In an earlier paper [10], Howard proved one divisibility of Perrin-Riou's Iwasawa main conjecture for Heegner points on elliptic curves. In this paper, that result is generalized to abelian varieties of GL₂ type (i.e., abelian varieties with real multiplication defined over totally real fields) under the hypothesis that the abelian variety is associated to a Hilbert modular form via a construction of Zhang.

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space X into $\mathbb{R}^{m}$ can be uniformly approximated by C^∞-smooth mappings with no critical points. This kind of result can be regarded as a sort of strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of X can be separated by a one-codimensional smooth manifold that is a level set of a smooth function with no critical points. In particular, every closed set in X can be uniformly approximated by open sets whose boundaries are C^∞-smooth one-codimensional submanifolds of X. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space X with any smooth manifold M modeled on X.

We associate to any irreducible germ $\mathcal{S}$ of a complex quasi-ordinary hypersurface an analytically invariant semigroup. We deduce a direct proof (without passing through their embedded topological invariance) of the analytical invariance of the normalized characteristic exponents. These exponents generalize the generic Newton-Puiseux exponents of plane curves. Incidentally, we give a toric description of the normalization morphism of the germ $\mathcal{S}$.

We continue the study of [34], proving concentration phenomena for the equation − ε² Δu + u = u^p in a smooth bounded domain Ω ⊆ $\mathbb{R}^n$ and with Neumann boundary conditions. The exponent p is greater than or equal to 1, and the parameter ε is converging to zero. For a suitable sequence ε_j → 0, we prove the existence of positive solutions u_j concentrating at the whole boundary of Ω or at some of its components.

We study transfer operators $\mathcal{M}^{(k)}$ associated to a finite family {ψ_ω} of $\mathcal{C}^{r}$ (r ≥ 1) transversal maps U_ω → $\mathbb{R}^n$, where U_ω ⊂ $\mathbb{R}^n$, with $\mathcal{C}^{r}$ compactly supported weights g_ω, acting on k-forms in $\mathbb{R}^n$. Using the definitions of sharp trace Tr^≯ and flat trace Tr^≭, the following formula holds between power series: $\mathrm{Det}^{\#}(1-z\mathcal{M})=\Pi_{k=0}^n \mathrm{Det}^{\flat}(1-z\mathcal{M}^{(k)})^{(-1)^k}$. Following ideas of Kitaev [17], we define kneading operators $\mathcal{D}_k$(z), which are kernel operators. Our main result is the equality (as formal power series)

$$ \mathrm{Det}^{\#}(1-z\mathcal{M})=\prod_{k=0}^{n-1} \mathrm{Det}^{\flat}\big(1+\mathcal{D}_k(z)\big)^{(-1)^{k+1}}.$$

We also show that a finite power of $\mathcal{D}_k$(z) is trace-class on L². This (partially) generalizes results obtained by Baladi, Kitaev, Ruelle, and Semmes in dimension one, complex and real [8], [10]).

Through the study of some elliptic and parabolic fully nonlinear partial differential equations, we establish conformal versions of the quermass-integral inequality, the Sobolev inequality, and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on {locally conformally flat} manifolds.