In this talk we will present a Weitzenbock formula for wedge product of two one forms and present two applications.
A classic result of Liouville tells us that any bounded entire function is constant. And modern complex geometry has allowed us to generalize this result to mappings between suitable complex manifolds. We shall take this principle even further to the realm of almost Hermitian geometry, using what tools and structures remain in the non-integrable case.
Control systems with symmetry that are not linearizable via feedback transformations may still be explicitly integrable via a "composition" of linearizations: the first from a quotient system and the second from restricting to integral manifolds of a contact system on a principal G-bundle. Such control systems with symmetry are called cascade feedback linearizable and this talk will mainly focus on the second of the above linearizations.
We will discuss some recent work on diagonalizing the Ricci tensor of invariant metrics on compact Lie groups, homogeneous spaces and cohomogeneity one manifolds, and connections to the Ricci flow.
There has been recent interest in understanding which solvmanifolds admit Ricci-negative metrics. There has been little investigation into the entire space of metrics for those manifolds known to admit Ricci-negative metrics. We will discuss some specific results for four-dimensional solvmanifolds and for almost Abelian solvmanifolds.
The earliest examples of isospectral manifolds have the same local geometries. In fact, isopectral pairs of manifolds arising from the group theoretic method of Sunada all have the same local geometries. However, examples from Gordon, Schueth, Sutton, and An-Yu-Yu demonstrate that in dimension five or higher, the local geometry is not a spectral invariant, even among locally homogeneous spaces. It is then interesting to investigate whether the local geometry of locally homogeneous manifolds is a spectral invariant in dimension four or lower. In this work, we provide strong evidence that the local geometry of locally homogeneous three-manifolds is indeed a spectral invariant. This is a joint work with Benjamin Schmidt and Craig Sutton.
For isometric group actions on a Riemannian manifolds, Grove and Searle introduced the notion of a core: a special submanifold with an action by a "smaller" group and having isometric orbit space. In positive curvature, when a core reduction to special actions (e.g. fixed point homogeneous) is possible, one can derive strong geometric and topological information about the orbit space, hence both the submanifold and the original manifold as well. We generalize the notion of a core to the context of singular Riemannian foliations. We prove that if a positively curved manifold carries a singular Riemannian foliation with nontrivial core, the resulting leaf space will have boundary, generalizing the isotropy lemma of Wilking. This is joint work with Diego Corro.
We attempt to define higher rank Seiberg-Witten equations on a closed, spin, oriented 4-manifolds, where a fixed spinor bundle \( \mathfrak{s} \) of X is twisted by a U(n)-vector bundle E (n>1). We shall show that with respect to a fixed unitary connection \( \nabla_0 \) of E and a parameter \( \epsilon>0 \), the moduli space of irreducible solutions to the U(n) Seiberg-Witten equations, \( \mathcal{M}^{\nabla_0, \epsilon}_{irr} \) is a finite dimensional compact manifold. This is a work in progress.
Manifold submetries generalize Riemannian submersions, isometric group actions, and isoparametric maps. Locally around a point, manifold submetries are determined by a manifold submetries from round spheres. In this talk, we will prove a surprising equivalence between the class of manifold submetries from round spheres and a special class of polynomial algebras, with applications, for example, to Invariant Theory. This is based on a joint work with Ricardo Mendes.
Within the class of nilpotent Lie algebras, filiform Lie algebras play a special role and have been well-studied. We will review the definition of filiform and summarize some basic results about filiform Lie algebras and associated homogeneous spaces. Then we will discuss, within the class of filiform Lie algebras, those that could be said to have the most symmetry. Finally we will describe recent results on the classification of N-graded filiform Lie algebras and their soliton inner products.
Square-tiled surfaces are finite branched covers of the standard square torus, with branching over exactly one point. These can be also thought of as surfaces obtained by taking finitely many Euclidean unit squares, and gluing the edges in parallel pairs. In this talk we will see topological and geometric statistics of random square-tiled surfaces using a simple combinatorial model.
In recent years, the problem of understanding the topology of spaces of Riemannian metrics which satisfy a particular curvature constraint has aroused considerable interest. There are many results concerning the non-triviality of such spaces: at the level of homotopy groups for example. Unsurprisingly, most progress has occurred in the case of positive scalar curvature. In particular, recent results of Ebert and Randal-Williams as well as those of the speaker show that such spaces can even display H-space and loop space structure. In this talk we will survey some of these results and then discuss some new results which are joint with David Wraith. These concern the analogous problem for the positive $k$-Ricci curvature (defined by Jon Wolfson).
In this talk we will be interested in Lie groups admitting (left-invariant) metrics with negative Ricci curvature. Note that structurally, the only pin- ching curvature behavior which is still not understood in the homogeneous case is Ric < 0. We will review the known results and then we will consi- der two situations. On the one hand, we will show a general construction of examples with compact Levi factor, which is unexpected in some sense and show that this construction actually gives many new examples. On the other hand, we will discuss the solvable case with focus in rank one solvable Lie groups.