Abstracts of past seminars
| Geometry and Topology Seminar Abstracts of past seminars |
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The current seminars are listed here
A list of past seminars is available for:
We give a survey of old and new results relating to the topology of non-compact hyperbolic 3-manifolds with finitely generated fundamental group. There are a number of interesting open problems, notably those of Ahlfors and Thurston. Important progress was has been made by Bonahon, Minsky and many others on these questions. We shall discuss an alternative approach to some of the work of Minsky which promises some simplification and generalisation.
Abstract: I'll start by discussing the general notion of a space of stability conditions on a derived category. This is an abstraction of Mike Douglas' idea of \Pi stability for D-branes. Then I'll discuss some features of this space in the special case of the derived category of the orbifold C3/Z3. This example is (totally unfinished) joint work with Alastair King.
Using a generalisation of Helley's theorem, I shall prove fixed-point results, beginning with a new proof that Out(Fn) (hence Aut(Fn) and SL(n,Z)) have Serre's property FA. The generalisation of Helley's Theorem has a remarkably easy proof.
Although the classical formalism of the Calculus of Variations is a venerable subject about which it seems not much can be said, it has some major drawbacks when dealing with intrinsic geometric problems. In this talk a formalism on the Calculus of Variations via Exterior Differential Systems will be discussed. Such a formalism is, while in greater generality than customary, particularly effective for intrinsic geometric problems, and it sheds new light on even the classical Lagrange Problems.
Abstract: A hyperbolic 3-manifold N has a convex core, the smallet convex submanifold of N which is homotopy equivalent to N. The boundary of the convex core is made up of surfaces, bent along a geodesic lamination. We consider the class of geometrically finite hyperbolic 3-manifolds and study the question of how the boundary of the convex core determines the geometry of N. In particular, we show in the case the bending locus is a finite collection of curves, that the lengths of these curves determines the structure on N. This work is in joint with C. Series.
This is joint with Michel Boileau.
We develop a theory of Voronoi decompositions in Riemannian geometry. This has applications to the topology of 3-manifolds. A sample application: The (infinite) set of diffeomorphism classes of closed orientable 3-manifolds which admit a Riemannian metric of volume at most V > 0 and |sec. curvature| < 1 contains at most finitely many irreducible, atoroidal 3-manifolds which do not satisfy Thurston's geometrization conjecture.
I'll start by explaining basic things about the Hilbert scheme - what it is, why it's smooth, etc. Then, I'll say something about the algebraic structures on it's cohomology.
We fix a combinatorial type of polyhedron and consider all the compact hyperbolic polyhedra with this combinatorial type. We are interested in the set of dihedral angles of these hyperbolic polyhedra. With the restriction that all dihedral angles be acute, Andreev's Theorem completely describes this space. We will give some properties of this space without any restriction on the dihedral angles.
In the past few years a number of classical GIT moduli spaces have been shown to admit complex hyperbolic structures (ball quotients). Most notable are the moduli spaces of Del Pezzo surfaces, in particular those of cubic surfaces (Allcock, Carlson, and Toledo) and of rational elliptic surfaces (Heckman and Looijenga). The techniques invariably involve carefully chosen auxillary period maps tailored to each case. A large class of moduli space/ball quotient examples is due to Deligne and Mostow, in their exploration of moduli of points on P1 and hypergeometric functions. At first glance, the new examples are not directly of Deligne-Mostow type, since the discrete groups do not appear on the various lists of Deligne-Mostow and Thurston. However, we show, by taking a view of hypergeometric functions based on intersection cohomology valued in local systems, that there are a host of such examples "functorially" obtained as locally symmetric subvarieties of the Deligne-Mostow examples. Furthermore, we show there are two "ancestral" Deligne-Mostow examples with associated "ancestral" hyperball quotients admitting a natural GIT description.
Let (flambda) be a one parameter family of holomorphic maps. Suppose f0 has a multiple fixed point at 0 and flambda has only simple fixed points for lambda \neq 0 . Let K be a connected compact set attracted by 0 under f0 . Then, for most values of lambda , K is attracted to an attractive fixed point for flambda. This statement does not involve polynomial vector field, the proof does, and requires a study of the geometry they create.
(Joint work with Vlad Markovic)
Let Omega be a simply connected proper open subset of the plane. We regard Omega as a subset of the 2-sphere, and the 2-sphere as the boundary of hyperbolic 3-space. Let X be the complement in the 2-sphere of Omega. Bill Thurston (1978) showed the importance of the hyperbolic convex hull of X, and, in particular, of the boundary in hyperbolic space of this convex hull, when considering Omega as a complex manifold. Sullivan's Theorem states that there is a universal constant K, such that Omega and the convex hull boundary are K-quasiconformally related by a homeomorphism which extends to the identity on the boundary. Thurston and Sullivan conjectured that one could take K=2. We have proved that this conjecture is false for the complement of the logarithmic spiral r=exp(i theta).
We fix an arbitrary group G and consider the set of all simplicial trees
on which G acts minimally (by simplicial homeomorphisms). Such trees are
generally very large and infinite, possibly with infinite branching at
each vertex. G-trees are interesting because they yield decompositions of
G as generalised free products of smaller groups. An example of such a
decomposition is the JSJ-decomposition of 3-manifold groups.
There is an equivalence relation on the set of G-trees which is generated
by certain local moves. We show that this coincides with another
equivalence relation defined in terms of the dynamics of individual
automorphisms. Using this result, one can prove uniqueness theorems for
G-trees. For example, if one requires a G-tree to satisfy certain
properties, then the tree may be uniquely determined from G alone.
One specific result is: if there is a G-equivariant quasi-isometry between two "strict" G-trees, then they are isometric, by a unique equivariant isometry.
i) Introduction. The definition and some examples of orientable(O),
symplectic(S), complex(C), Kahler(K) and projective(P) manifolds.
ii) Some implications:-
(a) P--->K (Fubini Study form)
(b) K--->S and K--->C (by definition)
(c) S---NOT-->K (Thurston´s example)
(d) C---NOT-->K, nor even S (eg. circle x (3-sphere) )
(e) O---NOT--> C (4-sphere)
(f) K---NOT---> P (certain complex tori)
There is an isomorphism between the Hilbert scheme of points on a surface X and the Hilbert scheme of S_n-orbits on X^n. I will explain something about how this is proved and what some its implications are.
Abstract: I will describe (but not explain) a remarkable correspondence, discovered by Vafa, between branes in type II string theories and rational curves on del Pezzo surfaces.
Higher-dimensional category theory is the study of n-categories, operads,
braided monoidal categories, and other such exotic structures. Although it
can be treated purely as an algebraic subject, it is inherently topological
in nature: the higher-dimensional diagrams one draws to represent these
structures can be taken quite literally as pieces of topology. Examples of
this are the braids in a braided monoidal category, and the pentagon which
appears in the definitions of both monoidal category and A_infinity space.
I will try to give a -afternoonish description of some of the dreams
people have for higher-dimensional category theory and its interactions with
topology. Grothendieck, for instance, suggested that tame topology should be
the study of n-groupoids; others have hoped that an n-category of cobordisms
between cobordisms between ... will provide a clean setting for TQFT; and
there is convincing evidence that the whole world of n-categories is a mirror
of the world of homotopy groups of spheres.
We outline a theory of noncommutative Riemannian and other differential-geometric structures coming out of quantum groups. The theory also applies to finite groups, which we elaborate. Differential structures are provided in this case by conjugacy classes and there is a natural Killing form metric, which we show for the permutation group $S_3$ to have constant positive curvature. In this way the noncommutative differential geometry of permutation groups resembles that of compact manifolds. We also fully compute the noncommutative de Rahm cohomology and Maxwell theory including sources, i.e. a theory of classical electromagnetism on $S_3$. Moreover, we solve the U(1) Yang-Mills theory (this differs from the U(1) Maxwell theory in noncommutative geometry), including the moduli spaces of flat connections. We show that the Yang-Mills action has a simple form in terms of Wilson loops in the permutation group
This talk is purely expository; I will explain three theorems which can be understood by studying the configuration space of distinct points in a manifold. The first two come from classical algebraic topology (the Barratt-Priddy- Quillen theorem, and the fact that the homology of an iterated loop space is a Poisson algebra). These results come from looking at the configuration space of points in R^m. The third result is a recent one of Kontsevich/Tamarkin, and gives a more subtle understanding of the configuration space of Euclidean space. The talk is aimed at first-year graduate students.
We examine the geometry of pseudocharacters (coarse homomorphisms to the reals) on groups, and discuss the relationship with quasi-actions on trees. We give examples of "exotic" quasiactions on trees by 3-manifold groups.
Abstract:
Let G be a finitely generated group. For each n we
define an to be the number of index n subgroups of G (this is
always finite). The theory of subgroup growth deals with the
behaviour of the an. For instance, one studies the growth rate of
the an, or uses them as the coefficients in a formal power series
or formal zeta function.
Recently the speaker, together with Alex Lubotzky and Ehud
Hrushovski, has studied the representation growth of G: here we
take an instead to be the number of irreducible complex
n-dimensional characters of G, modulo a natural equivalence
relation. Many of the methods from subgroup growth, including ideas
from model theory, can be applied here. I will discuss two
particular cases: when G is a finitely generated nilpotent group,
and when G is an arithmetic group. We conjecture that if G is an
arithmetic group then G satisfies the congruence subgroup property
if and only if the an grow at most polynomially.
Let C(F,G) denote the space of conjugacy classes of representations of a finitely generated group F into a reductive algebraic or Lie group G. If G=SU(2), then C( pi1(M) , G ) is the intersection of two (3g-3)-dimensional subvarieties of the (6g-6)-dimensional space C( pi1( Sg ) , G ), where Sg is a genus g surface. Counting the points of this intersection in a suitable way gives Casson's invariant for M. Here we describe an alternative way of counting C( pi1( M ) , G ) for arbitrary G. It is not clear whether the result is independent of the choice of Heegaard decomposition.
This talk discussed the recently established quantum multiplication on the rational cohomology of a closed symplectic manifold.
The most recent step in the Geometrization program for 3-manifolds is the proof of Thurston's orbifold theorem. We shall discuss the proof, obtained together with M. Boileau and B. Leeb, and we will focus on the case where the group of the orbifold is finite, which has been the last case to be solved. (Another proof has been obtained by D. Cooper, C. Hodgson and S. Kerckhoff).
This represents joint work with Steve Boyer (UQAM) and Bert Wiest
(PIMS postdoctoral fellow at UBC).
A group is left-orderable if there is a strict total ordering < of
its elements which is left-invariant: g < h iff fg < fh. If a
left-invariant ordering is also right-invariant, it is said to be
a biordering, and the group is said to be bi-orderable.
Left orderable groups are torsion-free, and have other good algebraic
properties. Spaces with orderable fundamental groups also enjoy special
topological conditions. To a newcomer to the subject, the number of
nonabelian, but orderable, groups is astonishing. For example, all
knot groups are left-orderable -- some, but not all, are bi-orderable.
This talk will discuss orderability of surface groups and of 3-manifold
groups, by which we mean the class of
fundamental groups of compact connected 3-manifolds (possibly with
boundary and/or nonorientable). We will see that orderability of these
groups is very common, but not universal. Sample results:
Theorem 1: Let M be any connected surface. Then its fundamental
group is bi-orderable, with the following two exceptions: if M is a Klein
bottle, its group is left-orderable, but not bi-orderable; if M is the
projective plane, its group has order two and is therefore not even
left-orderable.
Theorem 2: If M is a P^2 -irreducible compact, connected 3-manifold
with positive first Betti number, then the fundamental group of M is
left-orderable.
For the special case of Seifert-fibred 3-manifolds, we are able to
characterise those whose groups are left- and bi-orderable. Each of the
eight 3-manifold geometries has an example of a 3-manifold with that
geometry, whose group is left-orderable, as well as an example of
one whose group is not.
Let Hn denote hyperbolic n-space, and G a discrete subgroup of isometries of Hn. G is called GFERF if it is subgroup separable on all of its geometrically finite subgroups. The talk will discuss the proof that the groups PSL(2,Od) (where Od denotes the ring of integers in Q(\sqrt{-d})) are all GFERF. We will also discuss extensions to certain co-compact Kleinian groups
Thurston introduced a natural compactification of Teichmüller spaces of finite area hyperbolic surfaces. He defined a boundary at infinity to the Teichmüller space to be the space of projective measured laminations. We introduce a Thurston-type boundary to Teichmüller spaces of infinite area hyperbolic surfaces. The main tool in our work is the notion of Liouville currents. The Liouville map assigns to any point in the Teichmüller space a measure on the space of geodesics. We show that the Liouville map is differentiable in the appropriate sense. The description of the tangent map gives us a new topology on the space of measures. In the new topology, the Liouville map is a topological embedding of the Teichm\" uller space. The image of the Teichmüller space under the Liouville map is closed and unbounded. Further, the asymptotic rays to the image constitute a natural boundary to the Teichmüller space. We show that the space of asymptotic rays is homeomorphic to the space of projective bounded measured laminations on the surface.
Via taking families of hyperplane sections, one can reduce various parts of the geometry of complex projective surfaces to studying smooth diffeomorphisms of Riemann surfaces. Following work of Donaldson, this classical picture can be applied to study the more mysterious class of symplectic four-manifolds (which may not be projective). We will review these themes, and explain how one can - at least in principle - approach questions in symplectic topology via the study of mapping class groups and the moduli space of curves.
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