# Difference between revisions of "Geometry and Topology Seminar 2019-2020"

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===Ruifang Song (UW Madison)=== | ===Ruifang Song (UW Madison)=== | ||

− | '' | + | ''We introduce a system of differential equations associated to a smooth algebraic variety X acted by a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is holonomic and thus has finite dimensional solution space assuming G acts on X with finitely many orbits. When X is a partial flag variety, we show that this system gives the Picard-Fuchs system of Calabi-Yau hypersurfaces in X. In particular, when X is a toric variety, our construction recovers GKZ systems and extended GKZ systems, which play important roles in studying periods of Calabi-Yau hypersurfaces/complete intersections in toric varieties. In general, if X is a Fano variety, L is the anticanonical line bundle and G=Aut(X), this construction can be used to study the Picard-Fuchs system of Calabi-Yau hypersurfaces/complete intersections in X. '' |

===Valentin Ovsienko (University of Lyon)=== | ===Valentin Ovsienko (University of Lyon)=== |

## Revision as of 22:27, 19 October 2011

## Contents

- 1 Fall 2012
- 2 Abstracts
- 2.1 Gloria Mari Beffa (UW Madison)
- 2.2 Ke Zhu (University of Minnesota)
- 2.3 Antonio Ache (UW Madison)
- 2.4 John Mackay (Oxford University)
- 2.5 David Fisher (Indiana University)
- 2.6 Erwan Lanneau (University of Marseille, CPT)
- 2.7 Ruifang Song (UW Madison)
- 2.8 Valentin Ovsienko (University of Lyon)
- 2.9 Igor Zelenko (Texas A&M University)
- 2.10 Conan Leung (Chinese University of Hong Kong)
- 2.11 David Dumas (University of Illinois at Chicago)

## Fall 2012

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

## 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)

Obstruction-Flat Asymptotically Locally Euclidean Metrics

Given an even dimensional Riemannian manifold [math](M^{n},g)[/math] with [math]n\ge 4[/math], it was shown in the work of Charles Fefferman and Robin Graham on conformal invariants the existence of a non-trivial 2-tensor which involves [math]n[/math] derivatives of the metric, arises as the first variation of a conformally invariant and vanishes for metrics that are conformally Einstein. This tensor is called the Ambient Obstruction tensor and is a higher dimensional generalization of the Bach tensor in dimension 4. We show that any asymptotically locally Euclidean (ALE) metric which is obstruction flat and scalar-flat must be ALE of a certain optimal order using a technique developed by Cheeger and Tian for Ricci-flat metrics. We also show a singularity removal theorem for obstruction-flat metrics with isolated [math]C^{0}[/math]-orbifold singularities. In addition, we show that our methods apply to more general systems. This is joint work with Jeff Viaclovsky.

### 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)

*Hodge-de Rham theory for infinite dimensional bundles and local rigidity*

It is well known that every cohomology class on a manifold can be represented by a harmonic form. While this fact continues to hold for cohomology with coefficients in finite dimensional vector bundles, it is also fairly well known that it fails for infinite dimensional bundles. In this talk, I will formulate a notion of a harmonic cochain in group cohomology and explain what piece of the cohomology can be represented by harmonic cochains. I will use these ideas to prove a vanishing theorem that motivates a family of generalizations of property (T) of Kazhdan. If time permits, I will discuss applications to local rigidity of group actions.

### Erwan Lanneau (University of Marseille, CPT)

*Dilatations of pseudo-Anosov homeomorphisms and Rauzy-Veech induction*

In this talk I will explain the link between pseudo-Anosov homeomorphisms and Rauzy-Veech induction. We will see how to derive properties on the dilatations of these homeomorphisms (I will recall the definitions) and as an application, we will use the Rauzy-Veech-Yoccoz induction to give lower bound on dilatations. This is a common work with Corentin Boissy (Marseille).

### Ruifang Song (UW Madison)

*We introduce a system of differential equations associated to a smooth algebraic variety X acted by a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is holonomic and thus has finite dimensional solution space assuming G acts on X with finitely many orbits. When X is a partial flag variety, we show that this system gives the Picard-Fuchs system of Calabi-Yau hypersurfaces in X. In particular, when X is a toric variety, our construction recovers GKZ systems and extended GKZ systems, which play important roles in studying periods of Calabi-Yau hypersurfaces/complete intersections in toric varieties. In general, if X is a Fano variety, L is the anticanonical line bundle and G=Aut(X), this construction can be used to study the Picard-Fuchs system of Calabi-Yau hypersurfaces/complete intersections in X. *

### 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*

### Conan Leung (Chinese University of Hong Kong)

*TBA*

### David Dumas (University of Illinois at Chicago)

*TBA*