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

## Fall 2011

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) [local]
September 16 Ke Zhu (University of Minnesota) Yong-Geun
September 23 Antonio Ache (UW Madison) [local]
September 30 John Mackay (Oxford University) Tullia
October 7 David Fisher (Indiana University) Richard and Tullia
October 14 Erwan Lanneau (University of Marseille, CPT) Jean Luc
October 21 Ruifang Song (UW Madison) [local]
October 24 ( with Geom. analysis seminar) Valentin Ovsienko (University of Lyon) Gloria
November 4 Steven Simon (NYU) Max
November 18 Igor Zelenko (Texas A&M University) Gloria
December 1 Bing Wang (Simons Center for Geometry and Physics) [Jeff]
December 2 David Dumas (University of Illinois at Chicago) Richard
December 9 Brian Clarke (Stanford) Jeff

## 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.

Obstruction-Flat Asymptotically Locally Euclidean Metrics

Given an even dimensional Riemannian manifold $\displaystyle{ (M^{n},g) }$ with $\displaystyle{ n\ge 4 }$, 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 $\displaystyle{ n }$ 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 $\displaystyle{ C^{0} }$-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).

The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties

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 its solution space is finite dimensional 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. 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 in toric varieties. This is based on joint work with Bong Lian and Shing-Tung Yau.

### 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.

### Steven Simon (NYU)

Equivariant Analogues of the Ham Sandwich Theorem

The Ham Sandwich Theorem, one of the earliest applications of algebraic topology to geometric combinatorics, states that under generic conditions any n finite Borel measures on R^n can be bisected by a single hyperplane. Viewing this theorem as a Z_2-symmetry statement for measures, we generalize the theorem to other finite groups, notably the finite subgroups of the spheres S^1 and S^3, in each case realizing group symmetry on Euclidian space as group symmetries of its Borel measures by studying the topology of associated spherical space forms. Direct equipartition statements for measures are given as special cases. We shall also discuss the contributions of the tangent bundles of these manifolds in answering similar questions.

### Igor Zelenko (Texas A&M University)

On geometry of curves of flags of constant type

The talk is devoted to the (extrinsic) geometry of curves of flags of a vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$. We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for such problem. Under some natural assumptions on the subgroup $G$ and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure Linear Algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves.

Our motivation to study such equivalence problems comes from the new approach to the geometry of structures of nonholonomic nature on manifolds such as vector distributions, sub-Riemannian structure etc. This approach is based on the Optimal Control Theory and it consists of the reduction of the equivalence problem for such nonholonomic geometric structures to the (extrinsic) differential geometry of curves in Lagrangian Grassmannians and, more generally, of curves of flags of isotropic and coisotropic subspaces in a linear symplectic space with respect to the action of the Linear Symplectic Group. The application of the general theory to the geometry of such curves case will be discussed in more detail.

### Bing Wang (Simons Center for Geometry and Physics)

Uniformization of algebraic varieties

For algebraic varieties of general type with mild singularities, we show the Bogmolov-Yau inequality holds. If equality is attained, then this variety is a global quotient of complex hyperbolic space away from a subvariety. This is will be a technical version of the speaker's colloquium talk.

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

Real and complex boundaries in the character variety

The set of holonomy representations of complex projective structures on a compact Riemann surface is a submanifold of the SL(2,C) character variety of the fundamental group. We will discuss the real- and complex-analytic geometry of this manifold and its interaction with the Morgan-Shalen compactification of the character variety. In particular we show that the subset consisting of holonomy representations that extend over a given hyperbolic 3-manifold group (of which the surface is an incompressible boundary) is discrete.

TBA