arXiv research

The paper explores metrics on tree moduli spaces and a new topological group.

Counting HCMU sphere components using weighted trees.

The basin of infinity of a polynomial map $f : {\bf C} \arrow {\bf C}$ carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial t…

The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing an…

In this paper we construct Riemannian metrics and weight functions over Casson handles. We show that the corresponding Atiyah-Hitchin-Singer complexes are Fredholm for some class of Casson handles of bounded type. Using these, the Yang-Mills moduli spaces are constructed as finite dimensional smooth manifolds over Cass…

Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.

Study the boundary of Riemann surfaces with abelian automorphisms.

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the $ε$-thick part of moduli space of S equipped with the Teichmüller or Thurston's Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order $\log \frac{g+p}ε$. The same result also ho…

A conceptual framework for cluster analysis from the viewpoint of p-adic geometry is introduced by describing the space of all dendrograms for n datapoints and relating it to the moduli space of p-adic Riemannian spheres with punctures using a method recently applied by Murtagh (2004b). This method embeds a dendrogram …

We give a proof, using harmonic maps from disks to real trees, of Skora's theorem (Morgan-Otal (1993), Skora (1990), originally conjectured by Shalen): if G is the fundamental group of a surface of genus at least 2, then any small minimal G-action on a real tree is dual to the lift of a measured foliation. Analytic too…

Solves weighted bi-colored plane tree enumeration and applies to geometric problems.

The Hitchin component is a connected component of the character variety of reductive group homomorphisms from the fundamental group of a closed surface S of genus greater than 1 to the Lie group PSL_m(R). The Teichmuller space of S naturally embeds into the Hitchin component. The limit points in the Thurston compactifi…

The paper connects harmonic forms to tree maps and character varieties.

Let $X$ be a closed, four-dimensional, oriented, smooth manifold with a Riemannian metric, $g$, let $G$ be a compact Lie group, and $P$ be a principal $G$ bundle over $X$. D. Groisser and T. Parker (1987, 1989) and S. K. Donaldson (1990) conjectured that the moduli space of $g$-anti-self-dual connections on $P$, endowe…

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist $p$-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the $p$-adic projective line. The implications a…

Defines super stable maps and proves quotient superorbifolds for genus zero.

In this paper, we give an equivariant compactification of the space PFlat(S) of homothety classes of half-translation structures on a compact, connected, orientable surface S. We introduce the space PMix(S) of homothety classes of mixed structures on S, that are CAT(0) tree-graded spaces in the sense of Drutu and Sapir…

We propose a new statistical model suitable for machine learning of systems with long distance correlations such as natural languages. The model is based on directed acyclic graph decorated by multi-linear tensor maps in the vertices and vector spaces in the edges, called tensor network. Such tensor networks have been …

Recently, the infinitesimal moduli space of heterotic $G_2$ compactifications was described in supergravity and related to the cohomology of a target space differential. In this paper we identify the marginal deformations of the corresponding heterotic nonlinear sigma model with cohomology classes of a worldsheet BRST …

Let M and N be even-dimensional oriented real manifolds, and $u:M \to N$ be a smooth mapping. A pair of complex structures at M and N is called u-compatible if the mapping u is holomorphic with respect to these structures. The quotient of the space of u-compatible pairs of complex structures by the group of u-equivaria…

A bijection proves a polynomial volume for genus-0 hyperbolic surfaces with boundaries.

A closed formula is obtained for the integral $\int_{\mathcal{\bar{H}}_g^1}κ_{1}ψ^{2g-2}$ of tautological classes over the locus of hyperelliptic Weierstraß points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained. The calculation utilizes the homeomorphism between the moduli…

Paper derives a formula for the determinant of Dirichlet-to-Neumann operator on Riemann surfaces.

We generalize our previous results (Theorem 1 and Corollary 2 in arXiv:1412.4114) and Theorem 1 in arXiv:1502.00668) on the existence of an $L^2$-energy gap for Yang-Mills connections over closed four-dimensional manifolds and energies near the ground state (occupied by flat, anti-self-dual, or self-dual connections) t…

We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplect…

Mathematical proof of S-duality and universal isometries in q-map spaces.

Constructs moduli spaces for complex affine and dilation surfaces.

One way to obtain invariants of some Legendrian submanifolds in 1-jet spaces $J^1M$, equipped with the standard contact structure, is through the Morse theoretic technique of generating families. This paper extends the invariant of generating family cohomology by giving it a product $μ_2$. To define the product, moduli…

New Poisson structures found on Higgs bundle moduli spaces.

Study special Lagrangian moduli spaces with boundary.

It is shown that in the semi-classical approximation of the electroweak sector of the Standard Model the moduli space of vacua can be identified with the first de Rham cohomology group of space-time. This gives a slightly different physical interpretation of the occurrence of the well-known Ahoronov-Bohm effect. Moreov…

Constructs projective moduli spaces for Calabi-Yau pairs.

We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on $\mathbb{C}P^1$ and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenki…

Develops moduli theory for Calabi-Yau pairs, constructing a projective space.

Study of Hitchin moduli spaces over Teichmüller space.

The paper constructs moduli spaces for genus one fibered K3 surfaces.

Study compares constrained and decoupled moduli spaces of manifolds with particles and discs.

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson-Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker-Maruyama m…

Analyzes the moduli space of Higgs bundles to prove its quasi-projectivity.

Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.

We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized Kähler structures on maximal flag manifolds under $B$-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decompo…

Researchers compute the heterotic moduli-space metric up to $α'^2$.

Homological stability fails for 4-manifold moduli spaces, detected by new class.

Counting lattice points in moduli space of Klein surfaces.

We shall construct a moduli space of pairs of Kähler-Einstein structures and special lagrangians and obtain smoothness of the moduli space of these pairs. Further we show that the moduli space of these pairs is locally embedded in a certain relative cohomology group.

The paper studies the moduli space of Higgs pairs and their geometric properties.

This note finds explicit representatives for moduli space of parabolic bundles.

The study finds non-trivial elements in moduli spaces of curvature metrics.