Geometry and Topology Seminar 2019-2020

Fall 2012

The seminar will be held in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm

date	speaker	title	host(s)
September 9	Gloria Mari Beffa (UW Madison)	The pentagram map and generalizations: discretizations of AGD flows	[local]
September 16	Ke Zhu (University of Minnesota)	Thin instantons in G2-manifolds and Seiberg-Witten invariants	Yong-Geun
September 23	Antonio Ache (UW Madison)	TBA	[local]
September 30	John Mackay (Oxford University)	What does a random group look like?	Tullia
October 7	David Fisher (Indiana University)	TBA	Richard and Tullia
October 21	Ruifang Song (UW Madison)	The Picard-Fuchs equations of Calabi-Yau complete intersections in homogeneous spaces	[local]
October 24 ( with Geom. analysis seminar)	Valentin Ovsienko (University of Lyon)	The pentagram map and generalized friezes of Coxeter	Gloria
November 4	Steven Simon (NYU)	TBA	Max
November 18	Igor Zelenko (Texas A&M University)	TBA	Gloria

Abstracts

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.

Antonio Ache (UW Madison)

TBA

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)

TBA

Ruifang Song (UW Madison)

TBA

Valentin Ovsienko (University of Lyon)

The pentagram map and generalized friezes of Coxeter

The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map. In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.

Igor Zelenko (Texas A&M University)

TBA

Fall-2010-Geometry-Topology