Research

The dynamics of straight-line flows on compact translation surfaces (surfaces formed by gluing Euclidean polygons edge-to- edge via translations) has been widely studied due to their connections to polygonal billiards and Teichmuller theory. However, much less is known regarding straight-line flows on non-compact infinite translation surfaces. In this note we will consider straight-line flows on the Mucube -- an infinite Z^3 periodic half-translation square-tiled surface -- first discovered by Coxeter and Petrie and more recently studied by Athreya-Lee. We will give a complete characterization of the periodic directions for the straight-line flow on the Mucube -- first in terms of a genus one quotient and secondly in terms of an infinitely generated subgroup of SL_2(Z). Finally we will use the latter characterization to obtain the Veech group of the Mucube.    

In a 1916 paper, Ramanujan studied the additive convolution $S_{a,b}(n)$ of sum-of-divisors function)$s, $\sigma_a(n)$ and $\sigma_b(n), and proved an asymptotic formula for it when $a$ and $b$ are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real $a$ and $b$. Ramanujan's conjecture was subsequently proved in 1927 by Ingham.

In this paper we give two new proofs of Ramanujan's conjecture, each of which improves upon Ingham's result by obtaining a power savings in the error term. The first proof requires no special background, and is simpler than Ingham's. The second proof is more involved, but obtains lower order terms in the asymptotics for most ranges of the parameters.

Finally, we describe a connection to a counting problem in geometric topology that was studied in the second author's thesis and which served as our initial motivation in studying this sum.

A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. In this paper we consider a randomizing model for STSs and generalizations to branched covers of other simple translation surfaces which we call polygon-tiled surfaces. We obtain a local central limit theorem for the genus and subsequently obtain that the distribution of the genus is asymptotically normal. We also study holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. We show that asymptotically almost surely the set of holonomy vectors of a random STS contains the set of primitive vectors of $\mathbb{Z}_2$ and with probability approaching $1/e$, these sets are equal.

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between these properties and their frequency in each stratum of translation surfaces.

Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of $n$ squares and two cone points with angle $4\pi$ each, we set up and parametrize the classification into four diagrams. Our main result is to provide formulae for enumeration of square-tiled surfaces in these four diagrams, completing the detailed count for genus two. The formulae are in terms of various well-studied arithmetic functions, enabling us to give asymptotics for each diagram. Interestingly, two of the four cylinder diagrams occur with asymptotic density 1/4, but the other diagrams occur with different (and irrational) densities.

A continuous-time quantum walk on a graph $G$ is given by the unitary matrix $U(t)=exp(−itA)$, where $A$ is the Hermitian adjacency matrix of $G$. We say $G$ has pretty good state transfer between vertices $a$ and $b$ if for any $\epsilon>0$, there is a time $t$, where the $(a,b)$-entry of $U(t)$ satisfies $|U(t)_{a,b}|≥1−\epsilon$. This notion was introduced by Godsil (2011). The state transfer is perfect if the above holds for $\epsilon=0$. In this work, we study a natural extension of this notion called universal state transfer. Here, state transfer exists between every pair of vertices of the graph. We prove the following results about graphs with this stronger property: (1) Graphs with universal state transfer have distinct eigenvalues and flat eigenbasis (where each eigenvector has entries which are equal in magnitude). (2) The switching automorphism group of a graph with universal state transfer is abelian and its order divides the size of the graph. Moreover, if the state transfer is perfect, then the switching automorphism group is cyclic. (3) There is a family of prime-length cycles with complex weights which has universal pretty good state transfer. This provides a concrete example of an infinite family of graphs with the universal property. (4) There exists a class of graphs with real symmetric adjacency matrices which has universal pretty good state transfer. In contrast, Kay (2011) proved that no graph with real-valued adjacency matrix can have universal perfect state transfer. We also provide a spectral characterization of universal perfect state transfer graphs that are switching equivalent to circulants.