Georgia Institute of Technology College of Sciences

In this talk, we describe transforming a theoretical algorithm from structural graph theory into open-source software being actively used for real-world data analysis in computational biology. Specifically, we apply the r-dominating set algorithm for graph classes of bounded expansion in the setting of metagenome analysis. We discuss algorithmic improvements required for a practical implementation, alongside exciting preliminary biological results (on real data!). Finally, we include a brief retrospective on the collaboration process. No prior knowledge in metagenomics or structural graph theory is assumed. Based on joint work with T. Brown, D. Moritz, M. O’Brien, F. Reidl and T. Reiter.

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.

There is so much that the GT library can do for you, from providing research materials to assistance with data visualization to patent guidance. However, rather than trying to guess what you want from us, this year we asked! Based on the response to a short ranking survey I sent out last month, this session will cover: 1. How to find grants, fellowships, and travel money with the sponsorship database, Pivot. There are opportunities for postdocs and non US citizens too!2. How to use MathSciNet. We will cover navigating its classification index to actually getting the article you want. 3. How to find and download articles from our systems, Google Scholar, and from other libraries. And if we have time: 4. How to make a poster and cheaply print it.

Some basic problems, notions and results of the Ergodic theory will be introduced. Several examples will be discussed. It is also a preparatory talk for the next day colloquium where finite time properties of dynamical and stochastic systems will be discussed rather than traditional questions all dealing with asymptotic in time properties.