We prove a rigidity theorem for represented semi-simple Lie groups. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal $G$ orbit in $\mathbb{P}_{\mathfrak{g}}$) is extrinsically rigid to third order (with the exception of $\mathfrak{g} = \mathfrak{a}_1$).

In contrast, we show that the adjoint variety of $SL_3\mathbb{C}$ and the Segre product $Seg(\mathbb{P}^1 \times \mathbb{P}^n$) are flexible at order two. In the $SL_3\mathbb{C}$ example we discuss the relationship between the extrinsic projective geometry and the intrinsic path geometry.

We extend machinery developed by Hwang and Yamaguchi, Se-ashi, Tanaka and others to reduce the proof of the general theorem to a Lie algebra cohomology calculation. The proofs of the flexibility statements use exterior differential systems techniques.

The purpose of this note is to scrutinize the proof of Burago and Zalgaller regarding the existence of $PL$ isometric embeddings of $PL$ compact surfaces into $\mathbb{R}^3$. We conclude that their proof does not admit a direct extension to higher dimensions. Moreover, we show that, in general, $PL$ manifolds of dimension $n \ge 3$ admit no nontrivial $PL$ embeddings in $\mathbb{R}^{n+1}$ that are close to conformality. We also extend the result of Burago and Zalgaller to a large class of noncompact $PL$ 2-manifolds. The relation between intrinsic and extrinsic curvatures is also examined, and we propose a $PL$ version of the Gauss compatibility equation for smooth surfaces.

Minimal affine Kac-Moody groups act on affine twin buildings by isometries. However there is no way to extend this action to any completion of the Kac-Moody groups. To remedy this, we introduce in this paper affine twin cities, a new class of objects, whose elements behave like completions of twin buildings. Twin cities are defined as special arrays of affine buildings connected among themselves by twinnings. Corresponding to completed affine Kac-Moody groups they are characterized by the type of the affine buildings and by some kind of "regularity conditions" describing the completion. The isometry groups of affine twin cities are (completions of) affine Kac-Moody groups. We study applications of cities in infinite dimensional differential geometry by proving infinite dimensional versions of classical differential geometric results: For example, we show that points in an isoparametric submanifold in a Hilbert space correspond to all chambers in a city. In two sequels we will describe the theory of twin cities for formal completions.

We show a relationship between Chern–Simons 1- and 3-forms and harmonic forms on a principal bundle. Doing so requires one to consider an adiabatic limit. For the 3-form case, assume that $G$ is simple and the corresponding Chern–Weil 4-form is exact. Then, the Chern–Simons 3-form on the princpal bundle $G$-bundle, minus a canonical term from the base, is harmonic in the adiabatic limit.

In this paper we define and study the moduli space of metric-graph-flows in a manifold $M$. This is a space of smooth maps from a finite graph to $M$, which, when restricted to each edge, is a gradient flow line of a smooth (and generically Morse) function on $M$. Using the model of Gromov- Witten theory, with this moduli space replacing the space of stable holomorphic curves in a symplectic manifold, we obtain invariants, which are (co)homology operations in $M$. The invariants obtained in this setting are classical cohomology operations such as cup product, Steenrod squares, and Stiefel- Whitney classes. We show that these operations satisfy invariance and gluing properties that fit together to give the structure of a topological quantum field theory. By considering equivariant operations with respect to the action of the automorphism group of the graph, the field theory has more structure. It is analogous to a homological conformal field theory. In particular we show that classical relations such as the Adem relations and Cartan formulae are consequences of these field theoretic properties. These operations are defined and studied using two different methods. First, we use algebraic topological techniques to define appropriate virtual fundamental classes of these moduli spaces. This allows us to define the operations via the corresponding intersection numbers of the moduli space. Secondly, we use geometric and analytic techniques to study the smoothness and compactness properties of these moduli spaces. This will allow us to define these operations on the level of Morse-Smale chain complexes, by appropriately counting metric-graph-flows with particular boundary conditions.

We study the infinitesimal deformations of a proper nearly parallel $G_2$-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a subspace of co-closed $\Lambda^3_{27}$-eigenforms of the Laplace operator for the eigenvalue $8\mathrm{scal} /21$. We give a similar description for the space of infinitesimal Einstein deformations of a fixed nearly parallel $G_2$-structure. Moreover we show that there are no deformations on the squashed $S^7$ and on $\mathrm{SO}(5)/\mathrm{SO}(3)$, but that there are infinitesimal deformations on the Aloff-Wallach manifold $N(1, 1) = \mathrm{SU}(3)/U(1)$.

We study the generic Hodge groups $\mathrm{Hg}(\mathcal{X})$ of local universal deformations $\mathcal{X}$ of Calabi-Yau 3-manifolds with onedimensional complex moduli, give a complete list of all possible choices for $\mathrm{Hg}(\mathcal{X})_{\mathbb{R}}$ and determine the latter real groups for known examples.

For $n \ge 2$, we show that on the standard Poisson homogeneous space $\mathbb{S}^{2n−1}$ (including $SU (2) \approx \mathbb{S}3$), there exists a Poisson scaling $\phi_\lambda$ at any scale $\lambda \gt 0$ that is smooth on each symplectic leaf and continuous globally. A generalization to the case of the standard Bruhat-Poisson compact simple Lie groups endowed with a stronger topology is also valid.

We study the geometry of the natural map from the Hurwitz space $\overline{H}_{2k,k+1}$ to the moduli space $\overline{\mathcal{M}}_{2k}$. We calculate the cycle class of the Hurwitz divisor $D_2$ on $\overline{\mathcal{M}}_g$ for $g = 2k$ given by the degree $k + 1$ covers of $\mathbb{P}^1$ with simple ramification points, two of which lie in the same fibre. This has applications to bounds on the slope of the moving cone of $\overline{\mathcal{M}}_g$, the calculation of other divisor classes and motivated an algebraic proof for the formula of the Hodge bundle of the Hurwitz space.