Georgia Institute of Technology College of Sciences

Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this setup to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as "combinatorial shadows" of curves. This tropical relationship between graphs and algebraic curves has led to beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs.

An introduction for non-experts on real and finite Euler sums, also known as multiple zeta values.

(Alon-Tarsi Conjecture): For even n, the number of even nxn Latin squares differs from the number of odd nxn Latin squares. (Stones-Wanless, Kotlar Conjecture): For all n, the number of even nxn Latin squares with the identity permutation as first row and first column differs from the number of odd nxn Latin squares of this type. (Aharoni-Berger Conjecture): Let M and N be two matroids on the same vertex set, and let A1,...,An be sets of size n + 1 belonging to both M and N. Then there exists a set belonging to both M and N and meeting all Ai. We prove equivalence of the first two conjectures and a special case of the third one and use these results to show that Alon-Tarsi Conjecture implies Rota's bases conjecture for odd n and any system of n non-singular real valued matrices where one of them is non-negative and the remaining have non-negative inverses.Joint work with Ron Aharoni.

Theoretical aspects: If a smooth dynamical system on a compact invariant set is structurally stable, then it has the shadowing property, that is, any pseudo (or approximate) orbit has a true orbit nearby. In fact, the system has the Lipschitz shadowing property, that is, the distance between the pseudo and true orbit is at most a constant multiple of the local error in the pseudo orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement for discrete dynamical systems, that is, if a discrete dynamical system has the Lipschitz shadowing property, then it is structurally stable. In this talk this result will be reviewed and the analogous result for flows, obtained jointly with S. Pilyugin and S. Tikhomirov, will be described. Numerical aspects: This is joint work with Brian Coomes and Huseyin Kocak. A rigorous numerical method for establishing the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous system of ordinary differential equations is presented. Given a suitable approximate connecting orbit and assuming that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently ``good''. It turns out that inversion of the operator is equivalent to the solution of a boundary value problem for a nonautonomous inhomogeneous linear difference equation. A numerical procedure is given to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse (the latter determines how ``good'' the approximating orbit must be).