Let S be a complex smooth projective surface and L be a line bundle on S. G¨ottsche conjectured that for every integer r, the number of r-nodal curves in |L| is a universal polynomial of four topological numbers when L is sufficiently ample. We prove G¨ottsche’s conjecture using the algebraic cobordism group of line bundles on surfaces and degeneration of Hilbert schemes of points. In addition, we prove the G¨ottsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasimodular forms and two unknown series.

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This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles B, spatial chaos occurs when the spatial entropy h(B) is positive. B is called a minimal cycle generator if P(B) 6= ∅ and P(B′) = ∅ whenever B′ $ B, where P(B) is the set of all periodic patterns on Z2 generated by B. Given a set of Wang tiles B, write B = C1 ∪ C2 ∪ · · · ∪ Ck ∪ N, where Cj, 1 ≤ j ≤ k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C1 ∪ C2 ∪ · · · ∪ Ck. Then, the positivity of spatial entropy h(B) is completely determined by C1 ∪ C2 ∪ · · · ∪ Ck. Furthermore, there are 39 equivalent classes of marginal positive-entropy (MPE) sets of Wang tiles and 18 equivalent classes of saturated zero-entropy (SZE) sets of Wang tiles. For a set of Wang tiles B, h(B) is positive if and only if B contains an MPE set, and h(B) is zero if and only if B is a subset of an SZE set.

In this paper, we derive the CR Reilly'’s formula and its applications to studying of the fi…rst eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the …first Dirichlet eigenvalue estimate in a compact pseudo-hermitian manifold with boundary and the fi…rst eigenvalue estimate of the tangential sublaplacian on closed oriented embedded p-minimal hypersurfaces in a closed pseudohermitian manifold of vanishing torsion.

Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. In 1975, Erdős raised the question of determining $f(n)$ [see J. A. Bondy and U. S. R. Murty, Graph theory with applications, Elsevier, New York, 1976; MR0411988 (54 #117)]. We prove that for $n\geq 36\cdot 5t^2-4\cdot 5t+1$ one has $f(n)\geq n+9t-1$ . We conjecture that $\liminf (f(n)-n)/\sqrt n\leq\sqrt 3$ .