arXiv research

Two metrics on graph spaces compared to Weil-Petersson.

Paper studies metric ribbon graphs and provides a recursion for their volumes.

Incomplete pressure metric on bordered surface Teichmüller space proved.

Counting lattice points in moduli space of Klein surfaces.

We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the ou…

Holomorphic analogs of Feynman integrals are shown to be finite.

Critical graphs of quadratic differentials equidistribute in moduli space.

Classifies geometric graph manifolds with non-negative scalar curvature.

Let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$. It acts properly on the outer space $X_n$ of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces missing. In this paper, we construct complete geodesic metrics and complete piecewise s…

Constructs a combinatorial model for bordered Riemann surfaces with a compactification.

Study detects non-trivial elements in diffeomorphism groups via trivalent graphs.

Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.

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…

Random covers of hyperbolic surfaces follow a specific probability measure.

We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which $\sqcup_{g} M_{g}$ is open dense. The metrized graphs, which are oft…

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 spa…

Mathematical analysis of Riemann surfaces and their moduli spaces using hybrid Laplacians.

We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic…

Moduli space linked to Tait colorings of planar graphs.

A closed hyperbolic surface of genus $g\ge 2$ can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. The…

A metric defined on body moduli space.

We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul an…

Paper introduces hybrid curves moduli space and canonical measures.

The paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.

Study neural architectures on learned latent graphs using Schrödinger dynamics.

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

Exponential decay observed between two metrics on a moduli space.

Study on moduli spaces of non-negative curvature metrics on manifolds.

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

Study describes flat metric moduli spaces on 4D manifolds.

In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in math.AG/0402015, and used in math.QA/0412149 to construct open-closed topological conformal field theories. This dual version of the r…

New space for polarized manifolds with constant curvature metrics.

Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof …

Quantizes vortex moduli space using modified Quillen metric.

Study of flat metrics on orbifolds and their moduli spaces.

Study shows non-trivial higher homotopy groups in moduli space of metrics.

Study on moduli space of bi-invariant metrics in Lie groups.

The paper finds maximal metrics on Euclidean spaces.

Simply connected moduli space of Ricci flat metrics on K3 surfaces.

Study distinguishes components of moduli spaces for certain metrics.

Study classifies singular foliations on complex plane, finding finite moduli space.

Lie group structure on automorphisms of Courant algebroids, moduli space of metrics.

We construct the moduli space of r-jets at a point of Riemannian metrics on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via differential invariants. The moduli space is proved to be a differentiable space which admits a finite canonical stratification into smoo…

Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.

New proof of Weil-Petersson metric for Calabi-Yau varieties.

Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.

We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities of angles less than $2π$ along a time-like graph $Γ$. To each such space we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient …

Study shows infinitely many metrics with nonnegative sectional or positive Ricci curvature on specific 5D quotients.