Seminars, Colloquia, and Conferences
Colloquia
The colloquium meets on Fridays at 4:00pm in
BromfieldPearson 101, unless otherwise indicated.
Spring 2018
January 19, 2018
Gaussian Curvature and Gyroscopes
Mark Levi (The Pennsylvania State University)
Abstract:
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
February 16, 2018
Title  TBD
February 23, 2018
Title  TBD
March 2, 2018
Title  TBD
March 9, 2018
Skew Flat Fibrations
Michael Harrison (Lehigh University)
Abstract:
Is it possible to cover 3dimensional space by a collection of lines, such that no
two lines intersect and no two lines are parallel? More precisely, does there exist
a fibration of R^3 by pairwise skew lines? We give some examples and provide a
complete classification of such objects, by exhibiting a deformation retract from the
space of skew fibrations of R^3 to the subspace of Hopf fibrations. As a corollary of
the proof we obtain Gluck and Warner's classification of great circle fibrations of S^3.
We conclude with a discussion of skew fibrations in higher dimensions and some
surprising connections to the HurwitzRadon function and to vector fields on spheres.
March 16, 2018
Dual singularities in exceptional type nilpotent cones
Paul Levy (Lancaster University, UK)
Abstract:
It is wellknown 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 orderreversing involution on the poset of nilpotent
orbits. More generally, if g is any simple Lie algebra over C then
LusztigSpaltenstein duality is an orderreversing 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 viceversa). Subsequent work of KraftProcesi 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.
March 23, 2018
SPRING BREAK  no Colloquium
April 6, 2018
Developing a Coherent Approach to Multiplication in School Mathematics
Andrew Izsak (Tufts University)
Abstract:
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.
April 20, 2018
Uniform distribution, generalized polynomials and the theory of multiple recurrence
Vitaly Bergelson (Ohio State University)
Abstract:
A classical theorem
due to H. Weyl states that if P is a real polynomial such that at least one of
its coefficients (other than the constant term) is irrational, then the sequence
P(n), n=1,2,... is uniformly distributed mod 1. After briefly reviewing various
approaches to the proof of Weyl's theorem, we will discuss some modern
developments which involve "generalized polynomials", that is, functions which
are obtained from the conventional polynomials by the use of the greatest
integer function, addition and multiplication. As we shall see, there exists an
intrinsic connection between the generalized polynomials, dynamical systems on
nilmanifolds and the polynomial extensions of Szemeredi's theorem on arithmetic
progressions. We will conclude with formulating and discussing some natural open
problems and conjectures.
