Wai Yan Pong

Some applications of ordinal dimensions to the theory of differentially closed fields

Using the Lascar inequalities, we show that any finite rank δ-closed subset of a quasiprojective variety is definably isomorphic to an affine δ-closed set. Moreover, we show that if X is a finite rank subset of the projective space ℙ n and a is a generic point of ℙ n , then the projection from a is injective on X . Finally we prove that if RM = RC in DCF 0 , then RM = RU.

Sums of Consecutive Integers

Wai Yan Pong (wpong@csudh.edu) received his B.Sc. from the Chinese University of Hong Kong and his M.Sc. and Ph.D. from the University of Illinois at Chicago. He was a Doob Research Assistant Professor at the University of Illinois at Urbana-Champaign for three years. He then moved to California and is now teaching at California State University, Dominguez Hills. His research interests are in model theory and number theory.

LENGTH SPECTRA OF NATURAL NUMBERS

A natural number n can generally be written as a sum of m consecutive natural numbers for various values of m ≥ 1. The length spectrum of n is the set of these admissible m. Two numbers are spectral equivalent if they have the same length spectrum. We show how to compute the equivalence classes of this relation. Moreover, we show that these classes can only have either 1,2 or infinitely many elements.

Applications of differential algebra to algebraic independence of arithmetic functions

We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic functions and Dirichlet series by a theorem of Ax on differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature.

Locally finite homogeneous graphs

A connected graph G is said to be z-homogeneous if any isomorphism between finite connected induced subgraphs of G extends to an automorphism of G. Finite z-homogeneous graphs were classified in [17]. We show that z-homogeneity is equivalent to finite-transitivity on the class of infinite locally finite graphs. Moreover, we classify the graphs satisfying these properties. Our study of bipartite z-homogeneous graphs leads to a new characterization for hypercubes.

Quantifier-eliminable locally finite graphs

We identify the locally finite graphs that are quantifier-eliminable and their first order theories in the signature of distance predicates.

ON A RESULT OF ROSENLICHT

ABSTRACT We prove a model theoretic result about orthogonality in the theory of differentially closed fields. Using that, we deduce a result of Rosenlicht (Proposition 1).