Basic Notions Seminar
All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.
The Riemann-Roch Theorem for GraphsCaroline MellesUSNATime: 03:45 PMThe Riemann-Roch theorem is one of the fundamental results on algebraic curves. In 2007, Baker and Norine formulated and proved a Riemann-Roch theorem for graphs. Central to their proof is the notion of a non-special divisor on a graph. Non-special divisors can be created by certain combinatorial procedures. The number of equivalence classes of non-special divisors on a graph is the value of the Tutte polynomial of the graph evaluated at (1,0). The Riemann-Roch theorem for graphs will be discussed, with emphasis on the role of non-special divisors in the proof.
Successes and Failures of Linearization (Surprising Misbehavior of Piece-wise Smooth Systems)Irina PopoviciUSNATime: 12:00 PM1. A few standard results from non-linear ODEs associated with hyperbolic cases (when linearization works), and when small perturbations in the underlying system do not change the behavior of trajectories. 2. Standard results addressing the existence of cycles (linearization and perturbations of systems may lead to different trajectories). 3. Difficulties associated with systems that lack the C1 smoothness (only piece-wise smooth).