Aaron Abrams

Washington and Lee University | W&L · Department of Mathematics

We give a simple and complete description of those convex lattice polygons in the plane that can be dissected into lattice triangles of integer area. A new version of Sperner's Lemma plays a central role.

Given an n × n n\times n matrix with integer entries in the range [ − h , h ] [-h,h] , how close can two of its distinct eigenvalues be? The best previously known examples (Lu [Minimum eigenvalue separation, ProQuest LLC, Ann Arbor, MI, 1992. Thesis (Ph.D.)–University of California, Berkeley; Wilkinson [The algebraic eigenvalue problem, Monographs...

To any combinatorial triangulation T of a square, there is an associated polynomial relation pT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_T$$\end{document} among...

Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into triangles, the area of any one of the triangles of the dissection is integral over the ring generated by the are...

Given an $n\times n$ matrix with integer entries in the range $[-h,h]$, how close can two of its distinct eigenvalues be? The best previously known examples have a minimum gap of $h^{-O(n)}$. Here we give an explicit construction of matrices with entries in $[0,h]$ with two eigenvalues separated by at most $h^{-n^2/16+o(n^2)}$. Up to a constant in...

Monsky’s celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation f among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial p, also a relation among the areas of the triangles in such a dissection, that is invariant under ce...

To any combinatorial triangulation $T$ of a square, there is an associated polynomial relation $p_T$ among the areas of the triangles of $T$. With the goal of understanding this polynomial, we consider polynomials obtained from $p_T$ by choosing $l$ of its variables and specializing $p_T$ to these variables by zeroing out the remaining variables. W...

Monsky's celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $f$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial $p$, also a relation among the areas of the triangles in such a dissection, that is invariant unde...

Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sizes of sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For...

The rate of convergence of simple random walk on the Heisenberg group over $Z/nZ$ with a standard generating set was determined by Bump et al [1,2]. We extend this result to random walks on the same groups with an arbitrary minimal symmetric generating set. We also determine the rate of convergence of simple random walk on higher-dimensional versio...

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the closed 4-manifold, with homomorph...

We study the map from conductances to edge energies for harmonic functions on graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of conductances such that the associated harmonic function realizes those orientations and energies. We call the associated func...

Given a combinatorial triangulation of an n-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for the areas of the triangles in such drawings. We define a generalized notion of triangulation, and we show that...

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, whereas the homotopical Dehn function measures fillings of curves by disks. Because the two definitions involve different sorts of bou...

Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of Diaconis for the convergence to the stationary distribution in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phen...

This note gives a central limit theorem for the length of the longest subsequence of a random permutation which follows some repeating pattern. This includes the case of any fixed pattern of ups and downs which has at least one of each, such as the alternating case considered by Stanley in [2] and Widom in [3]. In every case considered the converge...

We study the Dehn function at infinity in the mapping class group, finding a polynomial upper bound of degree four. This is the same upper bound that holds for arbitrary right-angled Artin groups.

We consider a problem in parametric estimation: given samples from an unknown distribution, we want to estimate which distribution, from a given one-parameter family, produced the data. Following Schulman and Vazirani (2005), we evaluate an estimator in terms of the chance of being within a specified tolerance of the correct answer, in the worst ca...

We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $\{1,...,n+1\}$ with exactly $k$ parts. We compute the exponential generating function for the Euler charac...

We construct "pushing maps" on the cube complexes that model right-angled Artin groups (RAAGs) in order to study filling problems in certain subsets of these cube complexes. We use radial pushing to obtain upper bounds on higher divergence functions, finding that the k-dimensional divergence of a RAAG is bounded by r^{2k+2}. These divergence functi...

A permutation σ describing the relative orders of the first n iterates of a point x under a self-map f of the interval I=[0,1] is called an order pattern. For fixed f and n, measuring the points x ∈ I (according to Lebesgue measure) that generate the order pattern σ gives a probability distribution μ n (f) on the set of length n permutations. We st...

Everyone knows that buying lottery tickets is a bad investment. But do you know why? You will most likely lose your dollar, but there is a small chance that you will win big. In some lottery drawings, your expected rate of return is spectacularly high. Do you know how to identify a drawing that has a positive expected rate of return? When you find...

Let G be a regular graph and H a subgraph on the same vertex set. We give surprisingly compact formulas for the number of copies of H one expects to find in a random subgraph of G.

In order to study how well a finite group might be generated by repeated random multiplications, P. Diaconis suggested the following urn model. An urn contains some balls labeled by elements which generate a group G. Two are drawn at random with replacement and a ball labeled with the group product (in the order they were picked) is added to the ur...

A metamorphic robotic system is an aggregate of homogeneous,robot units which can individually and selectively locomote in such a way as to change the global shape of the system. We introduce a mathematical framework for deflning and analyzing general metamorphic,robots. With this formal structure, combined with ideas from geometric group theory, w...

J Hempel [Topology, 2001] showed that the set of distances of the Heegaard splittings (S,V, h^n(V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V in PML(S). Here h is a pseudo-Anosov homeomorphism of a surface S while V is the set of isotopy classes of simple closed curves in S bounding essential disks in...

Consider the set of functions fθ(x) = |θ - x| on R. Define a, Markov process that starts with a point x0 ∈ R. and continues with Xk+1 = fθk+1 (xk) with each θk+1 chosen from a fixed bounded distribution μ on R+. We prove the conjecture of Letac that if μ is not supported on a lattice, then this process has a unique stationary distribution πμ and an...

This paper is intended to provide concrete examples of concepts discussed elsewhere in this volume, especially splittings of groups and nonpositively curved cube complexes but also other things. The idea of the construction (configuration spaces) is not new, but this family of examples does not seem to be well known. Nevertheless they arise in a va...

A pair of random walks (R, S) on the vertices of a graph G is successful if two tokens moving one at a time can be scheduled (moving only one token at a time) to travel along R and S without colliding. We consider questions related to P. Winkler's clairvoyant demon problem, which asks whether for random walks R and S on G, Pr[(R, S)is successful] >...

We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The pro...

In searching the world for examples of interesting topological objects, it may not be obvious that an outstanding place to look is within the walls of an automated warehouse or factory. In this exposition we describe a class of topological spaces that arise naturally in this very context. The examples we construct actually arose simultaneously in t...

Let S subset of or equal to R-n, and let k is an element of N. Greenwell and Johnson [3] define <(chi)over cap>((k))(S) to be the smallest integer m (if such an integer exists) such that for every k x m array D = (d(ij)) of positive real numbers, S can be colored with the colors C-1,...,C-m such that no two points of S which are a (Euclidean) dista...

Given a family of probability distributions, and a small number of observed samples, we consider the problem of estimating which distribution produced the samples. The goal is to maximize the chance of being within a specied tolerance of the correct distribution, in the worst case.

Thesis (Ph. D. in Mathematics)--University of California, Berkeley, Spring 2000. Includes bibliographical references (leaves 63-67).