Research

Given a positive rational number n/d with d odd, its odd greedy expansion starts with the unit fraction with smallest odd denominator that is at most n/d, adds the unit fraction with smallest odd denominator so the sum is at most n/d, and continues as long as the sum is less than n/d. It is an open question whether this expansion always has finitely many terms. Given m-1 odd positive integers, we find all rational numbers whose odd greedy expansion has length m and begins with these numbers as denominators. Given m-2 compatible odd positive integers, we find a two-parameter family of rational numbers whose odd greedy expansion has length m and begins with these numbers as denominators. 

Given a positive rational number n/d with d odd, its odd greedy expansion starts with the largest odd denominator unit fraction at most n/d, adds the largest odd denominator unit fraction so the sum is at most n/d, and continues as long as the sum is less than n/d. Given a fixed positive integer n, we find all reduced fractions with numerator n whose odd greedy expansion has length 2. 

We describe the Macaulay2 package A1BrouwerDegrees for computing local and global A^1-Brouwer degrees and studying symmetric bilinear forms over the complex numbers, the real numbers, the rational numbers, and finite fields of characteristic not equal to 2.

We give an algorithm that produces all solutions of the equation \sum_{i=1}^n 1/x_i = 1 in integers of the form 2^a k^b, where k is a fixed positive integer that is not a power of 2, a is an element of {0,1,2\ that can vary from term to term, and b is a nonnegative integer that can vary from term to term. We also completely characterize the pairs (k,n) for which this equation has a nontrivial solution in integers of this form.

An arithmetical structure on a finite, connected graph without loops is an assignment of positive integers to the vertices that satisfies certain conditions. Associated to each of these is a finite abelian group known as its critical group. We show how to determine the critical group of an arithmetical structure on a star graph or complete graph in terms of the entries of the arithmetical structure. We use this to investigate which finite abelian groups can occur as critical groups of arithmetical structures on these graphs. 

An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group. Keyes and Reiter gave an operation that takes in an arithmetical structure on a finite, connected graph without loops and produces an arithmetical structure on a graph with one fewer vertex. We study how this operation transforms critical groups. We bound the order and the invariant factors of the resulting critical group in terms of the original arithmetical structure and critical group. When the original graph is simple, we determine the resulting critical group exactly. 

An arithmetical structure on a finite, connected graph G is a pair of vectors (d, r) with positive integer entries for which (diag(d)-A)=0, where A is the adjacency matrix of G and where the entries of r have no common factor. The critical group of an arithmetical structure is the torsion part of the cokernel of (diag(d)-A). In this paper, we study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two "prongs" at one end. We give a process for determining the number of arithmetical structures on the bident with n vertices and show that this number grows at the same rate as the Catalan numbers as n increases. We also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.

An arithmetical structure on the complete graph Kn  with n vertices is given by a collection of n positive integers with no common factor each of which divides their sum. We show that, for all positive integers c less than a certain bound depending on n, there is an arithmetical structure on Kn  with largest value c. We also show that, if each prime factor of c is greater than (n+1)^2/4, there is no arithmetical structure on Kn  with largest value c. We apply these results to study which prime numbers can occur as the largest value of an arithmetical structure on Kn .

A. Rosenfeld introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there are often approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP).

This paper has two parts, on Baumslag-Solitar groups and on general G-trees.

In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces.

In the second part we establish a universal lower bound of 1/12 for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group BS(2,3) show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions.

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface.

Krakowski and Regev found a basis of polynomial identities satisfied by the Grassmann algebra over a field of characteristic 0 and described the exact structure of these relations in terms of the symmetric group. Using this, they found an upper bound for the the codimension sequence of the T-ideal of polynomial identities of the Grassmann algebra. Working with certain matrices, they found the same lower bound, thus determining the codimension sequence exactly. In this paper, we compute the codimension sequence of the Grassmann algebra directly from these matrices, thus obtaining a proof of the codimension result of Krakowski and Regev using only combinatorics and linear algebra. We also obtain a corollary from our proof.

For any element A of the modular group PSL(2,Z), it follows from work of Bavard that scl(A) is greater than or equal to rot(A)/2, where scl denotes stable commutator length and rot denotes the rotation quasimorphism. Sometimes this bound is sharp, and sometimes it is not. We study for which elements A in PSL(2,Z) the rotation quasimorphism is extremal in the sense that scl(A) = rot(A)/2. First, we explain how to compute stable commutator length in the modular group, which allows us to experimentally determine whether the rotation quasimorphism is extremal for a given A. Then we describe some experimental results based on data from these computations.

Our main theorem is the following: for any element of the modular group, the product of this element with a sufficiently large power of a parabolic element is an element for which the rotation quasimorphism is extremal. We prove this theorem using a geometric approach. It follows from work of Calegari that the rotation quasimorphism is extremal for a hyperbolic element of the modular group if and only if the corresponding geodesic on the modular surface virtually bounds an immersed surface. We explicitly construct immersed orbifolds in the modular surface, thereby verifying this geometric condition for appropriate geodesics. Our results generalize to the 3-strand braid group and to arbitrary Hecke triangle groups.