Georgia Institute of Technology College of Sciences

In this talk we’ll discuss strong 4-colourings of graphs and prove two new cases of the Strong Colouring Conjecture. Let H be a graph with maximum degree at most 2, and let G be obtained from H by gluing in vertex-disjoint copies of K_4. We’ll show that if H contains at most one odd cycle of length exceeding 3, or if H contains at most 3 triangles, then G is 4-colourable. This is joint work with Greg Puleo.

My goal is to present a computer assisted proof of a non-trivial theorem in nonlinear dynamics, in full detail.  My (quite biased) definition of non-trivial is that there should be some infinite dimensional complications.  However, since I want to go through all the details, I need these complications to be as simple as possible.  So, I'll consider the Henon map, and prove that some 1 dimensional stable and unstable manifolds attached to a hyperbolic fixed point intersect transversally.  By Smale's theorem, this implies the existence of chaotic motions.  Recall that one can prove the existence chaotic dynamics for the Henon map more or less by hand using topological methods.  Yet transverse intersection of the manifolds is a stronger statement, and moreover the method I'll discuss generalizes to much more sophisticated examples where pen-and-paper fail.

The idea of the proof is to develop a high order polynomial expansion of the stable/unstable manifolds of the fixed point, to prove an a-posteriori theorem about the convergence and truncation error bounds for this expansion, and to check the hypotheses of this theorem using the computer.  All of this relies on the parameterization method of Cabre, Fontich, and de la Llave, and on finite numerical calculations using interval arithmetic to manage the inevitable roundoff errors. Once global enough representations of the local invariant manifolds are obtained and equipped with mathematically rigorous error bounds, it is a finite dimensional problem to establish that the manifolds intersect transversally.  

 In their seminal paper on combinatorial Hodge theory, Adiprasito, Huh, and Katz showed, among other things, that a very specific set of toric varieties has Chow rings that satisfy Poincaré duality, even though the varieties are not compact. In joint work with Farbod Shokrieh, we generalize this statement to all toric varieties whose fans are supported on a tropically smooth set. This has several consequences in tropical intersection theory; most notably it allows us to prove the long-suspected duality between tropical cycles and cocycles.

In my talk I will assume no prior knowledge of tropical intersection theory. I will define tropical cycles and cocycles explicitly and explain how they are connected to the intersection theory of toric varieties and the Chow rings of fans appearing in combinatorial Hodge theory. Finally, we will see how to use the duality statement mentioned above to define the tropical intersection product.

From a perspective of an applied algebraic geometer, I will address several use cases of algorithmic machinery that goes hand in hand with the language of tropical geometry. One of the examples originates in dynamical systems and may shed (tropical) light on a long-standing conjecture in celestial mechanics.