Georgia Institute of Technology College of Sciences

TBA

The fundamental data structures for numerical methods in algebraic geometry are called "witness sets." The term "trace test" refers to certain numerical methods which verify the completeness of such witness sets. It is natural to ask questions about the complexity of such a test and in what sense its output may be regarded as "proof." I will give a basic exposition of the trace test(s) with a view towards these questions

A classic theorem of Mader gives the extremal functions for graphs that do not contain the complete graph on p vertices as a minor for p up to 7. Motivated by the study of linklessly embeddable graphs, we present some results on the extremal functions of apex graphs with respect to the number of triangles, and on triangle-free graphs with excluded minors. Joint work with Robin Thomas.

The goal of this lecture is to explain and motivate the connection between Aubry-Mather theory (Dynamical Systems), and viscosity solutions of the Hamilton-Jacobi equation (PDE).This connection is the content of weak KAM Theory.The talk should be accessible to the “generic” mathematician. No a priori knowledge of any of the two subjects is assumed.The set-up of this theory is classical mechanical systems, in its Lagrangian formulation to take advantage of the action principle. This is the natural setting for Celestial Mechanics. Today it is also the setting for motions of satellites in the solar system.Hamilton found a reformulation of Lagrangian mechanics in terms of position and momentum instead of position and speed. In this formulation appears the Hamilton-Jacobi equation. Although this is a partial differential equation, its solutions allow to find solutions of the Hamiltonian (or Lagrangian) systems which are, in fact, governed by an ordinary differential equation.KAM (Kolmogorov-Arnold-Moser) theorem addressed at its beginning (Kolomogorov) the problem of stability of the solar system. It came as a surprise, since Poincare ́’s earlier work pointed to instability. In fact, some initial conditions lead to instability (Poincare ́) and some others lead to stability(Kolomogorov).Aubry-Mather theory finds some more substantial stable motion that survives outside the region where KAM theorem applies.The KAM theorem also provides global differentiable solutions to the Hamilton-Jacobi equation.It is known that the Hamilton-Jacobi equation usually does not have smooth global solutions. Lions & Crandall developed a theory of weak solutions of the Hamilton-Jacobi equation.Weak KAM theory explains how the Aubry-Mather sets can be obtained from the points where weak solutions of the Hamilton-Jacobi equation are differentiable.