arXiv research

Proves a stack of G-bundles with logarithmic connections is finite type.

Genus 3 Heegaard groups of lens space connected sums are finitely generated.

The kth finite subset space of a topological space X is the space exp_k X of non-empty subsets of X of size at most k, topologised as a quotient of X^k. Using results from our earlier paper (math.GT/0210315) on the finite subset spaces of connected graphs we show that the kth finite subset space of a connected cell com…

The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.

Paper proves edge-connectivity equals minimum degree for graphs with non-negative curvature.

Simply-connected shrinking Kähler-Ricci solitons are proven.

Smooth distributions on subcartesian spaces can be globally finitely generated.

Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.

Compact manifolds with boundary have finite type mapping class groups.

Study Kazdan-Warner equations on graphs using Brouwer degree theory.

Convex cores found for group actions on median spaces.

We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.

Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.

New result on finite gerbes simplifies classification of certain bundles.

In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…

A finitely presented group is weakly geometrically simply connected (wgsc) if it is the fundamental group of some compact polyhedron whose universal covering is wgsc i.e. it has an exhaustion by compact connected and simply connected sub-polyhedra. We show that this condition is almost-equivalent to Brick's qsf propert…

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense, and for any such operator we construct a partial connection on a (finite ra…

The homology of 4-manifold moduli spaces can be infinitely generated.

Finite vector bundles over complex manifolds are trivializable via finite covers.

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of …

Graph conditions ensure matching arc complexes are connected and hyperbolic.

Classifies connected shelves up to order six.

If a class of finitely generated groups Curly(G) is closed under isometric amalgamations along free subgroups, then every G in Curly(G) can be quasi-isometrically embedded in a group Hat(G) in Curly(G) that has no proper subgroups of finite index. Every compact, connected, non-positively curved space X admits an isomet…

The paper analyzes finite element methods on manifolds with approximate metrics.

Quandles can be regarded as generalizations of symmetric spaces. In the study of symmetric spaces, the notion of flatness plays an important role. In this paper, we define the notion of flat quandles, by referring to the theory of Riemannian symmetric spaces, and classify flat connected finite quandles.

Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a new connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central …

Paper solves whether zero sets are mapping degree sets.

The paper explores the structure of Reeb spaces for smooth functions on manifolds.

The paper characterizes simply connected quandles using cocycles with prime values.

We prove that any connected proper Dupin hypersurface in $\R^n$ is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in $\R^n$ that satisfies a certain finiteness condition. Hence any taut submanifo…

Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a $k$-differential with at least one pole of order at least $k…

The paper connects link symmetries to finite subgroups of O(3).

The study extends Huber's theorem to higher dimensions with curvature constraints.

Let $S$ be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets.

Spatial graphs are decomposed into planar forests and braids.

We show that for an m-connected cell complex X the space exp_k X of non-empty subsets of X of cardinality at most k is (m + k - 2)-connected

If X is a full, finitely generated, projective module over a non-commutative torus, the Yang-Mills functional attains its minimum exactly on the flat connections on X. We classify the flat connections on modules admitting integrable connections.

For most positive integer pairs $(a,b)$, the topological space $#a{\mathbb C \mathbb P}^2#b{\bar{\mathbb C \mathbb P^2}}$ is shown to admit infinitely many inequivalent smooth structures which dissolve upon performing a single connected sum with $S^2\times S^2$. This is then used to construct infinitely many non-equiva…

Let H_g denote the closed 3-manifold obtained as the connected sum of g copies of S^2 times S^1, with free fundamental group of rank g. We prove that, for a finite group G acting on H_g which induces a faithful action on the fundamental group, there is an upper bound for the order of G which is quadratic in g, but that…

Instantons on ALF spaces constructed from bow data.

The paper proves mapping class groups of closed surfaces are simply connected at infinity.

We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.

We provide an explicit section for a mapping class group sequence.

We consider local invariants of general connections (with torsion). The group of origin-preserving diffeomorphisms acts on a space of jets of general connections. Dimensions of moduli spaces of generic connections are calculated. Poincaré series of the geometric structure of connection is constructed, and shown to be a…

We prove the Farrell-Jones Conjecture for (non-connective) $A$-theory with coefficients and finite wreath products for hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties su…

Invariants for 4-manifolds from Hopf group-algebras.

We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.

Finite-time blow-up in Yang-Mills flow for small energy initial connections.