We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth (n + 1)-dimensional Riemannian manifolds, a theorem proved first by Pitts for 2 ≤ n ≤ 5 and extended later by Schoen and Simon to any n.

We use moduli spaces of instantons and Chern-Simons invariants of flat connections to prove that the Whitehead doubles of (2, 2n − 1) torus knots are independent in the smooth knot concordance group; that is, they freely generate a subgroup of infinite rank. 1.

In this paper, we study the Pontryagin numbers of 24 dimensional String manifolds. In partic- ular, we find representatives of an integral basis of the String cobrodism group at dimension 24, based on the work of Mahowald and Hopkins (The structure of 24 dimensional manifolds having normal bundles which lift to B O [8], from “Recent progress in homotopy theory” (D. M. Davis, J. Morava, G. Nishida, W. S. Wilson, N. Yagita, editors), Contemp. Math. 293, Amer. Math. Soc., Providence, RI, 89-110, 2002), Borel and Hirzebruch (Am J Math 80: 459–538, 1958) and Wall (Ann Math 75:163–198, 1962). This has immediate applications on the divisibility of various characteristic numbers of the manifolds. In particular, we establish the 2-primary divisibilities of the signature and of the modified signature coupling with the integral Wu class of Hopkins and Singer (J Differ Geom 70:329–452, 2005), and also the 3- primary divisibility of the twisted signature. Our results provide potential clues to understand a question of Teichner.

Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses K¨ahler identities, the Rumin-de Rham complex and spectral geometry.

We consider reparametrizations of Heisenberg nilflows.We show that if a Heisenberg nilflow is uniquely ergodic, all non-trivial timechanges within a dense subspace of smooth time-changes are mixing. Equivalently, in the language of special flows, we consider special flows over linear skew-shift extensions of irrational rotations of the circle. Without assuming any Diophantine condition on the frequency, we define a dense class of smooth roof functions for which the corresponding special flows are mixing whenever the roof function is not a coboundary. Mixing is produced by a mechanism known as stretching of ergodic sums. The complement of the set of mixing time-changes (or, equivalently, of mixing roof functions) has countable codimension and can be explicitly described in terms of the invariant distributions for the nilflow (or, equivalently,for the skew-shift), producing concrete examples of mixing time-changes.