DescriptionMorse Theory is a classical area in topology now about a hundred years old, which allows us to understand differentiable manifolds up to deformations. This theory has many applications and was used to prove outstanding problems in the 20th century, such as the high-dimensional Poincaré conjecture.In 1998, Robin Forman introduced a combinatorial Morse theory for cell complexes which allowed him to use many of the techniques from the smooth theory to study the topology of cell complexes. These cell complexes are a generalization of simplicial complexes where one looks at cells instead of simplices. The theory already works very well on simplicial complexes, and these provide an easy starting point to the theory. Morse functions on a cell complex simply assign to every cell a real number subject to a certain condition. One can then introduce the concept of a critical cell (as opposed to a critical point) and it is possible to recover the homotopy type of the cell complex from the critical cells. One of the nice features of this approach is that everything is discrete so that one does not have to worry about continuity or smoothness of functions defined on the cell complex. A lot of the results in smooth Morse theory carry over to the discrete setting. Moreover, there is a notion of piecewise linear manifold which is analogous to a smooth manifold, and whose topology is particularly interesting and accessible via discrete Morse theory. Group ProjectThe group project will revolve around learning basic concepts and results in discrete Morse Theory. By the end of the group project we would have learned about topics such as
Mode of Operation and Evidence of Learning for the group projectThe project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.Individual ProjectThe individual project will build on the knowledge we have gained in the group project and will explore additional advanced topics. A few examples of topics you will be able to investigate are:
Mode of Operation and Evidence of Learning for the individual projectThe project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.Prerequisites and CorequisitesAnalysis I and Linear Algebra I are necessary. Algebra II could be useful, but is not necessary.There are no corequisites, but you might consider Analysis and Topology III, and Codes and Knots III. Additional InformationIf you would like to discuss or receive more information about this project, please contact me at dirk.schuetz@durham.ac.ukResourcesThe standard main article is Morse theory for Cell Complexes, Advances in Mathematics 134 (1998), 90-145, by R. Forman. Other articles, including introductory expositions can be obtained from his website. |
email: D Schuetz