Department of

Mathematics


Seminar Calendar
for events the day of Thursday, April 1, 2021.

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Thursday, April 1, 2021

11:00 am in zoom,Thursday, April 1, 2021

On subRiemannian geodesics and billiard trajectories

Alvaro del Pino (Universiteit Utrecht)

Abstract: SubRiemannian Geometry studies triples consisting of a smooth manifold, a subbundle of the tangent bundle (a "tangent distribution"), and a metric along the distribution. Such a triple can be regarded as a degenerate limit of Riemannian manifolds, where the directions not in the distribution have been infinitely penalised. In such a setup, which is the framework in which Geometric Control Theory is phrased, a central question is to study the properties of the minimising curves. In many ways, this resembles the usual theory of geodesics in Riemannian Geometry, but various exotic behaviours appear. The underlying reason behind these behaviours is that, upon dualising, the subRiemannian metric becomes a degenerate Hamiltonian with fibrewise non-compact level sets. I will review the basic theory behind this setup, particularly the cotangent formulation, which goes back to work of Pontryagin. I aim to give a (very biased) overview of the area, emphasising various results about global topological properties of subRiemannian geodesics, as well as some intriguing open questions. If time allows, I will comment on recent work, joint with L. Dahinden, in which we study the subRiemannian billiard flow, which is the natural generalisation to manifolds with boundary.

3:00 pm in Zoom,Thursday, April 1, 2021

Minimal elements in the limit weak order

Christian Gaetz   [email] (Massachusetts Institute of Technology)

Abstract: The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule and cominuscule nodes of the Dynkin diagram, while the infinite Coxeter elements correspond to a single node, which we call the heavy node, in all affine types other than type A. No background on Weyl groups will be assumed. This talk is based on joint work with Yibo Gao. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.