We show that the counting of rational curves on a complete toric variety which are in general position relative to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory

We describe a systematic way of constructing effective divisors on the moduli space of stable curves having exceptionally small slope. We show that every codimension 1 locus in ${\stackrel{̲}{M}}_g$ consisting of curves failing to satisfy a Green-Lazarsfeld syzygy-type condition provides a counterexample to the Harris-Morrison slope conjecture. We also introduce a new geometric stratification of the moduli space of curves somewhat similar to the classical stratification given by gonality but where the analogues of hyperelliptic curves are the sections of $K3$ surfaces

We construct a family of ${\mathrm{BC}}_n$-symmetric biorthogonal abelian functions generalizing Koornwinder's orthogonal polynomials (see [10]) and prove a number of their properties, most notably analogues of Macdonald's conjectures. The construction is based on a direct construction for a special case generalizing Okounkov's interpolation polynomials (see [13]). We show that these interpolation functions satisfy a collection of generalized hypergeometric identities, including new multivariate elliptic analogues of Jackson's summation and Bailey's transformation

We prove the following conjecture recently formulated by Jakobson, Nadirashvili, and Polterovich (see [15, Conjecture 1.5, page 383]). On the Klein bottle $\mathbb{K}$, the metric of revolution ${\displaystyle g_0=\frac{9+(1+8\cos {}^{2}v){}^2}{1+8\cos {}^{2}v}({\mathrm{du}}^2+\frac{{\mathrm{dv}}^2}{1+8\cos {}^{2}v}),}$ $0\le u<\pi /2$, $0\le v<\pi$, is the unique extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof leads us to study a Hamiltonian dynamical system that turns out to be completely integrable by quadratures