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Alden Walker

Publications and source records attributed to Alden Walker.

At least 19 recordsLinked to original sources
arXiv2026

Laminations and External Angles for Similarity Pairs

A {\em similarity pair} is the dynamical system in $\mathbb{C}$ generated by two maps $f:z \to sz-1$ and $g:z \to sz+1$ for $|s|<1$. Associated to the dynamical system is an attractor $\Lambda$. The Barnsley--Harrington Mandelbrot set $\mathcal{M}$ is the set of $s\in \mathbb{D}$ for which $\Lambda$ is connected. Let $K$ denote the filled set of a connected $\Lambda$. For $s\in \partial \mathcal{M}$ we show that the (partially defined) action of the semigroup on $\partial K$ is topologically conjugate to a (discontinuous) piecewise linear action of constant slope. Conditional on a conjecture (satisfied for `most' $s\in \partial \mathcal{M}$) we give a necessary and sufficient condition in terms of the dynamics on $\partial K$ for $K$ to contain cut points, and we describe the set of all such cut points in terms of infinite walks in a directed graph $\textrm{IG}$ obtained by an explicit recursive algorithm. The structure of the `dynamical cut point set' for a 2-dimensional family of piecewise linear actions (containing those coming from $s\in \partial \mathcal{M}$) recovers and generalizes the Douady--Hubbard--Thurston quadratic minor lamination for the abstract Mandelbrot set.

math.DS↗
arXiv2024

Enabling full-speed random access to the entire memory on the A100 GPU

We describe some features of the A100 memory architecture. In particular, we give a technique to reverse-engineer some hardware layout information. Using this information, we show how to avoid TLB issues to obtain full-speed random HBM access to the entire memory, as long as we constrain any particular thread to a reduced access window of less than 64GB.

cs.PF↗
arXiv2020

Conjugation Curvature in Solvable Baumslag-Solitar Groups

For an element in $BS(1,n) = \langle t,a | tat^{-1} = a^n \rangle$ written in the normal form $t^{-u}a^vt^w$ with $u,w \geq 0$ and $v \in \mathbb{Z}$, we exhibit a geodesic word representing the element and give a formula for its word length with respect to the generating set $\{t,a\}$. Using this word length formula, we prove that there are sets of elements of positive density of positive, negative and zero conjugation curvature, as defined by Bar Natan, Duchin and Kropholler.

math.GR↗
arXiv2020

A new proof of the growth rate of the solvable Baumslag-Solitar groups

We exhibit a regular language of geodesics for a large set of elements of $BS(1,n)$ and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of $BS(1,n)$, which was initially computed by Collins, Edjvet and Gill in [5]. Our methods are based on those we develop in [8] to show that $BS(1,n)$ has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan, Duchin and Kropholler in [1].

math.GR↗
arXiv2018

Extreme points in limit sets

Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.

math.DS↗
arXiv2018

The Gromov boundary of the ray graph

The ray graph is a Gromov hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor set. As a consequence, we prove that the Gromov boundary of the ray graph is homeomorphic to a quotient of a subset of the circle. This version contains some updates and corrections.

math.GT↗
arXiv2018

Two simultaneous actions of big mapping class groups

We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of the paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the circle. The third part focuses on some fruitful relations between the dynamics of the two actions. For example, we prove that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). In addition, we are able to construct non trivial quasimorphisms on many subgroups of big mapping class groups, even if they are not acylindrically hyperbolic.

math.GT↗
arXiv2017

Estimating the Operating Characteristics of Ensemble Methods

In this paper we present a technique for using the bootstrap to estimate the operating characteristics and their variability for certain types of ensemble methods. Bootstrapping a model can require a huge amount of work if the training data set is large. Fortunately in many cases the technique lets us determine the effect of infinite resampling without actually refitting a single model. We apply the technique to the study of meta-parameter selection for random forests. We demonstrate that alternatives to bootstrap aggregation and to considering \sqrt{d} features to split each node, where d is the number of features, can produce improvements in predictive accuracy.

stat.ML↗
arXiv2017

The energy landscape of a simple neural network

We explore the energy landscape of a simple neural network. In particular, we expand upon previous work demonstrating that the empirical complexity of fitted neural networks is vastly less than a naive parameter count would suggest and that this implicit regularization is actually beneficial for generalization from fitted models.

stat.ML↗
arXiv2016

The empirical size of trained neural networks

ReLU neural networks define piecewise linear functions of their inputs. However, initializing and training a neural network is very different from fitting a linear spline. In this paper, we expand empirically upon previous theoretical work to demonstrate features of trained neural networks. Standard network initialization and training produce networks vastly simpler than a naive parameter count would suggest and can impart odd features to the trained network. However, we also show the forced simplicity is beneficial and, indeed, critical for the wide success of these networks.

stat.ML↗
arXiv2014

Roots, Schottky semigroups, and a proof of Bandt's Conjecture

In 1985, Barnsley and Harrington defined a ``Mandelbrot Set'' $\mathcal{M}$ for pairs of similarities --- this is the set of complex numbers $z$ with $0<|z|<1$ for which the limit set of the semigroup generated by the similarities $x \mapsto zx$ and $x \mapsto z(x-1)+1$ is connected. Equivalently, $\mathcal{M}$ is the closure of the set of roots of polynomials with coefficients in $\lbrace -1,0,1 \rbrace$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ``holes'' in $\mathcal{M}$, and conjectured that these holes were genuine. These holes are very interesting, since they are ``exotic'' components of the space of (2 generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique of traps to construct and certify interior points of $\mathcal{M}$, and use them to prove Bandt's Conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in $\mathcal{M}$.

math.DS↗
arXiv2014

Stable immersions in orbifolds

We prove that in any hyperbolic orbifold with one boundary component, the product of any hyperbolic fundamental group element with a sufficiently large multiple of the boundary is represented by a geodesic loop that virtually bounds an immersed surface. In the case that the orbifold is a disk, there are some conditions. Our results generalize work of Calegari-Louwsma and resolve a conjecture of Calegari.

math.GT↗
arXiv2014

Surface subgroups from linear programming

We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions of free groups. A special case is the HNN extension associated to the endomorphism of a rank 2 free group sending a to ab and b to ba; this example (and the random examples) answer in the negative well-known questions of Sapir. We further show that the unit ball in the Gromov norm (in dimension 2) of a double of a free group along a collection of subgroups is a finite-sided rational polyhedron, and that every rational class is virtually represented by an extremal surface subgroup. These results are obtained by a mixture of combinatorial, geometric, and linear programming techniques.

math.GR↗
arXiv2013

Stable commutator length in free products of cyclic groups

We give an algorithm to compute stable commutator length in free products of cyclic groups which is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.

math.GT↗
arXiv2013

Random rigidity in the free group

We prove a rigidity theorem for the geometry of the unit ball in random subspaces of the scl norm in B_1^H of a free group. In a free group F of rank k, a random word w of length n (conditioned to lie in [F,F]) has scl(w)=log(2k-1)n/6log(n) + o(n/log(n)) with high probability, and the unit ball in a subspace spanned by d random words of length O(n) is C^0 close to a (suitably affinely scaled) octahedron. A conjectural generalization to hyperbolic groups and manifolds (discussed in the appendix) would show that the length of a random geodesic in a hyperbolic manifold can be recovered from the bounded cohomology of the fundamental group.

math.GR↗
arXiv2011

Integer hulls of linear polyhedra and scl in families

The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is a ratio of quasipolynomials, and that unit balls in the scl norm quasi-converge in finite dimensional surgery families.

math.CO↗
arXiv2011

Ziggurats and rotation numbers

We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle, and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's theorem (i.e. the conjecture of Jankins-Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces.

math.DS↗