We study property ($T$) and the fixed-point property for actions on$L$^{$p$}and other Banach spaces. We show that property ($T$) holds when$L$^{2}is replaced by$L$^{$p$}(and even a subspace/quotient of$L$^{$p$}), and that in fact it is independent of 1≤$p$<∞. We show that the fixed-point property for$L$^{$p$}follows from property ($T$) when 1<$p$< 2+$ε$. For simple Lie groups and their lattices, we prove that the fixed-point property for$L$^{$p$}holds for any 1<$p$<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces.

We prove the Zorich–Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichmüller ow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface.

Let$χ$be an analytic vector field defined in a real-analytic manifold of dimension three. We prove that all the singularities of$χ$can be made elementary by a finite number of blowing-ups in the ambient space.

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the$n$th derivative tends to infinity, as $$n\to\infty$$ . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.

We show that closed, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by (Gromov in Inst Hautes Études Sci Publ Math 56:5,1982).