arXiv research

The paper studies complex affine structures near irregular singularities.

Proves heat expansion for Laplacian on a singularity.

Given a closed oriented PL four-manifold $X$ and a closed surface $B$ embedded in $X$ with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of $X$ branched along $B$. For $X$ simply-connected, we deduce a necessary condition on the intersection form of a simply-connected i…

Study infinite-dimensional Toda manifold at irregular singularity, revealing non-uniqueness of formal solutions.

We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular…

Paper connects Painlevé VI equation to irregular systems, solving monodromy data.

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne-Simpson…

T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irr…

Study local wild mapping class groups for irregular connections on complex curves.

We study triangulations $\cal T$ defined on a closed disc $X$ satisfying the following condition: In the interior of $X$, the valence of all vertices of $\cal T$ except one of them (the irregular vertex) is $6$. By using a flat singular Riemannian metric adapted to $\cal T$, we prove a uniqueness theorem when the valen…

With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an ess…

Consider a dihedral cover $f: Y\to X$ with $X$ and $Y$ four-manifolds and $f$ branched along an oriented surface embedded in $X$ with isolated cone singularities. We prove that only a slice knot can arise as the unique singularity on an irregular dihedral cover $f: Y\to S^4$ if $Y$ is homotopy equivalent to $\mathbb{CP…

We give a topological model for a polynomial map from $\C^n$ to $\C$ in the neighborhood of a fiber with isolated singularities. This is motivated out of the ``unfolding of links'' described earlier by the first author and Lee Rudolph. The topological model gives a useful encoding of the local and global monodromy for …

Study of symplectic groupoids from tt*-Toda equations.

In previous work, the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type. In this paper, the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point …

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the connection. We prove that…

Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus. Later, the first author applies this to the particular case of coverings of $\mat…

Paper proves short-time existence for network flow, providing detailed insights.

We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of vari…

Study on the asymptotic geometry of Higgs bundles over projective line.

New Calabi-Yau metrics found on complex symmetric spaces.

Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.

This paper deals with moduli spaces of framed principal bundles with connections with irregular singularities over a compact Riemann surface. These spaces have been constructed by Boalch by means of an infinite-dimensional symplectic reduction. It is proved that the symplectic structure induced from the Atiyah--Bott fo…

This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.

Solves Deligne-Simpson problem for special connections on Gm.

In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equ…

The Ooguri-Vafa space is a 4-dimensional incomplete hyperkähler manifold, defined on the total space of a singular torus fibration with one singular nodal fiber. It has been proposed that the Ooguri-Vafa hyperkähler metric should be part of the local model of the hyperkähler metric of the Hitchin moduli spaces, near th…

Study on flat connections with controlled irregularity.

Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.

We demonstrate how the complex integral formula for the Airy functions arises from Penrose's twistor contour integral formula. We then use the Lax formulation of the isomonodromy problem with one irregular singularity of order four to show that the Airy equation arises from the anti-self-duality equations for conformal…

We show that any 4-manifold admitting a $(g;k_1,k_2,0)$-trisection is an irregular 3-fold cover of the 4-sphere whose branching set is a surface in $S^4$, smoothly embedded except for one singular point which is the cone on a link. A 4-manifold admits such a trisection if and only if it has a handle decomposition with …

In this note, we investigate pluri-half-anticanonical systems on the so called LeBrun twistor spaces. We determine its dimension, the base locus, structure of the associated rational map, and also structure of general members, in precise form. In particular, we show that if n>2 and m>1, the base locus of the system |mK…

To detect the irregular trade behaviors in the stock market is the important problem in machine learning field. These irregular trade behaviors are obviously illegal. To detect these irregular trade behaviors in the stock market, data scientists normally employ the supervised learning techniques. In this paper, we empl…

EDICT learns evidential distributions for irregular time series, improving predictions and uncertainty quantification.

New algorithm for Coxeter connections with maximally ramified singularities.

We show that there are no irregular Sasaki-Einstein structures on rational homology 5-spheres. On the other hand, using K-stability we prove the existence of continuous families of non-toric irregular Sasaki-Einstein structures on odd connected sums of $S^2 \times S^3$.

Study deformation spaces of irregular isomonodromy systems on Riemann surfaces.

We construct a new five parameter family of constant mean curvature trinoids with two asymptotically Delaunay ends and one irregular end.

LLapDiff models irregular multivariate time series without step-by-step integration.

CRUs model irregular time series with continuous hidden states.

For a complex polynomial in two variables we study the morphism induced in homology by the embedding of an irregular fiber in a regular neighborhood of it. We give necessary and sufficient conditions for this morphism to be injective, surjective. Particularly this morphism is an isomorphism if and only if the correspon…

The book is devoted to study so-called irregular subsets of the Grassmannian manifold $G^{n}_{k}(V)$ (this class of sets was introduced by author). In the previous variant of the book we restrict ourself only to the case when $V$ is an $n$-dimensional vector space under the field $R$. Now we consider irregular subsets …

ACSSM models irregular time series with continuous dynamics.

Neural Laplace Control tackles offline RL for continuous-time delayed systems with irregular observations.

Reconstructs fundamental groups from liquid local systems.

We start by a review of the chronology of mathematical results on the Dirichlet-to-Neumann map which paved the way towards the physics of transformational acoustics. We then rederive the expression for the (anisotropic) density and bulk modulus appearing in the pressure wave equation written in the transformed coordina…

Machine learning methods such as convolutional neural networks (CNNs) are becoming an integral part of scientific research in many disciplines, spatial vector data often fail to be analyzed using these powerful learning methods because of its irregularities. With the aid of graph Fourier transform and convolution theor…

daep learns from irregular, multimodal astronomical data.