Inspired by the sound localization system of the barn owl, we define a new class of neural codes, called periodic codes, and study their basic properties. Periodic codes are binary codes with a special patterned form that reflects the periodicity of the stimulus. Because these codes can be used by the owl to localize sounds within a convex set of angles, we investigate whether they are examples of convex codes, which have previously been studied for hippocampal place cells. We find that periodic codes are typically not convex, but can be completed to convex codes in the presence of noise. We introduce the convex closure and Hamming distance completion as ways of adding codewords to make a code convex, and describe the convex closure of a periodic code. We also find that the probability of the convex closure arising stochastically is greater for sparser codes. Finally, we provide an algebraic method using the neural ideal to detect if a code is periodic. We find that properties of periodic codes help to explain several aspects of the behavior observed in the sound localization system of the barn owl, including common errors in localizing pure tones.

We investigate Milnor invariants of various braids arising from the signed sorting networks.

Using some regular polynomials with prescribed Galois groups, we construct explicit infinite families of unramified extensions over quadratic fields by specializations satisfying certain congruence conditions.

Iterated mapping has seen a lot of success lately in many problems such as bit retrieval, diffraction signal reconstruction, and graph coloring. We add another application of iterated mapping, namely finding solutions to the no-three-in-a-line problem. Given an $n\times n$ grid, we utilize iterated mapping to find $2n$ points such that any straight line (of any slope) drawn will not intersect three of the selected points.

Using exhaustive techniques and results from Lackenby and many others, we compute the tunnel numbers of all 1655 alternating 11- and 12-crossing knots and of 881 nonalternating 11- and 12-crossing knots. We also find all 5525 Montesinos knots with 14 or fewer crossings.

We explore the properties of nonpiecewise syndetic sets with positive upper density, which we call discordant, in countably infinite amenable (semi-)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden’s theorem and Szemerédi’s theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.

The 2-dimensional motion of a particle subject to Brownian motion and ambient shear flow transportation is considered. Numerical experiments are carried out to explore the relation between the shear strength, box size, and the particle’s expected first hitting time of a given target. The simulation is motivated by biological settings such as reproduction processes and the workings of the immune system. As the shear strength grows, the expected first hitting time converges to the expected first hitting time of the 1-dimensional Brownian motion. The dependence of the hitting time on the shearing rate is monotone, and only the form of the shear flow close to the target appears to play a role. Numerical experiments also show that the expected hitting time drops significantly even for quite small values of shear rate near the target.

A guessing game with two secret numbers is a game played between a questioner and a responder. The two players first agree upon the set, $N$, in which the game will be played, as well as the number of questions, $Q$, which will be asked by the questioner. The responder first chooses two distinct numbers from $N$. The questioner then asks questions of the form “How many of your chosen numbers are in the set $A\subseteq N$?” to which the responder answers truthfully. The goal for the questioner is to determine the responder’s two numbers using at most $Q$ questions. We study a continuous version of this game where $N$ is the closed interval of real numbers from 0 to 1. We introduce tools to study this game and use them to examine strategies for the questioner using a geometric approach. We establish a condition that must be satisfied by optimal strategies and give a strategy that can be made arbitrarily close to optimal.

We study the differentiation operator acting on discrete function spaces, that is, spaces of functions defined on an infinite rooted tree. We discuss, through its connection with composition operators, the boundedness and compactness of this operator. In addition, we discuss the operator norm and spectrum and consider when such an operator can be an isometry. We then apply these results to the operator acting on the discrete Lipschitz space and weighted Banach spaces, as well as the Hardy spaces defined on homogeneous trees.