Georgia Institute of Technology College of Sciences

Fix k vertices in a graph G, say a_1,...,a_k, if there exists a cycle that visits these vertices with this specified order, we say such a cycle is (a_1,a_2,...,a_k)-ordered. It is shown by Thomas and Wollan that any 10k-connected graph is k-linked, therefore any 10k-connected graph has an (a_1,a_2,...,a_k)-ordered for any a_1,...,a_k. However, it is possible that we can improve this bound when k is small. It is shown by W. Goddard that any 4-connected maximal planar graph has an (a_1,...,a_4)-ordered cycle for any choice of 4 vertices. We will present a complete characterization of 4-ordered cycle in planar graphs. Namely, for any four vertices a,b,c,d in planar graph G, if there is no (a,b,c,d)-ordered cycle in G, then one of the follows holds: (1) there is a cut S separating {a,c} from {b,d} with |S|\leq 3; (2) roughly speaking, a,b,d,c "stay" in a face of G with this order.

The discussion will focus on some recent advances in improving performance of rendering 3D scenes. First, a Monte Carlo method based upon the Metropolis algorithm is described. Then a method of using spherical harmonics to generate vectors and matrices which allow efficient high-quality rendering in real time will be described. Finally, a discussion will be made of possible future areas for improving the efficiency of such algorithms.

We will study a simple dynamical system with two driving vector fields on the unit interval. The driving vector fields point to opposite directions, and we will follow the trajectory induced by one vector field for a random, exponentially distributed, amount of time before switching to the regime of the other one. Thanks to the simplicity of the system, we obtain an explicit formula for its invariant density. Basically exploiting analytic properties of this density, we derive versions of the law of large numbers, the central limit theorem and the large deviations principle for our system. If time permits, we will also discuss some ideas on how to prove existence of invariant densities, both in our one-dimensional setting and for more general systems with random switching. The talk will rely to a large extent on my Master's thesis I wrote last year under the guidance and supervision of Yuri Bakhtin.

The talk will begin with an elementary geometric discussion of Riemann-Roch theory for sub-lattices of the integer lattice orthogonal to some positive vector. A pair of necessary and sufficient conditions for such a lattice to have the Riemann-Roch property will be presented. By studying a certain chip firing game on a directed graph related to the lattice spanned by the rows of its Laplacian I will describe a combinatorial method for checking whether a directed graph has the Riemann-Roch property. The talk will conclude with a presentation of arithmetical graphs, which after the application of a simple transformation, may be viewed as a special class of directed graphs. Examples from this class demonstrate that either, both or neither of the Riemann-Roch conditions may be satisfied for a directed graph. This is joint work with Arash Asadi.