\( \newcommand{\Sp}{\operatorname{Sp}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\GL}{\operatorname{GL}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\SU}{\operatorname{SU}} \newcommand{\PU}{\operatorname{PU}} \newcommand{\Pin}{\operatorname{Pin}} \newcommand{\CAT}{\operatorname{CAT}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Out}{\operatorname{Out}} \newcommand{\vcd}{\operatorname{vcd}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\Flat}{\operatorname{Flat}} \newcommand{\CP}{\mathbb{CP}} \newcommand{\CC}{\mathbb{C}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\calG}{\mathcal{G}} \newcommand{\calO}{\mathcal{O}} \newcommand{\from}{\colon} \newcommand{\cross}{\times} \)

Geometry and Topology Seminar

Warwick Mathematics Institute, Term II, 2017-2018

Please contact Saul Schleimer if you would like to speak or to suggest a speaker. A list of all events at WMI can be found here.

Thursday January 11, 15:00, room MS.03

None (None)


Abstract: None

Thursday January 18, 15:00, room MS.03

Chris Leininger (UIUC)

Strict domination and hyperbolic manifolds

Abstract: A strict domination between two closed hyperbolic manifolds is a $c$-Lipschitz map with $c < 1$, which has nonzero degree. Through the work of Gueritaud-Kassel and Tholozan, this has some interesting connections to locally homogeneous spaces and volumes. Strict dominations arise quite naturally in dimension two from holomorphic branched coverings, thanks to the Schwarz-Pick Theorem. After discussing this motivation, I will describe work with Grant Lakeland, building on a example suggested by Ian Agol, providing a general construction of strict domination in dimensions three and four.

Thursday January 25, 15:00, room MS.03

Elia Fioravanti (Oxford)

Superrigidity of actions on finite rank median spaces

Abstract: Finite rank median spaces simultaneously generalise real trees and finite dimensional $\CAT(0)$ cube complexes. Requiring a group to act on a finite rank median space is in general much more restrictive than only asking for an action on a general median space. This is particularly evident for irreducible lattices in products of rank-one simple Lie groups: they admit proper cocompact actions on infinite rank median spaces, but any action on a finite rank median space must fix a point.

Our proof of the latter fact is based on a generalisation of a superrigidity result of Chatterji-Fernos-Iozzi. We will sketch the techniques that go into this, focussing on analogies and differences between cube complexes and median spaces.

Thursday February 1, 15:00, room MS.03

Viveka Erlandsson (Bristol)

Measures on geodesic currents and counting curves on surfaces

Abstract: A famous result by Mirzakhani gives the asymptotic growth of the number of simple curves of bounded length $L$, as $L$ grows, on a hyperbolic surface (later generalised to curves of bounded self intersection number). Based on these results, in joint work with Souto, we showed that the asymptotics hold also for any Riemannian metric on the surface. We did so by studying certain mapping class group invariant measures on the space of geodesic currents. Invariant measures of the space of measured laminations were classified by Lindenstrauss and Mirzakhani, and in joint work with Mondello we extend this classification to the larger space of currents. In this talk I will discuss these two results and their connection.

Thursday February 8, 15:00, room MS.03

Michael Shapiro (Bath)

The Heisenberg group has rational growth in all generating sets

Abstract: Given a group \(G\) and a finite generating set \(\calG\) the (spherical) growth function \( f_\calG(x) = a_0 + a_1 x + a_2 x^2 + \ldots\) is the series whose coefficients \(a_n\) count the number of group elements at distance \(n\) from the identity in the Cayley graph \(\Gamma_\calG(G)\). For hyperbolic groups and virtually abelian groups, this is always the series of a rational function regardless of generating set. Many other groups are known to have rational growth in particular generating sets.

In joint work with Moon Duchin, we show that the Heisenberg group also has rational growth in all generating sets.

The first ingredient in this result is to compare the group metric, which we can see as a metric on the integer Heisenberg group with a metric on the real Heisenberg group. This latter is induced by a norm in the plane which is in turn induced by a projection of the generating set. The second ingredient is a wondrous theorem of Max Bensen regarding summing the values of polynomical over sets of lattice points in families of polytopes. We are able to bring these two ingredients together by showing that every group element has a geodesic whose projection into the plane fellow-travels a well-behaved set of polygonal paths.

Thursday February 15, 15:00, room MS.03

Nicholas Lindsay (King's College London)

\(S^1\)-invariant symplectic hypersurfaces in dimension 6 and the Fano condition

Abstract: A symplectic Fano manifold is a compact symplectic manifold where the first Chern class is a positive multiple of the cohomology class of the symplectic form. In dimension 4, these manifolds are necessarily symplectomorphic to del Pezzo surfaces, by results of Ohta, Ono and others. In dimensions 12 and above, some non-Kähler examples where found by Fine and Panov. They also conjectured that 6-dimensional symplectic Fano manifolds with a Hamiltonian circle action are symplectomorphic to Fano 3-folds.

In this talk, I will discuss a joint work with Dmitri Panov, where we show that 6-dimensional symplectic Fano manifolds with a Hamiltonian circle action are simply connected and satisfy \(c_1 \cdot c_2 = 24\). This may be interpreted as positive evidence for the aforementioned conjecture. The proof involves constructing a symplectic hypersurface in a certain class of symplectic 6-manifolds with a Hamiltonian circle action.

Wednesday February 21, 16:00, room MS.03

Bert Wiest (Rennes)

An analogue of the curve complex for Garside groups

Abstract: Garside groups are a family of groups with particularly nice algorithmic properties, containing in particular all Artin groups of spherical type. The most famous examples are the braid groups. In this talk, I will present a simple construction which associates to every Garside group $G$ a locally infinite, delta-hyperbolic graph on which $G$ acts; we call it the "additional length complex". I will show that these complexes share important features with curve complexes – in fact, the additional length complex of the braid group $B_n$ is conjectured to be quasi-isometric to the curve complex of the $n$-times punctured disk. Our construction has the potential to be adapted to many other contexts. (Joint work with Matthieu Calvez.)

Thursday March 1, 15:00, room MS.03

Marko Berghoff (Humboldt)

Feynman amplitudes and moduli spaces of graphs

Abstract: Moduli spaces of graphs show up in various areas of mathematics, for instance in geometric group theory or tropical geometry. Moreover, recent results by Bloch and Kreimer hint at a connection between the combinatorial structure of such spaces and questions regarding the analytic structure of Feynman amplitudes, a long-standing open problem in quantum field theory. In this talk I will give a short introduction to Feynman amplitudes and show how to interpret them as integrals over “semi-discrete” volume forms on an appropriate moduli space of colored graphs. In the case relevant for physics most of these integrals are divergent and need to be renormalized to find their finite physical value. I will explain how these divergences are encoded in the structure of the moduli space and how a Borel-Serre bordification of it can be used to solve the renormalization problem. If time permits I discuss some further properties of these moduli spaces of colored graphs and how similar constructions à la Outer space could help to improve our understanding of Feynman amplitudes viewed as boundary values of complex functions.

Thursday March 1, 16:00, room MS.03

Mark Hagen (Bristol)

Hyperplane-essential actions and taco moves

Abstract: Various well-known facts, for example Stallings' ends theorem and the Nielsen realisation theorem for $Out(F_n)$, boil down to passing from a cubulation of a group to a splitting. I will make this interpretation precise, and also explain how some open problems are reducible to more general statements about passing from actions on cube complexes to actions on trees. These questions include the Kropholler-Roller conjecture and the question of finding a Nielsen realisation theorem for outer automorphism groups of as wide a class of right-angled Artin groups as possible. There is a general procedure, called "panel collapse", for passing from a cubulation of a group $G$ to a "lower-complexity" one. The talk will outline this construction, explain how to recover the two theorems mentioned above from the construction, and discuss the extra challenges posed by the above two open problems. This talk is on some joint work with Nicholas Touikan, and on some joint work with Henry Wilton.

Thursday March 8, 15:00, room MS.03

Rodolfo Gutierrez (Jussieu)

Rauzy–Veech groups of flat surfaces

Abstract: The Rauzy–Veech induction is a powerful renormalization procedure for (half-)translation flows. By tracking the changes it induces in homology, we define the Rauzy–Veech monoid (or group) of a connected component of a stratum of Abelian or quadratic differentials. This monoid was proven to be pinching and twisting by Avila and Viana, which implies the Kontsevich–Zorich conjecture stating the simplicity of the Lyapunov spectrum of almost every translation flow with respect to the Masur–Veech measures.

In this talk, I will present a full classification of the Rauzy–Veech groups of Abelian strata: they are explicit finite-index subgroups of their ambient symplectic groups. This is strictly stronger than pinching and twisting and solves a conjecture of Zorich about the Zariski-density of such groups. Moreover, some techniques can be extended to the quadratic case to prove that the indices of the "plus" and/or "minus" Rauzy–Veech groups of certain connected components of quadratic strata are also finite. This proves that the Lyapunov spectra of such strata are simple, which was not previously known.

Wednesday March 14, 14:00-16:00, room MS.03

Yair Minsky (Yale)

The geometry and topology of Teichmüller space

Abstract: We will discuss a number of aspects of the geometry of the Teichmüller space of Riemann surfaces: its asymptotic geometry, as seen through Thurston's compactification by measured laminations; its coarse geometry, as captured by the relative hyperbolicity properties of the complex of curves; and the dynamical geometry of its geodesic flows. The interactions between these points of view are important for understanding the different roles Teichmüller space plays, for example in the structure of hyperbolic 3-manifolds. After an overview of these topics we will spend some time talking about the Weil-Petersson geodesic flow, which remains a delicate and difficult to understand part of the theory, and where we are still left with more questions than answers.

Thursday March 15, 15:00-17:00, room MS.03

Yair Minsky (Yale)

The geometry and topology of Teichmüller space

Abstract: As above.

Information on past talks. This line was last edited Monday, October 2, 2017 02:30:50 PM BST