Difference between revisions of "Colloquia 2012-2013"
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|[http://www.gregorybuck.com/ Greg Buck] (St. Anselm) | |[http://www.gregorybuck.com/ Greg Buck] (St. Anselm) | ||
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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. | 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. | 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. |
Revision as of 11:31, 5 November 2010
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) | TBA | Shi |
wed dec 1 | Peter Markowich (Cambridge and Vienna) | TBA | Shi (Wasow Lecture) |
dec 10 | Benson Farb (Chicago) | TBA | Jean-Luc |
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.