Abstract: In 1950's R.Thom and H.Whitney began to study singularity theory of differentiable maps between manifolds. One can study the theory from a local or global viewpoint, both of which are interesting and attractive independently. In 1955 Thom introduced the notion of the "Thom polynomial of singularities" in order to study the universally global behavior of a generic smooth map. This is a polynomial written by cohomology classes which is Poincare dual to the homology class represented by the closure of the singular point locus. The purpose of this talk is to discuss the Thom polynomials and other obstruction classes by referring to recent progress.

Tentative List of Participants:
Scott Ahlgren, University of Illinois at Urbana-Champaigne Kathrin Bringmann, University of Minnesota - Minneapolis Jan Hendrik Bruinier, Technical University at Darmstadt, Germany YoungJu Choie Pohang, Technical University, Korea Youn-Seo Choi, Korean Institute for Advanced Studies, Seoul, Korea Amanda Folsom, University of Wisconsin at Madison Sharon Garthwaite, Bucknell University, Lewisburg Pavel Guerzhoy, University of Hawaii at Manoa Paul Jenkins, University of California at Los Angeles, Winfried Kohnen, University of Heidelberg, Germany Karl Mahlburg, Massachusets Institute of Technology Ken Ono, University of Wisconsin at Madison Robert Osburn, University College, Dublin, Ireland David Penniston, Furman University, Greenville Sander Zwegers, University College, Dublin, Ireland

Abstract: We consider the period solutions of the coupled van der Pol equation system. The fact that the single van der Pol equation has a unique limit cycle which is obitally stable is well known and proved by Poincare-Bendixson theorem. The coupled van der Pol equation system we consider constructs the four-dimensional space. Therefore we can not apply Poincare-Bendixson theorem to our system. 

In our system we have two distinctive solutions: in-phase and out-of-phase solutions. We prove that the periodic solution of our coupled van der Pol equation system is in-phase or out-of-phase solution. Also we talk about the application of this system to robotics area: generation of walking patterns.

Abstract: The starting point of Lagrangian and Hamiltionian mechanics is the observation that the form of Newton's third law F=ma is not invariant under general coordinate changes when the force F is the gradient of a potential F = grad(V). Lagrangian and Hamiltonian mechanics are two equivalent but essentially different solutions to make the form of the equations invariant under general coordinate changes. However, they do not have exactly the same features. The Lagrangian point of view seems to be better adapted for (quantum) field theories, whereas the Hamiltonian point of view seems to be better adapted for non-relativistic quantum mecahnics. In this talk I will argue that most (if not all) features of Lagrangian mechanics are also present in symplectic geometry (a generalization of Hamiltonian mechanics) and that one obtains a better understanding of Lagrangian mechanics when seen this way. No prior knowledge on mechanics other than Newton's law is presupposed, nor mathematics beyond the beginning graduate level (though some knowledge of differential geometry and fiber bundles will help in understanding the last part of my talk).