Geometry and Topology Seminar

Thursday October 3, 15:00, room MS.04

Max Karoubi (Universitée Paris Diderot)

Algebraic and Hermitian K-theory of operator algebras

Abstract: (Joint work with Mariusz Wodzicki.) The main purpose of the lecture is to establish the real case of "Karoubi's Conjecture" in algebraic K-theory. The complex case was proved in 1990-91 by Suslin and Wodzicki and enables us to compute the algebraic K-theory of many rings in functional analysis. Compared to the case of complex algebras, the real case poses additional difficulties. This is due to the fact that topological K-theory of real Banach algebras has period 8 instead of 2. The method we employ to overcome these difficulties can also be used for complex algebras, and provides some simplifications to the original proofs.

Thursday October 10, 15:00, room MS.04

Dmitry Jakobson (McGill University)

Conformal invariants from nodal sets

Abstract: (Joint work with Yaiza Canzani, Rod Gover and Raphael Ponge.) We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant’s Nodal Domain Theorem. We also show that on any manifold of dimension n >= 3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n >= 3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds.

Friday October 25, 15:00, room MS.03

Andrew Ranicki (Edinburgh)

Surgery theory and braids

Abstract: Surgery theory is usually applied to manifolds of dimension $\geq 4$. However, it can also be applied to one-dimensional manifolds. A braid is a one-dimensional submanifold of $\RR^3$ with boundary, which is obtained by weaving together neighbouring parallel strands. The closure of a braid is a link, a closed one-dimensional submanifold of $\RR^3$. The weaving operation is a particular type of surgery, replacing $S^0 \cross D^1$ by $D^1 \cross S^0$. The talk will describe the effect of the weaving operation in terms of algebraic surgery on a 1-dimensional chain complex with Poincaré duality. This will give a new perspective on the earlier work of Etienne Ghys and Maxime Bourrigan on the signatures of the closures of braids, and of Julia Collins on the Seifert matrix of the canonical Seifert surface of the closure of a braid.

Thursday October 31, 15:00, room MS.04

Robert Gray (University of East Anglia)

On groups encoded by the structure of the idempotents in the full linear monoid

Abstract: Many semigroups that arise in nature are idempotent generated. For instance, J. A. Erdos (1967) proved that every non-invertible matrix of the full linear monoid $M_n(F)$ of all $n$ by $n$ matrices over a field $F$ is expressible as a product of idempotent matrices. The set of idempotents $E$ of an arbitrary semigroup has the structure of a so called biordered set. These structures were studied in detail in work of Nambooripad (1979) and Easdown (1985). There is a free object $IG(E)$ in the category of idempotent generated semigroups with biordered set $E$. A question that has been of interest in the literature is: which groups can arise as maximal subgroups of these free idempotent generated semigroups? Early results on this problem led to a conjecture that all such groups must be free. The first counterexample to this conjecture was given by Brittenham, Margolis and Meakin (2009), where it was shown that the free abelian group of rank two can arise. Their approach shows how in general each maximal subgroup of $IG(E)$ is isomorphic to the fundamental group of a certain two-complex constructed using the idempotents, called the Graham--Houghton complex. Gray and Ruskuc (2012) then went on to show that every group arises in this context. In contrast, less is known about the the structure of the maximal subgroups of free idempotent generated semigroups on naturally occurring biordered sets. Further extending their topological approach, Brittenham, Margolis and Meakin (2010) showed that the rank one component of the free idempotent generated semigroup of the biordered set of $M_n(F)$ has maximal subgroup isomorphic to the multiplicative subgroup of $F$. In this talk I will present some joint work with Igor Dolinka in which we extend this result, showing that general linear groups arise as maximal subgroups in higher rank components.

Thursday November 7, 15:00, room MS.04

Haluk Sengun (Warwick)

Torsion homology of arithmetic hyperbolic three-manifolds

Abstract: Torsion homology of arithmetic manifolds has received much attention from number theorists recently. After discussing the importance of torsion from the perspective of number theory, I will talk on new joint work with N. Bergeron and A. Venkatesh which relates the topological complexity of homology cycles to the asymptotic growth of torsion homology in the case of arithmetic hyperbolic three-manifolds.

Thursday November 14, 15:00, room MS.04

Vaibhav Gadre (Warwick)

Pseudo-Anosov maps and the curve complex

Abstract: Analogous to the classification of maps of the two-torus, Thurston provided a classification for mapping classes of a general closed orientable surface as finite order, reducible or pseudo-Anosov. The complex of curves is the graph with vertices given by isotopy classes of simple closed curves on the surface with edges between a pair of vertices when the curves can be realized disjointly on the surface. The mapping class group has a natural action on the complex of curves. In this talk, we will consider some aspects of the action of pseudo-Anosovs on the complex of curves.

Thursday November 21, 15:00, room MS.04

Jeffrey Giansiracusa (Swansea)

Generalized ribbon graph decompositions of moduli spaces

Abstract: The moduli space of surfaces equipped with almost complex structure is homotopy equivalent to the moduli space of ribbon graphs. This classic result admits a generalisation that I will explain. For a class of types of geometric structures that surfaces can carry, including spin structures and principal $G$-bundles, I will describe how the resulting moduli spaces admit generalised ribbon graph decompositions that can be described elegantly in terms of operads. The moduli spaces of surfaces form a modular operad that turns out to be freely generated by the sub cyclic operad of moduli spaces of discs.

Thursday November 28, 15:00, room MS.04

No seminar (NA)

NA

Abstract: There is no seminar planned for today. However, participants might be interested in the one-day meeting on Graphs, Geometry and Probability.

Thursday December 5, 15:00, room MS.04

Tilman Bauer (KTH)

An algebro-geometric view of unstable cohomology operations

Abstract: One of the main tools to compute homotopy groups of spaces is the unstable Adams spectral sequence, whose input is the cohomology of the space in question as an algebra over the Steenrod algebra. Abstractly, this can be done for other cohomology theories $K$, in which case the spectral sequence will converge to the homotopy groups of the $K$-nilpotent completion. This is particularly interesting for the case of Morava $K$-theory or related theories, which are associated with the chromatic filtration; these spectral sequences tend to be much more approachable. I will talk about an interpretation of unstable operations in terms of formal algebraic geometry and use it for a much simpler description of the unstable Adams spectral sequence.

Tuesday December 10, 15:00, room B3.03

Francoise Dal'bo (Université Rennes 1)

Growth of nonuniform lattices in pinched negatively curved manifolds

Abstract: (Joint with the ETDS seminar.) Let $X$ be a complete and simply connected Riemannian manifold with sectional curvature bounded between two negative constants. If $X$ admits a uniform lattice, the volume entropy $h(X)$ of $X$ coincides with the critical exponent of the Poincaré series associated to the lattice, and the growth function $f(x,R) = \Vol(B(x,R))$ is asymptotically equivalent to $\exp(h(X)R)$. What happens for nonuniform lattices?