Starting from just a sheet of paper, by folding, stacking, crumpling, tearing, we will explore a rich variety of phenomena, from magic tricks and geometry to elasticity and the traditional Japanese art of origami. Much of the lecture consists of actual table-top demos, which you can try later with friends and family.
Pentagramma Mirificum (the miraculous pentagram) is a beautiful geometric construction studied by Napier and Gauss. Its algebraic description yields the simplest instance of cluster transformations, a remarkable family of recurrences which arise in diverse mathematical contexts, from representation theory and enumerative combinatorics to theoretical physics and classical geometry (Euclidean, spherical, or hyperbolic).
Moduli spaces parameterize continuous deformations of mathematical objects. Their study (starting with ideas of Riemann in the 19th century) has revealed many surprises: universal structures, connections between different subjects, and wild leaps. I will present some sense of the motivations and results related to the study of moduli spaces from the perspective of examples ranging from configurations of mechanical linkages to the moduli of varieties and sheaves.
High school students learn how to express the solution of a quadratic equation in one unknown in terms of its three coefficients. Why does this formula matter? We offer an answer in terms of discriminants and data. This lecture invites the audience to a journey towards non-linear algebra.
Starting from the popular card game SET, we will look at questions concerning the maximum possible size of sets without three-term progressions in various different mathematical settings (namely, subsets of the numbers from 1 to some number N, and subsets of so-called vector spaces over finite fields). These are fundamental questions in additive combinatorics, with connections to several other mathematical areas. We will discuss what is known about these questions, as well as some related problems.
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