In this talk I will discuss a rather unique collection of tools and how they have been used to understand the spread of Influenza virus in the State of Montana. With flu counts from each county over a 10 year period some patterns emerge, which explain some vectors of the disease spread. Archetypal analysis then […]
It is well known that a real number is badly approximable if and only if the partial quotients in its continued fraction expansion are bounded. Motivated by a recent wonderful paper by Ngoc Ai Van Nguyen, Anthony Poels and Damien Roy (where the authors give a simple alternative solution of Schmidt-Summerer's problem) we found an […]
Gordian Knots According to the legend of Phrygian Gordium, Alexander the Great cut the ``Gordian Knot’’ and eventually went on to rule Asia thereby fulfilling an ancient prophecy. Where there are several descriptions of the precise nature of the Gordian Knot and Alexander’s action, an explicit mathematical treatment (the theory of thick knots) and the reasons […]
Gordian Knots According to the legend of Phrygian Gordium, Alexander the Great cut the ``Gordian Knot’’ and eventually went on to rule Asia thereby fulfilling an ancient prophecy. Where there are several descriptions of the precise nature of the Gordian Knot and Alexander’s action, an explicit mathematical treatment (the theory of thick knots) and the reasons […]
Knotting in living organisms is a feature that is visible to the careful observer of biological life. Since the 1970’s, with the increasing power of electron microscopes, scientists have been able to capture images of such structures in living organisms at near atomic levels. We will explore the mathematics of knotting that has provided tools study these […]
The talk will concentrate on open questions related to the optimal bounds for the discrepancy of an $N$-point set in the $d$-dimensional unit cube. The so-called star-discrepancy measures the difference between the actual and expected number of points in axis-parallel rectangles, and thus measures the equidistribution of the set. This notion has been explored by H. Weyl, K. Roth, and many others, […]
Many problems, arising in discrete and metric geometry, signal processing, physics, etc, can be reformulated as questions of optimizing discrete or continuous measures. We shall review some of such conjectures, as well as approaches to determining optimal (or at least good) point distributions and measures, and connections to other problems, such as discrepancy, sphere packings […]
Biological invasions often have outsized consequences for the invaded ecosystem and represent an interesting challenge to model mathematically. Landscape heterogeneity, non-local or time-dependent spreading mechanisms, coarse data, and air or […]
For k >= 2, the k-coloring graph C(G) of a base graph G has a vertex set consisting of the proper k-colorings of G with edges connecting two vertices corresponding to two different colorings of G if those two colorings differ in the color assigned to a single vertex of G. A base graph whose […]
Opioid addiction has become a national health crisis in recent years, with involvement in 66% of all drug overdose deaths in 2016 and high economic costs. In contrast to the dynamics of a classic disease or illicit drug epidemic, opioid addiction has its roots in legal, prescription medication - a fact which greatly increases the […]
TOPIC: The Mathematics of Reapportionment and Census Data Every ten years, the United States Census Bureau conducts a count of all persons living in the United States; one of those population counts will be carried out this year (2020). This Census is mandated by the US Constitution; it counts all people residing in the United […]
In water-limited regions, competition for water resources results in the formation of vegetation patterns; on sloped terrain, one finds that the vegetation typically aligns in stripes or arcs. The dynamics […]
