arXiv subjects

David Glickenstein

Publications and source records attributed to David Glickenstein.

At least 19 recordsLinked to original sources

arXiv2026

Mathematics Graduate Training in the Age of AI

Generative AI changes the conditions under which graduate mathematics is learned, assessed, written, and defended. The central claim of this paper is that mathematics graduate programs should respond to the moment by clarifying what graduate mathematics education is trying to teach and assess. In most ways, the goals of mathematics education have not changed. Rather, with changing tools it has become more essential than ever to make clear the goals of mathematical training. We use the term mathematical judgment to refer to the capacity to evaluate mathematics (e.g., claims, definitions, examples, proofs, analogies, computations, uses of tools, research directions) as mathematically sound, useful, well-posed, and appropriately justified. The recommendation is to center training on mathematical judgment, and we examine possible policies for graduate programs in Mathematics to this end.

math.HO↗

arXiv2025

Convergence of discrete conformal mappings on surfaces

Discrete conformal mappings based on circle packing, vertex scaling, and related structures has had significant activity since Thurston proposed circle packing as a way to approximate conformal maps in the 1980s. The first convergence result of Rodin-Sullivan (1987) proved that circle packing maps do indeed converge to conformal maps to the disk. Recent results have shown convergence of maps of other discrete conformal structures to conformal maps as well. We give a general theorem of convergence of discrete conformal mappings between surfaces that allows for a variety of discrete conformal structures and manifolds with or without boundary. The mappings are a composition of piecewise linear discrete conformal mappings and Riemannian barycentric coordinates, called barycentric discrete conformal maps. Estimates of the barycentric discrete conformal maps allow extraction of convergent subsequences and estimates for the pullback of the Riemannian metric, proving conformality. The theorem requires assumptions on fullness of simplices to prevent degenerate triangles and a local discrete conformal rigidity generalizing hexagonal rigidity of circle packings.

math.DG↗

arXiv2024

Persistent Classification: A New Approach to Stability of Data and Adversarial Examples

There are a number of hypotheses underlying the existence of adversarial examples for classification problems. These include the high-dimensionality of the data, high codimension in the ambient space of the data manifolds of interest, and that the structure of machine learning models may encourage classifiers to develop decision boundaries close to data points. This article proposes a new framework for studying adversarial examples that does not depend directly on the distance to the decision boundary. Similarly to the smoothed classifier literature, we define a (natural or adversarial) data point to be $(γ,σ)$-stable if the probability of the same classification is at least $γ$ for points sampled in a Gaussian neighborhood of the point with a given standard deviation $σ$. We focus on studying the differences between persistence metrics along interpolants of natural and adversarial points. We show that adversarial examples have significantly lower persistence than natural examples for large neural networks in the context of the MNIST and ImageNet datasets. We connect this lack of persistence with decision boundary geometry by measuring angles of interpolants with respect to decision boundaries. Finally, we connect this approach with robustness by developing a manifold alignment gradient metric and demonstrating the increase in robustness that can be achieved when training with the addition of this metric.

cs.LG↗

arXiv2023

An Exact Kernel Equivalence for Finite Classification Models

We explore the equivalence between neural networks and kernel methods by deriving the first exact representation of any finite-size parametric classification model trained with gradient descent as a kernel machine. We compare our exact representation to the well-known Neural Tangent Kernel (NTK) and discuss approximation error relative to the NTK and other non-exact path kernel formulations. We experimentally demonstrate that the kernel can be computed for realistic networks up to machine precision. We use this exact kernel to show that our theoretical contribution can provide useful insights into the predictions made by neural networks, particularly the way in which they generalize.

cs.LG↗

arXiv2021

Determinant of the finite volume Laplacian

The finite volume Laplacian can be defined in all dimensions and is a natural way to approximate the operator on a simplicial mesh. In the most general setting, its definition with orthogonal duals may require that not all volumes are positive; an example is the case corresponding to two-dimensional finite elements on a non-Delaunay triangulation. Nonetheless, in many cases two- and three-dimensional Laplacians can be shown to be negative semidefinite with a kernel consisting of constants. This work generalizes work in two dimensions that gives a geometric description of the Laplacian determinant; in particular, it relates the Laplacian determinant on a simplex in any dimension to certain volume quantities derived from the simplex geometry.

math.DG↗

arXiv2018

Multi-Level Steiner Trees

In the classical Steiner tree problem, given an undirected, connected graph $G=(V,E)$ with non-negative edge costs and a set of \emph{terminals} $T\subseteq V$, the objective is to find a minimum-cost tree $E' \subseteq E$ that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of $ρ= \ln(4)+\varepsilon < 1.39$. In this paper, we study a natural generalization, the \emph{multi-level Steiner tree} (MLST) problem: given a nested sequence of terminals $T_{\ell} \subset \dots \subset T_1 \subseteq V$, compute nested trees $E_{\ell}\subseteq \dots \subseteq E_1\subseteq E$ that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under various names including Multi-level Network Design, Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two simple $O(\ell)$-approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most $2\ell$ Steiner tree computations. We compare these heuristics experimentally on various instances of up to 500 vertices using three different network generation models. We also present various integer linear programming (ILP) formulations for the MLST problem, and compare their running times on these instances. To our knowledge, the composite algorithm achieves the best approximation ratio for up to $\ell=100$ levels, which is sufficient for most applications such as network visualization or designing multi-level infrastructure.

cs.DS↗

arXiv2018

DimReader: Axis lines that explain non-linear projections

Non-linear dimensionality reduction (NDR) methods such as LLE and t-SNE are popular with visualization researchers and experienced data analysts, but present serious problems of interpretation. In this paper, we present DimReader, a technique that recovers readable axes from such techniques. DimReader is based on analyzing infinitesimal perturbations of the dataset with respect to variables of interest. The perturbations define exactly how we want to change each point in the original dataset and we measure the effect that these changes have on the projection. The recovered axes are in direct analogy with the axis lines (grid lines) of traditional scatterplots. We also present methods for discovering perturbations on the input data that change the projection the most. The calculation of the perturbations is efficient and easily integrated into programs written in modern programming languages. We present results of DimReader on a variety of NDR methods and datasets both synthetic and real-life, and show how it can be used to compare different NDR methods. Finally, we discuss limitations of our proposal and situations where further research is needed.

cs.HC↗

arXiv2017

Euclidean formulation of discrete uniformization of the disk

Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to the circle. The main proofs of the uniformization use hyperbolic volumes (Andreev) or hyperbolic circle packings (by Beardon and Stephenson). We reformulate these problems into a Euclidean context, which allows more general discrete conformal structures and boundary conditions. The main idea is to replace the disk with a double covered disk with one side forced to be a circle and the other forced to have interior curvature zero. The entire problem is reduced to finding a zero curvature structure. We also show that these curvatures arise naturally as curvature measures on generalized manifolds (manifolds with multiplicity) that extend the usual discrete Lipschitz-Killing curvatures on surfaces.

math.GT↗

arXiv2017

A bug's eye view: the Riemannian exponential map on polyhedral surfaces

We explore the perspective of a bug living on the two-dimensional surface of a polyhedron. Images of various kinds of effects like lensing and cloaking are shown via color pictures of three viewpoints: the first person perspective of the bug, a map of the bug's viewpoint, and a look at the bug on the embedded polyhedron from a three-dimensional exterior viewer. The pictures were constructed by computing the exponential map of a polyhedron by cutting and rotating faces into the tangent plane of the bug.

math.DG↗

arXiv2017

PVC Polyhedra

We describe how to construct a dodecahedron, tetrahedron, cube, and octahedron out of pvc pipes using standard fittings.

math.HO↗

arXiv2016

Asymptotic behavior of $β$-polygon flows

In this article we investigate a family of nonlinear evolutions of polygons in the plane called the $β$-polygon flow and obtain some results analogous to results for the smooth curve shortening flow: (1) any planar polygon shrinks to a point and (2) a regular polygon with five or more vertices is asymptotically stable in the sense that nearby polygons shrink to points that rescale to a regular polygon. In dimension four we show that the shape of a square is locally stable under perturbations along a hypersurface of all possible perturbations. Furthermore, we are able to show that under a lower bound on angles there exists a rescaled sequence extracted from the evolution that converges to a limiting polygon that is a self-similar solution of the flow. The last result uses a monotonicity formula analogous to Huisken's for the curve shortening flow.

math.DG↗

arXiv2016

Distortion estimates for barycentric coordinates on Riemannian simplices

We define barycentric coordinates on a Riemannian manifold using Karcher's center of mass technique applied to point masses for n+1 sufficiently close points, determining an n-dimensional Riemannian simplex defined as a "Karcher simplex." Specifically, a set of weights is mapped to the Riemannian center of mass for the corresponding point measures on the manifold with the given weights. If the points lie sufficiently close and in general position, this map is smooth and injective, giving a coordinate chart. We are then able to compute first and second derivative estimates of the coordinate chart. These estimates allow us to compare the Riemannian metric with the Euclidean metric induced on a simplex with edge lengths determined by the distances between the points. We show that these metrics differ by an error that shrinks quadratically with the maximum edge length. With such estimates, one can deduce convergence results for finite element approximations of problems on Riemannian manifolds.

math.NA↗

arXiv2015

Soliton metrics for two-loop renormalization group flow on 3D unimodular Lie groups

The two-loop renormalization group flow is studied via the induced bracket flow on 3D unimodular Lie groups. A number of steady solitons are found. Some of these steady solitons come from maximally symmetric metrics that are steady, shrinking, or expanding solitons under Ricci flow, while others are not obviously related to Ricci flow solitons.

math.DG↗

arXiv2014

Duality structures and discrete conformal variations of piecewise constant curvature surfaces

A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Additional structure is defined either by giving a geometric structure to the Poincare dual of the triangulation or by assigning a discrete metric, a way of assigning length to oriented edges. This notion leads to a notion of discrete conformal structure, generalizing the discrete conformal structures based on circle packings and their generalizations studied by Thurston and others. We define and analyze conformal variations of piecewise constant curvature 2-manifolds, giving particular attention to the variation of angles. We give formulas for the derivatives of angles in each background geometry, which yield formulas for the derivatives of curvatures. Our formulas allow us to identify particular curvature functionals associated with conformal variations. Finally, we provide a complete classification of discrete conformal structures in each of the background geometries.

math.GT↗

arXiv2010

Regge's Einstein-Hilbert Functional on the Double Tetrahedron

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Regge's Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein-Hilbert-Regge functional on the space of metrics and on discrete conformal classes of metrics.

math.DG↗

arXiv2009

Ricci flow on three-dimensional, unimodular metric Lie algebras

We give a global picture of the Ricci flow on the space of three-dimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a two-dimensional dynamical system for the evolution of structure constants of the metric Lie algebra with respect to an evolving orthonormal frame. This system is amenable to direct phase plane analysis, and we find that the fixed points and special trajectories in the phase plane correspond to special metric Lie algebras, including Ricci solitons and special Riemannian submersions. These results are one way to unify the study of Ricci flow on left invariant metrics on three-dimensional, simply-connected, unimodular Lie groups, which had previously been studied by a case-by-case analysis of the different Bianchi classes. In an appendix, we prove a characterization of the space of three-dimensional, unimodular, nonabelian metric Lie algebras modulo isometry and scaling.

math.DG↗

arXiv2009

Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds

A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This variation generalizes variations within the class of circles with fixed intersection angles (such as circle packings) as well as other formulations of conformal variation of piecewise flat manifolds previously suggested. We describe the angle derivatives of the angles in two and three dimensional piecewise flat manifolds, giving rise to formulas for the derivatives of curvatures. The formulas for derivatives of curvature resemble the formulas for the change of scalar curvature under a conformal variation of Riemannian metric. They allow us to explicitly describe the variation of certain curvature functionals, including Regge's formulation of the Einstein-Hilbert functional (total scalar curvature), and to consider convexity of these functionals. They also allow us to prove rigidity theorems for certain analogues of constant curvature and Einstein manifolds in the piecewise flat setting.

math.DG↗

arXiv2007

Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross curvature flows

In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the limit. We also review Lott's results on the long term behavior of three-dimensional homogeneous solutions of Ricci flow, demonstrating the coordinates we choose and reviewing the groupoid technique. We find cross curvature soliton metrics on Sol and Nil, and show that the cross curvature flow of SL(2,R) limits to Sol.

math.DG↗