Georgia Institute of Technology College of Sciences

For integers k>=1 and n>=2k+1, the bipartite Kneser graph H(n,k) is defined as the graph that has as vertices all k-element and all (n-k)-element subsets of {1,2,...,n}, with an edge between any two vertices (=sets) where one is a subset of the other. It has long been conjectured that all bipartite Kneser graphs have a Hamilton cycle. The special case of this conjecture concerning the Hamiltonicity of the graph H(2k+1,k) became known as the 'middle levels conjecture' or 'revolving door conjecture', and has attracted particular attention over the last 30 years. One of the motivations for tackling these problems is an even more general conjecture due to Lovasz, which asserts that in fact every connected vertex-transitive graph (as e.g. H(n,k)) has a Hamilton cycle (apart from five exceptional graphs). Last week I presented a (rather technical) proof of the middle levels conjecture. In this talk I present a simple and short proof that all bipartite Kneser graphs H(n,k) have a Hamilton cycle (assuming that H(2k+1,k) has one). No prior knowledge will be assumed for this talk (having attended the first talk is not a prerequisite). This is joint work with Pascal Su (ETH Zurich).

Part II of last week's talk.

Complex-oriented cohomology theories are a class of generalized cohomology theories with special properties with respect to orientations of complex vector bundles. Examples include all ordinary cohomology theories, complex K-theory, and (our main theory of interest) complex cobordism.In two talks on these cohomology theories, we'll construct and discuss some examples and study their properties. Our ultimate goal will be to state and understand Quillen's theorem, which at first glance describes a close relationship between complex cobordism and formal group laws. Upon closer inspection, we'll see that this is really a relationship between C-oriented cohomology theories and algebraic geometry.

The conventional point of view is that the Lagrangian is a scalar object, which through the Euler-Lagrange equations provides us with one single equation. However, there is a key integrability property of certain discrete systems called multidimensional consistency, which implies that we are dealing with infinite hierarchies of compatible equations. Wanting this property to be reflected in the Lagrangian formulation, we arrive naturally at the construction of Lagrangian multiforms, i.e., Lagrangians which are the components of a form and satisfy a closure relation. Then we can propose a new variational principle for discrete integrable systems which brings in the geometry of the space of independent variables, and from this principle derive any equation in the hierarchy.