
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) 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.

 Thursday February 22, 15:00, room MS.03 Bert Wiest (Rennes) TBA Abstract: TBA

 Thursday March 1, 15:00, room MS.03 Marko Berghoff (Humboldt) TBA Abstract: TBA

 Thursday March 1, 16:00, room MS.03 Mark Hagen (Bristol) TBA Abstract: TBA

 Thursday March 8, 15:00, room MS.03 Rodolfo Gutierrez (Jussieu) TBA Abstract: TBA

 Wednesday March 14, 14:00-16:00, room MS.03 Yair Minsky (Yale) TBA Abstract: TBA

 Thursday March 15, 15:00-17:00, room MS.03 Yair Minsky (Yale) TBA Abstract: TBA

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