arXiv research

MAPS algorithm creates reliable prediction intervals for high-dimensional data.

Construct pseudo-Anosovs from expanding interval maps, reconciling Thurston's construction.

New tree structure for pseudo-Anosovs from interval maps.

Extends Fisher's Discriminant Analysis for interval-valued data.

We study the ergodic properties of compositions of interval exchange transformations and rotations. We show that for any interval exchange transformation T, there is a full measure set of αin [0, 1) so that T composed with R_α is uniquely ergodic, where R_α is rotation by α.

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch which we call non-classical interval exchanges, form a subclass of linear involutions without flips. They are analogs of classical interval exchanges, and are…

Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmueller geodesic flow on the moduli space of …

In [Mas82] and [Vee78] it was proved independently that almost every interval exchange transformation is uniquely ergodic. The Birkhoff ergodic theorem implies that these maps mainly have uniformly distributed orbits. This raises the question under which conditions the orbits yield low-discrepancy sequences. The case o…

We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions non-classical interval exchanges. They are related to measured foliations on orienta…

We prove that a continuum $X$ is tree-like (resp. circle-like, chainable) if and only if for each open cover $\U_4=\{U_1,U_2,U_3,U_4\}$ of $X$ there is a $\U_4$-map $f:X\to Y$ onto a tree (resp. onto the circle, onto the interval). A continuum $X$ is an acyclic curve if and only if for each open cover $\U_3=\{U_1,U_2,U…

Stochastic gradient Langevin dynamics (SGLD) is a computationally efficient sampler for Bayesian posterior inference given a large scale dataset. Although SGLD is designed for unbounded random variables, many practical models incorporate variables with boundaries such as non-negative ones or those in a finite interval.…

We study the J-flow on the toric manifolds, through study the transition map between the moment maps induced by two Kähler metrics, which is a diffeomorphism between polytopes. This is similar to the work of Fang-Lai, under the assumption of Calabi symmetry, they study the monotone map between two intervals. We get a p…

Modernizes Thurston's proof of entropy theorem for traintrack maps.

The paper constructs braiding structures for a specific subfactor.

In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise". We give ev…

Researchers extend regularity of $p$-harmonic maps into spheres for a new range of $p$.

Study Agol cycles in pseudo-Anosov 3-braids.

Local gluing connects flow lines in finite time intervals.

Study of SL(2,R) representations on a once-punctured torus, showing Cantor set spectrum.

VLM judges rank well but score poorly; task difficulty and annotation quality affect interval width.

It is proved that for every fractal continuous mapping F: I\to I^2 of the unit interval onto the unit square there is a pair of points x,y\in I, such that |F(x)-F(y)|^2\ge 5|x-y|.

Study finds weak solutions for complex map flows with optimal lifespan.

In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corol…

The family of translation surfaces $(X_g,ω_g)$ constructed by Arnoux and Yoccoz from self-similar interval exchange maps encompasses one example from each genus $g$ greater than or equal to $3$. We triangulate these surfaces and deduce general properties they share. The surfaces $(X_g,ω_g)$ converge to a surface $(X_\i…

A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($Σ^{1,1}$-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ($Σ^{1,0}$-singularities). Two fold maps are fold bordant if the…

We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions. Specifically, any element of the fundamental group of a surface S gives rise to a mapping class on the punctured surface…

Develops adiabatic theory for ACW flow on surfaces.

New bicombings found for mapping class groups and Teichmüller spaces.

The paper connects Nahm's equations to rational maps between projective spaces.

Let $S$ be a hyperbolic surface. We investigate the topology of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval $(κ_1,κ_2)$. Such a space falls into one of four qualitatively distinct classes, according to whet…

The forcing relation of braids has been introduced for a 2-dimensional analogue of the Sharkovskii order on periods for maps of the interval. In this paper, by making use of the Nielsen fixed point theory and a representation of braid groups, we deduce a trace formula for the computation of the forcing order.

We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal self-maps of intervals. In particular, the Master Teapot is connected, contains the unit cylinder, and its intersection with a set $\mathbb{D} \times \{c\}$ grows monotonically with $c$.…

In modeling multivariate time series, it is important to allow time-varying smoothness in the mean and covariance process. In particular, there may be certain time intervals exhibiting rapid changes and others in which changes are slow. If such time-varying smoothness is not accounted for, one can obtain misleading inf…

CASCADE improves uncertainty communication in Parkinson's disease medication management.

ICP improves prediction intervals for continuous outcomes at lower computational cost.

A map from 3-manifold skein to Lagrangian skein via holomorphic curve counting.

We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are in the same Rauzy class or not, without actually computing them. We prove a simil…

We show that a group of diffeomorphisms $\D$ on the open unit interval $I,$ equipped with the topology of uniform convergence on any compact set of the derivatives at any order, is non regular: the exponential map is not defined for some path of the Lie algebra. this result extends to the group of diffeomorphisms of fi…

We prove that the upper metric mean dimension of $C^0$-generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, coincides with the dimension of the manifold. In the case of continuous interval maps we also show that each level set for the metric mean dimension is $C^0$-d…

Having a regression model, we are interested in finding two-sided intervals that are guaranteed to contain at least a desired proportion of the conditional distribution of the response variable given a specific combination of predictors. We name such intervals predictive intervals. This work presents a new method to fi…

We provide single-model estimates of aleatoric and epistemic uncertainty for deep neural networks. To estimate aleatoric uncertainty, we propose Simultaneous Quantile Regression (SQR), a loss function to learn all the conditional quantiles of a given target variable. These quantiles can be used to compute well-calibrat…

The paper studies curves in Riemannian manifolds using total variation flow.

Prediction intervals are a valuable way of quantifying uncertainty in regression problems. Good prediction intervals should be both correct, containing the actual value between the lower and upper bound at least a target percentage of the time; and tight, having a small mean width of the bounds. Many prior techniques f…

New methods for ordinal classification of interval-valued data and functional data.

Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the se…

Develops active intervals for geodesics in Teichmüller space.

Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the sho…