We prove that the product of finitely many Toeplitz operators on the Hardy space is zero if and only if at least one of the operators is zero. We use some new vector-valued techniques that not only lead to a vector-valued version of this result but also appear to be needed in the scalar case

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, $V_k$ and $R_k$, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of $V_k$ and $R_k$ are analytically isomorphic if the group is $1$-formal; in particular, the tangent cone to $V_k$ at $1$ equals $R_k$. These new obstructions to $1$-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at $1$ to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given

Let us modify the scatterer configuration of a planar, finite-horizon Lorentz process in a bounded domain. Sinai asked in 1981 whether, for the diffusively scaled variant of the modified process, convergence to Brownian motion still holds. The main result of this work answers Sinai's question in the affirmative. Other types of local perturbations are also investigated: finite-horizon periodic Lorentz processes in the half strip or in the half plane (in these models, the local perturbation is the boundary condition) and finite-horizon, periodic Lorentz processes with a small, compactly supported external field in the strip. The corresponding limiting processes are Brownian motions with suitable boundary conditions and the skew Brownian motion on the line. The proofs combine Stroock and Varadhan's martingale method in [SV1] with our recent work in [DSV]

We study the equivariant $K$-group of the affine flag manifold with respect to the Borel group action. We prove that the structure sheaf of the (infinite-dimensional) Schubert variety in the K-group is represented by a unique polynomial, which we call the affine Grothendieck polynomial

In this article, we study the topology of the space $I_{\omega }$ of complex structures compatible with a fixed symplectic form $\omega$, using the framework of Donaldson. By comparing our analysis of the space $I_{\omega }$ with results of McDuff on the space $J_{\omega }$ of compatible almost complex structures on rational ruled surfaces, we find that $I_{\omega }$ is contractible in this case.

We then apply this result to study the topology of the symplectomorphism group of a rational ruled surface, extending results of Abreu and McDuff