Georgia Institute of Technology College of Sciences

For every natural number n, if we start with sufficiently many points in R^d in general position there will always exist n points in convex position. The problem of determining quantitative bounds for this statement is known as the Erdős-Szekeres problem, and is one of the oldest problems in Ramsey theory. We will discuss some of its history, along with the recent developments in the plane and in higher dimensions.

The Burau representation is a kind of homological representation of braid groups that has been around for around a century. It remains mysterious in many ways and is of particular interest because of its relation to quantum invariants of knots and links such as the Jones polynomial. In recent work, I came across a relationship between this representation and a moduli space of Euclidean cone metrics on spheres (think e.g. convex polyhedra) first examined by Thurston. After introducing the relevant definitions, I'll explain a bit about this connection and how I used the geometric structure on this moduli space to exactly identify the kernel of the Burau representation after evaluating its formal parameter at complex roots of unity. There will be many pictures!

In the early 60’s J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. Twenty years later, V. Arnold discovered a similar phenomenon on the sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of object where a similar type of behavior takes place: area-preserving maps of the cylinder. loosely speaking, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to “drift". This observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. This is a joint work with Mark Levi.

Discrete delay population models are often considered as a compromise between single-species models and more advanced age-structured population models, C.W. Clark, J. Math. Bio. 1976. This talk is based on a recent work (S. Streipert and G.S.K. Wolkowicz, 2023), where we provide a procedure for deriving discrete population models for the size of the adult population at the beginning of each breeding cycle and assume only adult individuals reproduce. This derivation technique includes delay to account for the number of breeding cycles a newborn individual remains immature and does not contribute to reproduction. These models include a survival probability (during the delay period) for the immature individuals, since these individuals have to survive to reach maturity and become members of what we consider the adult population. We discuss properties of this class of discrete delay population models and show that there is a critical delay threshold. The population goes extinct if the delay exceeds this threshold. We apply this derivation procedure to two well-known population models, the Beverton–Holt and the Ricker population model. We analyze their dynamics and compare it to existing delay models.