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Mathematics Colloquium


1. Symmetry-Breaking in Passive Tracers Advected by Laminar Shear Flow
2. Geometric Structures: story of a PhD
3. Monodromy of ODEs via projective structures

Date: Friday, September 28, 2018
Place and Time: Room 101, Love Building, 3:35-4:25 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstracts. 1. The dispersion of passive tracers in laminar shear flow is a classic problem in fluid dynamics. The boost in the tracer dispersion due to the interplay between advection and diffusion was first described mathematically and observed experimentally by Sir G. I. Taylor in 1953 for laminar flow in a straight circular pipe. This effective diffusivity sets in after the typical diffusive timescale and as such it is considered a long-time asymptotic result. This talk focuses on what happens at short and intermediate timescales for pipes of various geometries, before the effect of Taylor Dispersion is observed. At these timescales the tracer distribution is strongly asymmetric and understanding how its behavior relates to the cross-sectional geometry of the pipe is largely unexplored. Through analysis, simulation, and experiment, we explore the role different geometries play in controlling emerging longitudinal asymmetries in the cross-sectionally averaged distribution. Our results highlight the significance of the aspect ratio of the channel in controlling asymmetries and have potential implications for the design of fluidic channels in microfluidic devices. Ongoing and future directions will be discussed.

2. In this talk we will take the audience on a journey of my PhD history. The common topic being geometric structures in low dimensional topology, we will touch hyperbolic 3-manifolds, convex projective real surfaces and CR geometry. This will lead us to some of my current projects and interests.

3. Hilbert's XXI problem asks which representations of the fundamental group of a complex domain arise as the monodromy of some holomorphic ODE on it. Beyond Deligne's very general answer from the 1970s, not much is known about this problem for a fixed class of ODEs. We will present a recent approach in rank 2 via the study of the moduli space of branched complex projective structures.