arXiv research

Proves equivalence of local and global $L^p$-Brunn-Minkowski inequalities.

We show that every periodic virtual knot can be realized as the closure of a periodic virtual braid and use this to study the Alexander invariants of periodic virtual knots. If $K$ is a $q$-periodic and almost classical knot, we show that its quotient knot $K_*$ is also almost classical, and in the case $q=p^r$ is a pr…

We prove that the first reduced cohomology with values in a mixing Lp-representation, p larger than 1, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group h…

Let S be a closed, oriented surface of genus at least 2, and consider the extension 1 -> pi_1 S -> MCG(S,p) -> MCG(S) -> 1, where MCG(S) is the mapping class group of S, and MCG(S,p) is the mapping class group of S punctured at p. We prove that any quasi-isometry of MCG(S,p) which coarsely respects the cosets of the no…

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donalds…