Spring 2018 Colliquia
January 19, 2018
Mark Levi (The Pennsylvania State University)
Gaussian Curvature and GyroscopesAbstract: Some counterintuitive mechanical phenomena, such as the gyroscopic effect, turn out to be related to Gaussian curvature. This applies, for instance, to the Lagrange top, which is described in virtually every advanced text on classical mechanics, but without reference to differential geometry. I will describe this in more detail, for non–specialists, and will mention some interesting related things discovered in the last couple of decades.
March 9, 2018
Michael Harrison (Lehigh University)
Skew Flat FibrationsMarch 16, 2018
Paul Levy (Lancaster University, UK)
Dual singularities in exceptional type nilpotent conesAbstract: It is well-known that nilpotent orbits in sl(n,C) correspond bijectively with the set of partitions of n, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type A there is an order-reversing involution on the poset of nilpotent orbits. More generally, if g is any simple Lie algebra over C then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in g to the set of special nilpotent orbits in the Langlands dual Lie algebra g^L. It was observed by Kraft and Procesi that the duality in type A is manifested in the geometry of the nullcone. In particular, if two orbits OO_1 < OO_2 are adjacent in the partial order then so are their duals OO_1^t > OO_2^t, and the isolated singularity attached to the pair (OO_1,OO_2) is dual to the singularity attached to (OO_2^t,OO_1^t): a Kleinian singularity of type A_k is swapped with the minimal nilpotent orbit closure in sl(k+1,C) (and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits. In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs OO_1<OO_2 of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.
April 6, 2018
Andrew Izsak (Tufts University)
Developing a Coherent Approach to Multiplication in School MathematicsAbstract: Numerous national reports on the state of education in school mathematics have highlighted the need to strengthen instruction in topics related to multiplication. I will focus on one critical aspect of this problem, the mathematical preparation of teachers. In particular, I will report on an approach developed with colleagues at the University of Georgia to teaching multiplication topics related to future middle and high school teachers. The approach is based on reasoning from an explicit, quantitative meaning for multiplication and mathematical drawings such as number lines and strip diagrams with the goal of seeing common structure across topics from multiplication to proportions using both whole numbers and fractions.