogs as well. We will analyze these as:

• models for physical objects in order to discuss mechanical properties,
• limits of discrete structures which preserve the key features from the smooth category,
• points in certain domains of functionals that reveal information on embeddedness and smoothness,
• topological objects in their own right which exhibit a deep relation to 3-manifolds.

Tutorial Speakers

• Ken Millett (University of California, Santa Barbara)
• Jun O'Hara (Chiba University)
• Peter Schroeder (California Institute of Technology)
• Jennifer Schultens (University of California, Davis)
• Gert van der Heijden (Imperial College London)

Workshop (June 24 - 28, 2019)

The workshop will present the latest developments in the fields discussed at the summer school and beyond. In particular, we are interested in gathering experts working with curves and surfaces in topology, analysis, approximation, and modeling for applications in microbiology and elasticity. The mix of speakers and topics will foster transdisciplinary collaborations.

Topics to be explored include nonlinear mechanics, topological fluid dynamics, low-dimensional topology, quantum invariants, homology theory, categorification, contact geometry, geometric curvature energies, discretization of geometric flows, and discrete differential geometry.