Geometry and Topology Seminar 2019-2020
The seminar will be held in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
|September 9||Gloria Mari Beffa (UW Madison)||[local]|
|September 16||Ke Zhu (University of Minnesota)||Yong-Geun|
|September 30||John Mackay (Oxford University)||Tullia|
|October 7||David Fisher (Indiana University)||Richard and Tullia|
|November 4||Steven Simon (NYU)||Max|
|November 18||Igor Zelenko (Texas A&M University)||Gloria|
Gloria Mari Beffa (UW Madison)
The pentagram map and generalizations: discretizations of AGD flows
GIven an n-gon one can join every other vertex with a segment and find the intersection of two consecutive segments. We can form a new n-gon with these intersections, and the map taking the original n-gon to the newly found one is called the pentagram map. The map's properties when defined on pentagons are simple to describe (it takes its name from this fact), but the map turns out to have a unusual number of other properties and applications.
In this talk I will give a quick review of recent results by Ovsienko, Schwartz and Tabachnikov on the integrability of the pentagram map and I will describe on-going efforts to generalize the pentagram map to higher dimensions using possible connections to Adler-Gelfand-Dikii flows. The talk will NOT be for experts and will have plenty of drawings, so come and join us.
Ke Zhu (University of Minnesota)
Thin instantons in G2-manifolds and Seiberg-Witten invariants
For two nearby disjoint coassociative submanifolds $C$ and $C'$ in a $G_2$-manifold, we construct thin instantons with boundaries lying on $C$ and $C'$ from regular $J$-holomorphic curves in $C$. It is a high dimensional analogue of holomorphic stripes with boundaries on two nearby Lagrangian submanifolds $L$ and $L'$. We explain its relationship with the Seiberg-Witten invariants for $C$. This is a joint work with Conan Leung and Xiaowei Wang.
John Mackay (Oxford University)
What does a random group look like?
Twenty years ago, Gromov introduced his density model for random groups, and showed when the density parameter is less than one half a random group is, with overwhelming probability, (Gromov) hyperbolic. Just as the classical hyperbolic plane has a circle as its boundary at infinity, hyperbolic groups have a boundary at infinity which carries a canonical conformal structure.
In this talk, I will survey some of what is known about random groups, and how the geometry of a hyperbolic group corresponds to the structure of its boundary at infinity. I will outline recent work showing how Pansu's conformal dimension, a variation on Hausdorff dimension, can be used to give a more refined geometric picture of random groups at small densities.
David Fisher (Indiana University)
Igor Zelenko (Texas A&M University)