arXiv subjects

Charles Daly

Publications and source records attributed to Charles Daly.

7 recordsLinked to original sources

arXiv2026

Symplectic Tiling Billiards on Complete Affine Tori

In 2023, Richard Schwartz introduced a new dynamical system which is a marriage of two types of familiar billiards, tiling billiards and symplectic billiards. In this paper we investigate this dynamical system played on tilings of the plane which arise from non-Euclidean geometries on the torus. We review the affine analogue of the flat conformal structures on the torus through the work of Oliver Baues and William Goldman, and define an open subset of this deformation space corresponding to markings of complete affine tilings of the plane. We make this definition precise, and provide algebraic conditions on the symmetries of the tiling to define it. We then analyze the dynamics of symplectic tiling billiards played on these types of tilings and investigate the long-term dynamics of the system to prove a stability result concerning divergent trajectories. The divergence is defined in terms of geometric invariants arising from the tiling symmetry group. We argue that in some sense this divergence is a consequence of the tiles of a non-Euclidean tiling becoming 'thin' as one moves far away in the tiling. To do so we introduce a notion of thinness that is well adapted to the non-Euclidean affine tilings.

math.DS↗

arXiv2026

Asymptotics of the Meandering Number of a Cyclic Permutation

A cyclic meander is an embedded oriented loop in the plane intersecting a fixed infinite line, or circle, transversely in a linearly ordered set of $2n$ points. By keeping track of the order in which the loop visits these points, the cyclic meander induces a cyclic permutation on these marked points. Correspondingly, given a permutation on $n$ letters, one can ask whether or not a cyclic meander induces the permutation in this manner, and if not, what is the most efficient way of doing so if we allow more points of intersection? This process gives a way of associating to a permutation on $n$ letters a measurement of complexity of the permutation in question. The principal result of this work shows that the maximum of this quantity, the \emph{meander number}, over all cyclic permutations on $n$ letters, is bounded above and below quadratically in $n$. This result resolves a conjecture of Schwartz~\cite{richtpss} in relation to his work on the topological salesman problem. We conclude this work by proving the existence of families of cyclic permutations on $n$ letters whose meander numbers realize a continuum of growth rates between linear and quadratic.

math.CO↗

arXiv2025

Minimal Faithful Representations of Split Extensions by Abelian Groups

Every finite group admits a representation which is 'most efficient' in the sense that the representation is faithful and the dimension is minimal. We call such representations minimal faithful representations. The image of this representation retains all the information about the abstract group and can come with some additional geometric structure. For example, a minimal faithful representation could preserve a symplectic form or act on a fixed basis by permutations. It is the purpose of this paper to address these ideas for finite semi-direct products of groups split by an abelian group whose splitting homomorphism is faithful. We calculate the minimal faithful dimensions of such representations over both the complex and real numbers, and we express them in terms of group invariants to argue that in some sense these groups are extensions of tori. In the real case, the dimension of a minimal faithful representation and the structure it preserves encode the possible types of orbifold singularities of the group in question. We apply our framework to some families of groups to calculate their minimal faithful dimensions and analyze some of their orbifold singularities.

math.RT↗

arXiv2024

Projective Rigidity of Once-Punctured Torus Bundles via the Twisted Alexander Polynomial

In this paper we provide a means of certifying infinitesimal projective rigidity relative to the cusp for hyperbolic once punctured torus bundles in terms of twisted Alexander polynomials of representations associated to the holonomy. We also relate this polynomial to an induced action on the tangent space of the character variety of the free group of rank 2 into PGL(4,R) that arises from the holonomy of a hyperbolic once-punctured torus bundle. We prove the induced action on the tangent space of the character variety is the same as the group theoretic action that arises in the Lyndon Hochschild Serre spectral sequence on cohomology.

math.GT↗

arXiv2024

Computer Assisted Projective Rigidity

In this paper we provide a computer assisted proof that about two thousand surgeries far away from the ideal point in the hyperbolic Dehn filling space of the figure-eight knot complement are infinitesimally projectively rigid. We also prove that for projective deformations of the figure-eight knot complement sufficiently close to the complete hyperbolic structure, the induced map on the first cohomology of the longitude of the boundary torus is non-zero. This paper provides a complementary piece to the results of Heusener and Porti who showed that for each k in Z, there is a sufficiently large Nk for which every k/n-Dehn filling on the figure-eight knot complement for n larger than Nk is infinitesimally projectively rigid. In the process of the proof, we provide explicit representations of the figure-eight knot complement in PSO(3,1) which are rational in the real and imaginary parts of the shapes of the ideal tetrahedra used to glue the knot complement together.

math.GT↗

arXiv2020

Closed Affine Manifolds with an Invariant Line

A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.

math.DG↗

arXiv2020

Algebraic k-systems of curves

A collection $ Δ$ of simple closed curves on an orientable surface is an algebraic $ k $-system if the algebraic intersection number $\langle α,β\rangle$ is equal to $k $ in absolute value for every $ α, β\in Δ$ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic $k$-system of curves on a surface of genus $g$ is $2g+1$ when $g\ge 3$ or $k$ is odd, and $2g$ otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as $g^2$.

math.GT↗