arXiv research

Develops a method to make predictions more informative without sacrificing accuracy.

Let $G$ be a Lie group, with an invariant non-degenerate symmetric bilinear form on its Lie algebra, let $π$ be the fundamental group of an orientable (real) surface $M$ with a finite number of punctures, and let $\bold C$ be a family of conjugacy classes in $G$, one for each puncture. A finite-dimensional construction…

Given a compact orientable surface $Σ$, let $\Cal S(Σ)$ be the set of isotopy classes of essential simple loops on $Σ$. We determine a complete set of relations for a function from $\Cal S(Σ)$ to $\bold Z$ to be a geometric intersection number function. As a consequence, we obtain explicit equations in $\bold R^{\Cal S…

Given a compact orientable surface with finitely many punctures $Σ$, let $\Cal S(Σ)$ be the set of isotopy classes of essential unoriented simple closed curves in $Σ$. We determine a complete set of relations for a function from $\Cal S(Σ)$ to $\bold R$ to be the geodesic length function of a hyperbolic metric with geo…

New homology theory connects graph domination to subtle algebraic structures.

ST-GCN improves rs-fMRI prediction accuracy by modeling spatio-temporal graph connectivity.

A simplified model for brain activity measurement.

The Margulis constant for Kleinian groups is the smallest constant $c$ such that for each discrete group $G$ and each point $x$ in the upper half space ${\bold H}^3$, the group generated by the elements in $G$ which move $x$ less than distance c is elementary. We take a first step towards determining this constant by p…

Background: Functional magnetic resonance imaging (fMRI) provides non-invasive measures of neuronal activity using an endogenous Blood Oxygenation-Level Dependent (BOLD) contrast. This article introduces a nonlinear dimensionality reduction (Locally Linear Embedding) to extract informative measures of the underlying ne…

Functional connectivity refers to the temporal statistical relationship between spatially distinct brain regions and is usually inferred from the time series coherence/correlation in brain activity between regions of interest. In human functional brain networks, the network structure is often inferred from functional m…

This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichmüller theory, the mapping class groups, $SL(2, \bold C)$ representation variety of surface groups, and Thurston's theory o…

An end sum is a non-compact analogue of a connected sum. Suppose we are given two connected, oriented $n$-manifolds $M_1$ and $M_2$. Recall that to form their connected sum one chooses an $n$-ball in each $M_i$, removes its interior, and then glues together the two $S^{n-1}$ boundary components thus created by an orien…

A physically natural potential energy for simple closed curves in $\bold R^3$ is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot oc…

We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincaré characteristic zero) in ${\bold R}^3$ of constant mean curvature which meet planes $Π_1$ and $Π_2$ in constant contact angles $γ_1$ and $γ_2$ and bound, together with those planes, a…

Geometrization Theorem solves complex geometry problems.

Understanding human fetal neurodevelopment is of great clinical importance as abnormal development is linked to adverse neuropsychiatric outcomes after birth. Recent advances in functional Magnetic Resonance Imaging (fMRI) have provided new insight into development of the human brain before birth, but these studies hav…

Given any connected, open 3-manifold $U$ having finitely many ends, a non-compact 3-manifold $M$ is constructed having the following properties: the interior of $M$ is homeomorphic to $U$; the boundary of $M$ is the disjoint union of finitely many planes; $M$ is not almost compact; $M$ is eventually end-irreducible; th…

New interpretation of Mayer-Vietoris sequence using überhomology.

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of $PSL(2,{\bold C})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to dete…

We survey what is known about minimal surfaces in $\bold R^3$ that are complete, embedded, and have finite total curvature. The only classically known examples of such surfaces were the plane and the catenoid. The discovery by Costa, early in the last decade, of a new example that proved to be embedded sparked a great…

Brain networks in fMRI are typically identified using spatial independent component analysis (ICA), yet mathematical constraints such as sparse coding and positivity both provide alternate biologically-plausible frameworks for generating brain networks. Non-negative Matrix Factorization (NMF) would suppress negative BO…

New method transforms complex stochastic equations into simpler ones for efficient simulation.

A new deep learning method using Boolean logic reduces training and inference energy.

Optimizes financial decisions with illiquid assets using Kelly criterion.

The paper characterizes brain states and transitions using functional MRI data.

The paper proves the behavior of the second fundamental form for Kaehler submanifolds in Euclidean space.

Estimation of reliable whole-brain connectivity is a crucial step towards the use of connectivity information in quantitative approaches to the study of neuropsychiatric disorders. When estimating brain connectivity a challenge is imposed by the paucity of time samples and the large dimensionality of the measurements. …

Given a small corpus $\mathcal D_T$ pertaining to a limited set of focused topics, our goal is to train embeddings that accurately capture the sense of words in the topic in spite of the limited size of $\mathcal D_T$. These embeddings may be used in various tasks involving $\mathcal D_T$. A popular strategy in limited…

Kolmogorov-Arnold Networks promise scalable performance in high dimensions.

The inertia subgroup $I_n(π)$ of a surgery obstruction group $L_n(π)$ is generated by elements which act trivially on the set of homotopy triangulations $\Cal S(X)$ for some closed topological manifold $X^{n-1}$ with $π_1(X)=π$. This group is a subgroup of the group $C_n(π)$ which consists of the elements which can be …

Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic acc…

The IB theory explains how ML systems reduce data dimensions while preserving predictive power.

We consider the following signal recovery problem: given a measurement matrix $Φ\in \mathbb{R}^{n\times p}$ and a noisy observation vector $c\in \mathbb{R}^{n}$ constructed from $c = Φθ^* + ε$ where $ε\in \mathbb{R}^{n}$ is the noise vector whose entries follow i.i.d. centered sub-Gaussian distribution, how to recover …