We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new nonvanishing criterion for the contact Ozsváth-Szabó invariant

The representation dimension is an invariant introduced by Auslander to measure how far a representation infinite algebra is from being representation finite. In 2005, Rouquier gave the first examples of algebras of representation dimension greater than three. Here we give the first general method for establishing lower bounds for the representation dimension of given algebras or families of algebras. The classes of algebras for which we explicitly apply this method include (but do not restrict to) most of the previous examples of algebras of large representation dimension, for some of which the lower bound is improved to the correct value

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a hyperbolic $n$-manifold of finite volume under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold.

Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal covering space $T^1({\mathbb{H}}^n)$ is mapped into a proper subsphere of the ideal boundary sphere $\partial {\mathbb{H}}^n$ under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in ${\mathbb{H}}^n$ covering the closed immersed submanifold.

In particular, if the visual map does not send a lift of the curve into a proper subsphere of $\partial {\mathbb{H}}^n$, then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bundle of the manifold with respect to the normalized natural Riemannian measure.

The proof uses dynamical properties of unipotent flows on homogeneous spaces of $\mathrm{SO}(n,1)$ of finite volume

Extending earlier results for analytic curve segments, in this article we describe the asymptotic behavior of evolution of a finite segment of a $C^n$-smooth curve under the geodesic flow on the unit tangent bundle of a hyperbolic $n$-manifold of finite volume. In particular, we show that if the curve satisfies certain natural geometric conditions, then the pushforward of the parameter measure on the curve under the geodesic flow converges to the normalized canonical Riemannian measure on the tangent bundle in the limit. We also study the limits of geodesic evolution of shrinking segments.

We use Ratner's classification of ergodic invariant measures for unipotent flows on homogeneous spaces of $\mathrm{SO}(n,1)$ and an observation relating local growth properties of smooth curves and dynamics of linear $\mathrm{SL}(2,\mathbb{R})$-actions

Let $G=\mathrm{Sp}(2n,\mathbb{C})$ be a complex symplectic group. We introduce a ($G\times ({\mathbb{C}}^{\times }{)}^{\ell +1}$)-variety ${\mathfrak{N}}_{\ell }$, which we call the $\ell$-exotic nilpotent cone. Then, we realize the Hecke algebra $\mathbb{H}$ of type $C_n^{(1)}$ with three parameters via equivariant algebraic $K$-theory in terms of the geometry of ${\mathfrak{N}}_2$. This enables us to establish a Deligne-Langlands–type classification of simple $\mathbb{H}$-modules under a mild assumption on parameters. As applications, we present a character formula and multiplicity formulas of $\mathbb{H}$-modules