Georgia Institute of Technology College of Sciences

The question of what conditions guarantee that a symplectic$S^1$ action is Hamiltonian has been studied for many years. Sue Tolmanand Jonathon Weitsman proved that if the action is semifree and has anon-empty set of isolated fixed points then the action is Hamiltonian.Furthermore, Cho, Hwang, and Suh proved in the 6-dimensional case that ifwe have $b_2^+=1$ at a reduced space at a regular level $\lambda$ of thecircle valued moment map, then the action is Hamiltonian. In this paper, wewill use this to prove that certain 6-dimensional symplectic actions whichare not semifree and have a non-empty set of isolated fixed points areHamiltonian. In this case, the reduced spaces are 4-dimensional symplecticorbifolds, and we will resolve the orbifold singularities and useJ-holomorphic curve techniques on the resolutions.

In 1994, Dumortier, Roussarie and Rousseau launched a program aiming at proving the finiteness part of Hilbert’s 16th problem for the quadratic system. For the program, 121 graphics need to be proved to have finite cyclicity. In this presentation, I will show that 4 families of HH-graphics with a triple nilpotent singularity of saddle or elliptic type have finite cyclicity. Finishing the proof of the cyclicity of these 4 families of HH-graphics represents one important step towards the proof of the finiteness part of Hilbert’s 16th problem for quadratic systems. This is a joint work with Professor Christiane Rousseau and Professor Huaiping Zhu.

Experimental neuroscience is achieving rapid progress in the ability to collect neural activity and connectivity data. This holds promise to directly test many theoretical ideas, and thus advance our understanding of "how the brain works." How to interpret this data, and what exactly it can tell us about the structure of neural circuits, is still not well-understood. A major obstacle is that these data often measure quantities that are related to more "fundamental" variables by an unknown nonlinear transformation. We find that combinatorial topology can be used to obtain meaningful answers to questions about the structure of neural activity. In this talk I will first introduce a new method, using tools from computational topology, for detecting structure in correlation matrices that is obscured by an unknown nonlinear transformation. I will illustrate its use by testing the "coding space" hypothesis on neural data. In the second part of my talk I will attempt to answer a simple question: given a complete set of binary response patterns of a network, can we rule out that the network functions as a collection of disconnected discriminators (perceptrons)? Mathematically this translates into questions about the combinatorics of hyperplane arrangements and convex sets.

We will give an overview of ideas that go into solution of Yamabe problem: Given a compact Riemannian manifold (M,g) of dimension n > 2, find a metric conformal to g with constant scalar curvature.