Colloquia 2012-2013

Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Fall 2010

date	speaker	title	host(s)
sept 3	Timo Seppalainen (Madison)	Scaling exponents for a 1+1 dimensional directed polymer	local
sept 10	Moe Hirsch (Madison)	Actions of Lie groups and Lie algebras on manifolds	local
sept 17	Uri Andrews (Madison)	Computable stability theory	local
sept 24	Margo Anderson (UW-Milwaukee)	The politics of numbers	Jordan (Math and... seminar)
oct 1	Matthew Finn (U. of Adelaide)	Hot spots	Jean-Luc
wed oct 6	Robert Krasny (U. of Michigan)	Computing vortex sheet motion	Shi
oct 8	Anita Wager (Madison)	Bridging In and Out-of-School Mathematics: A Framework for Incorporating Students' Culture	Steffen
oct 15	Felipe Voloch (U. Texas Austin)	Local-Global principles for integral points on curves	Nigel
oct 22	Markus Banagl (U. Heidelberg)	On the Stability of Intersection Space Cohomology Under Deformation of Singularities	Maxim
oct 29	Irina Mitrea (IMA)	An Optimal Metrization Theorem for Topological Groupoids	WIMAW
wed nov 3	Tom Hales (Pittsburgh)	Introduction to Formal Proofs	Nigel (Distinguished lecture)
thu nov 4	Tom Hales (Pittsburgh)	Towards a Formal Proof of Kepler conjecture on Sphere Packings	Nigel (Distinguished lecture)
nov 5	Tom Hales (Pittsburgh)	Proof Assistants in Practice	Nigel (Distinguished lecture)
nov 12	Greg Buck (St. Anselm)	Measuring Entanglement Complexity	Jean-Luc
nov 19	Jeff Xia (Northwestern)	Partially Hyperbolic Dynamical Systems	Shi
wed dec 1	Peter Markowich (Cambridge and Vienna)	On Wigner and Bohmian Measures	Shi (Wasow Lecture)
dec 3	Saverio Spagnolie (UCSD)	Locomotion at low and intermediate Reynolds numbers	Jean-Luc (job talk)
mon dec 6	Brian Street (Madison)	A quantitative Frobenius Theorem, with applications to analysis	local (job talk)
dec 10	Benson Farb (Chicago)	TBA	Jean-Luc
mon dec 13 3 pm	Yun Jin (Harvard)	TBA	Timo (job talk)

Abstracts

Robert Krasny Computing Vortex Sheet Motion

Vortex sheets are used in fluid dynamics to model thin shear layers in slightly viscous flow. Examples include a mixing layer subject to Kelvin-Helmholtz instability and the trailing wake of an aircraft. One of the earliest simulations in computational fluid dynamics used the point vortex method to compute vortex sheet motion and the results seemed to confirm Prandtl's idea that vortex sheets roll up smoothly into concentrated spirals. However, later simulations with higher resolution encountered difficulty due to the fact that the initial value problem is ill-posed and a singularity forms at a finite time from smooth initial data. I'll describe the fundamental contributions on this topic by Louis Rosenhead, Garrett Birkhoff, and Derek Moore, and then discuss more recent regularized simulations past the critical time. The results support a conjecture by Dale Pullin on self-similarity, but chaotic dynamics intervenes unexpectedly. Finally I'll describe a new panel method for vortex sheet motion in 3D flow which uses a treecode to gain efficiency. A simulation of vortex ring dynamics will be shown and an application of the treecode in molecular dynamics will be briefly indicated.

Anita Wager Bridging In and Out-of-School Mathematics: A Framework for Incorporating Students' Culture

This presentation will examine a professional development designed to explore a broadened notion of teaching for understanding that considers the cultural and socio-political contexts in which children live and learn. The goal of the study was to identify how teachers, in the process of learning to consider their mathematics pedagogy through an equity lens, construed the relationships among mathematics achievement and culture. An analysis of the features teachers focused on when they incorporated the ideas of mathematics teaching for understanding with students' out-of-school mathematical knowledge revealed four related practices: (a) identifying embedded mathematical practices prominent in contexts, (b) addressing cultural activities using school mathematics, (c) creating teacher initiated situated settings, and (d) using cultural contexts for problems. The practices provide a framework to address an ongoing issue in mathematics education: how to incorporate students out-of-school experiences in the classroom.

Markus Banagl On the Stability of Intersection Space Cohomology Under Deformation of Singularities

In many situations, it is homotopy theoretically possible to associate to a singular space in a natural way a cell complex, its intersection space, whose cohomology possesses Poincare duality, but turns out to be a new cohomology theory for singular spaces, not isomorphic in general to intersection cohomology or L2-cohomology. An alternative description of the new theory by a de Rham complex is available as well. The theory has a richer internal algebraic structure than intersection cohomology and addresses questions in type II string theory. While intersection cohomology is stable under small resolutions, the new theory is often stable under deformations of singularities. The latter result is joint work with Laurentiu Maxim.

Greg Buck Measuring Entanglement Complexity

DNA, hair, shoelaces, vortex lines, rope, proteins, integral curves, thread, magnetic flux tubes, cosmic strings and extension cords; filaments come in all sizes and with diverse qualities. Filaments tangle, with profound results: DNA replication is halted, field energy is stored, polymer materials acquire their remarkable properties, textiles are created, shoes stay on feet.

We classify entanglement patterns by the rate with which entanglement complexity grows with the length of the filament. We show which rates are possible and which are expected in arbitrary circumstances. We identify a fundamental phase transition between linear and non-linear entanglement rates. We also find (perhaps surprising) relationships between total curvature, bending energy and entanglement.

Jeff Xia Partially Hyperbolic Dynamical Systems

We introduce a new class of topological invariants for partially hyperbolic dynamical systems and discuss some recent results obtained using these invariants. The talk will be aimed at general audience.

Peter Markowich On Wigner and Bohmian Measures

We present the most important approaches to the semiclassical analysis of the Schrödinger equation, based on Wigner measures and WKB techniques. These approaches are then compared to the Bohmian approach to semiclassical analysis, based on a highly singular system pf phase space ODEs, or equivalently on a highly nonlinear and singular self-consistent Vlasov equation.

Saverio Spagnolie Locomotion at low and intermediate Reynolds numbers

Many microorganisms propel themselves through fluids by passing either helical waves (typically prokaryotes) or planar waves (typically eukaryotes) along a filamentous flagellum. Both from a biological and an engineering perspective, it is of great interest to understand the role of the waveform shape in determining an organism's locomotive kinematics, as well as its hydrodynamic efficiency. We will begin by discussing polymorphism in bacterial flagella, and will compare experimentally measured biological data on swimming bacteria to optimization results from accurate numerical simulations. For eukaryotic flagella, it will be shown how the optimal sawtoothed solution due to Lighthill is regularized when energetic costs of internal bending and axonemal sliding are included in a classical efficiency measure. Finally, the locomotive dynamics of bodies at intermediate Reynolds numbers will be discussed, where a number of surprising and counter-intuitive behaviors can be seen even in very simple systems.

Brian Street A quantitative Frobenius Theorem, with applications to analysis

This talk concerns the classical Frobenius theorem from differential geometry, about involutive distributions. For many problems in harmonic analysis, one needs a quantitative version of the Frobenius theorem. In this talk, we state such a quantitative version, and discuss various applications. Topics we will touch on include singular integrals, regularity of some linear PDEs, sub-Riemannian geometry, and singular Radon transforms.