We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably
less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds
which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing
symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several
recent results involving fillability, planarity, and non-separating contact type embeddings. The cobordisms are built from symplectic
handles of the form Σ × D and Σ × [−1, 1] × S1, which have symplectic cores and can be attached to contact 3-manifolds
along sufficiently large neighborhoods of transverse links and preLagrangian tori. We also sketch a construction of J-holomorphic
foliations in these cobordisms and formulate a conjecture regarding maps induced on Embedded Contact Homology with twisted
In this paper we study the analytic torsion and the L2-torsion of compact locally symmetric manifolds. We consider the analytic
torsion with respect to representations of the fundamental group which are obtained by restriction of irreducible representations of
the group of isometries of the underlying symmetric space. The main purpose is to study the asymptotic behavior of the analytic
torsion with respect to sequences of representations associated to rays of highest weights.
The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations.
It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature
flow, thus giving a Lyapunov functional. Therefore, the entropy of the initial hypersurface bounds the entropy at all future
singularities. We show here that not only does the round sphere have the lowest entropy of any closed singularity, but there is a
gap to the second lowest.
Let G be a connected Lie group. We show that all characteristic classes of G are bounded—when viewed in the cohomology of the
classifying space of the group G with the discrete topology—if and only if the derived group of the radical of G is simply connected
in its Lie group topology. We also give equivalent conditions in terms of stable commutator length and distortion.
We give a local classification of generalized complex structures. About a point, a generalized complex structure is equivalent to
a product of a symplectic manifold with a holomorphic Poisson manifold. We use a Nash-Moser type argument in the style of
Conn’s linearization theorem.