Related papers: The strong Kervaire invariant problem in dimension…

Let G be a real form of a complex reductive group. Suppose that we are given involutions \sigma and \theta of G. Let H=G^\sigma denote the fixed group of \sigma and let K=G^\theta denote the fixed group of \theta. We are interested in…

Proving for triangulations an extended version of the 4-colour theorem by induction, we manage to exclude the case which led to the failure of Kempe's attempted proof. The new idea is to claim the existence of a "nice" 4-colouring, in which…

Three nonequivalent real forms of the complex twisted N=2 supersymmetric Toda chain hierarchy (solv-int/9907021) in real N=1 superspace are presented. It is demonstrated that they possess a global twisted N=2 supersymmetry. We discuss a new…

We study a modified version of an equation of the continuous Toda type in 1+1 dimensions. This equation contains a friction-like term which can be switched off by annihilating a free parameter $\ep$. We apply the prolongation method, the…

Let $\Sigma$ be a finite regular cell complex with $\emptyset \in \Sigma$, and regard it as a partially ordered set (poset) by inclusion. Let $R$ be the incidence algebra of the poset $\Sigma$ over a field $k$. Corresponding to the Verdier…

Given a smooth projective variety $X$ of Kodaira dimension zero, we show that there exists a constant $m$ depending on two invariants of the general fiber of the Albanese map, such that $|mK_X|\neq\emptyset$ .

Consider the equation div$(\varphi^2 \nabla \sigma)=0$ in $\mathbb{R}^N,$ where $\varphi>0$. It is well-known that if there exists $C>0$ such that $\int_{B_R}(\varphi \sigma)^2 dx\leq CR^2$ for every $R\geq 1$ then $\sigma$ is necessarily…

Dimension four provides a peculiarly idiosyncratic setting for the interplay between scalar curvature and differential topology. Here we will explain some of the peculiarities of the four-dimensional realm via a careful discussion of the…

We study the problem of prescribing $\sigma_k$-curvature for a conformal metric on the standard sphere $\mathbb{S}^n$ with $2 \leq k < n/2$ and $n \geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are…

This letter is an attempt to carry out a first-principle computation in M-theory using the point of view that the eleven-dimensional membrane gives the fundamental degrees of freedom of M-theory. Our aim is to derive the exact BPS $R^4$…

This is a survey of the rationality problem in invariant theory. It also contains some new results, in particular in Chapter 2 on moduli spaces of plane curves with a theta-characteristic, and a detailed account of the relation of the…

Given a smooth, closed, oriented 4-manifold X and alpha in H_2(X,Z) such that alpha.alpha > 0, a closed 2-form w is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles. The number…

Five-brane distributions with no strong coupling problems and high symmetry are studied. The simplest configuration corresponds to a spherical shell of branes with S^3 geometry and symmetry. The equations of motions with delta-function…

We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d $\mathcal{N}=(0,2)$ theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of…

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial…

We introduce the notion of a flat extension of a connection $\theta$ on a principal bundle. Roughly speaking, $\theta$ admits a flat extension if it arises as the pull-back of a component of a Maurer-Cartan form. For trivial bundles over…

We prove that, under a simple condition on the cohomology ring, every closed 4-manifold has mod 2 Seiberg-Witten simple type. This result shows that there exists a large class of topological 4-manifolds such that all smooth structures have…

We first construct a real family of $SL(2,\mathbb{R})$-invariant symbol composition product $\{\sharp_\theta\}_{\theta\in,\mathbb{R}}$ on the analogue of the Schwartz space $S(\mathbb{D})$ on the hyperbolic plane…

Given a Chevalley group $\mathcal{G}$ of classical type and a Borel subgroup $\mathcal{B} \subseteq \mathcal{G}$, we compute the $\Sigma$-invariants of the $S$-arithmetic groups $\mathcal{B}(\mathbb{Z}[1/N])$, where $N$ is a product of…

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some…