Thursdays, 15:30 - 16:30

Canadian Geometry and Topology seminar at Fields

Controlled algebra and topological data analysis

The talk will be an introduction to controlled algebra, giving among other an indication of how to prove topological invariance of Whitehead torsion. At the end I will discuss how these ideas may be useful in topological data analysis.

Counting loxodromic elements

In a group of isometries of a delta-hyperbolic space, a loxodromic element is one which acts with positive translation length, i.e. with north-south dynamics. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications to relatively hyperbolic groups and graph products, including right-angled Artin groups and right-angled Coxeter groups.

This is joint work with I. Gekhtman and S. Taylor.

Cubical dimension of small cancellation groups

Groups satisfying $C'(1/6)$ small cancellation condition are known to act properly and cocompactly on CAT(0) cube complexes. The cubical dimension of a group $G$ is the infimum $n$ such that $G$ acts properly on an $n$-dimensional CAT(0) cube complex. We will discuss a construction that for each n provides an example of a $C(6)$ group with the cubical dimension bounded below by $n$.

Perelman's functionals on compact manifolds with isolated conical singularities

We extend the theory of the Perelman's functionals on compact smooth manifolds to compact manifolds with isolated conical singularities. For the $\lambda$-functional, this is essentially an eigenvalue problem for a Schrödinger operator with singular potential. We obtain a certain asymptotic behavior of eigenfunctions near the singularities. This asymptotic behavior plays an important role for deriving the variation formulas of the $\lambda$-functional and other applications. Moreover, we show that the infimum of the $W$-functional over a suitable weighted Sobolev space on compact manifolds with isolated conical singularities is finite, and the minimizing function exists. We also obtain a certain asymptotic behavior for the minimizing function near the singularities.

This is a joint work with Professor Xianzhe Dai.

Surface systems and triple linking numbers

Stable equivalence of smoothly knotted surfaces

It is well known that there are smoothly inequivalent, objects in 4-dimensions that are topologically equivalent. Fairly general results exist stating that such objects become smoothly equivalent after some number of stabilizations. Until this past summer the only thing known about the number of stabilizations needed was an infinite collection of examples where one stabilization was enough.

This talk will present the proof of a theorem demonstrating that when the easiest topological invariants are trivial two smooth surfaces become smoothly isotopic after just one stabilization.

This is joint work with Kim, Melvin, Ruberman and Schwartz.

Coverings, groupoids, and embedding braids in mapping class groups

Given a finite-sheeted branched cover between surfaces with boundary, it is natural to ask whether every mapping class of the covering surface projects to the base surface, and whether every mapping class of the base surface lifts to the covering surface.

In this talk we will address these questions, with automorphisms of fundamental groupoids playing an integral role. As a consequence of the main result, we construct new embeddings of the braid group in the mapping class group of a surface.

This is joint work with Alan McLeay.

Some open problems in gauge theory, symplectic geometry, and low-dimensional topology

I will discuss a variety of open problems and partial results (due to myself and others), each relating to at least two of the fields appearing in the title. Some of the more famous open problems will include Witten's conjecture and the Atiyah-Floer conjecture; however, I will spend most of the time discussing more tractable variants of these, and how they relate.

Concordance of knots in homology spheres and the solvable filtration

The classical topological knot concordance group $\mathcal{C}$ consists knots in $S^3$ modulo those which bound flat disks in the 4-ball. There is a natural generalization, $\widehat{\mathcal{C}}$, the setting of knots in homology spheres. Recently, Adam Levine showed that in the smooth (as opposed to topological) setting the map $\mathcal{C} \rightarrow \widehat{\mathcal{C}}$ is not a surjection, producing knots in homology spheres not smoothly concordant to any knot in $S^3$. During this talk we will produce strong evidence that the opposite is true topologically. Namely we show that modulo any term of the solvable filtration of knot concordance introduced by Cochran-Orr-Teichner, the map $\mathcal{C} \rightarrow \widehat{\mathcal{C}}$ an isomorphism. Since every known invariant of topological concordance vanishes at some level of this filtration, our result implies that any possible difference between $\mathcal{C}$ and $\widehat{\mathcal{C}}$ not be detected by any currently existent technology.

On gauge theoretic invariants of homology $S^1 \times S^3$

I will survey some recent joint work on invariants of smooth 4-manifolds with homology of $S^1 \times S^3$. In our older work with Tom Mrowka and Daniel Ruberman we defined a Seiberg-Witten invariant of such manifolds. I will present a new formula (from our recent work with Jianfeng Lin and Daniel Ruberman) which expresses this invariant in terms of the monopole Floer homology and leads to some interesting topological applications.

$\tau$-invariants for knots in rational homology spheres

Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the three sphere called $\tau(K)$, which they also showed is a lower bound for the 4-ball genus. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, $K$, in a rational homology sphere, $Y$, we obtain a collection of $\tau$-invariants, one for each spin-c structure on $Y$. In addition, these invariants can be used to obtain a lower bound on the genus of a surface with boundary $K$ properly embedded in a negative definite 4-manifold with boundary $Y$.

On Loch Ness monsters and wild singularities - a Halloween-inspired introduction to infinite translation surfaces

Finite translation surfaces are obtained by gluing finitely many polygons along parallel edges of the same length. I will introduce a generalization of this concept and explain some issues that make the study more difficult. This involves more complicated kinds of singularities, infinite topological type, and flows that get trapped. Along the talk, there will be lots of examples and pictures.

Complements of closed hypersurfaces in $S^4$

We show that if a 3-manifold $M$ has an embedding in $S^4$ such that both complementary regions have abelian fundamental groups then the latter groups are cyclic, $\mathbb{Z} \oplus{\mathbb{Z}/n\mathbb{Z}}$ for some $n>1$, $\mathbb{Z}^2$ or $\mathbb{Z}^3$. If an homology handle $M$ with fundamental group $\pi$ embeds in $S^4$ then the Blanchfield pairing is neutral. We show that an homology handle has an embedding with both complementary fundamental groups abelian if and only if the commutator subgroup $\pi'$ is perfect, and the embedding is then essentially unique.

Metric transforms yielding Gromov hyperbolic spaces

A real valued function of one variable $\varphi$ is called a metric transform if for every metric space $(X,d)$ the composition $d_\varphi = \varphi\circ d$ is also a metric on $X$. In this talk we present a complete characterization of the class of approximately nondecreasing, unbounded metric transforms $\varphi$ such that the half line $[0,\infty)$ with the transformed Euclidean metric $\varphi(|x-y|)$ is Gromov hyperbolic. As an application we obtain a type of rigidity with respect to metric transformations of roughly geodesic Gromov hyperbolic spaces. This is based on joint work with Andrew Nicas.

Concordance and the Gordon-Litherland lattice

Stability of ALE Ricci-flat manifolds under Ricci-flow

We prove that if an ALE Ricci-flat manifold $(M,g)$ is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable. This is joint work with Alix Deruelle.

Geometric inequalities for black hole initial data sets

The standard picture of gravitational collapse leads (using heuristic arguments) to an inequality relating the ADM mass and angular momentum of axisymmetric spacetimes. Such geometric inequalities have been rigorously proved for axisymmetric, asymptotically flat maximal initial data for the vacuum Einstein equations in dimension d=4. After a broad review, I will discuss recent work on extending this class of inequalities to d>4, where a number of qualitative differences arise (e.g. black holes can have non-spherical horizon topology). In particular, I will discuss how a lower bound for the mass, in terms of a (regularized) harmonic energy functional is obtained. The unique minimizer of this energy is then shown to correspond to extreme black hole initial data for fixed angular momenta. This provides a variational characterization of extreme black hole.

Branched Covers of Quasipositive Links and L-Spaces.

We show that if $L$ is an oriented strongly quasipositive link other than the trivial knot or a link whose Alexander polynomial is a positive power of $(t-1)$, or a quasipositive link with non-zero smooth 4-ball genus, then the Alexander polynomial and signature function of $L$ determine an integer $n(L) \geq 1$ such that $\Sigma_n(L)$, the n-fold cyclic cover of $S^3$ branched over $L$, is not an L-space for $n \geq n(L)$. If $K$ is a strongly quasipositive knot with monic Alexander polynomial such as an L-space knot, we show that $\Sigma_n(K)$ is not an L-space for $n \geq 6$, and that the Alexander polynomial of K is a non-trivial product of cyclotomic polynomials if $\Sigma_n(K)$ is an L-space for some $n = 2,3,4,5$. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi- alternating quasipositive links. They also allow us to classify strongly quasipositive alternating links and strongly quasipositive alternating 3-strand pretzel links. This is joint work with Michel Boileau and Cameron Gordon.