Topology Festival

Pseudo-Anosov flows are an important class of dynamical systems on three-dimensional manifolds; their dynamical properties are deeply connected to three-dimensional geometry and topology. Given a pseudo-Anosov flow on a manifold M, there is an associated action of the fundamental group of M on a bifoliated plane, a topological plane equipped with two transverse foliations. This action has been shown to contain all the topological information about the flow. In this talk, we will first introduce pseudo-Anosov flows and their associated bifoliated planes via a classic example: a geodesic flow on a compact hyperbolic surface. We will discuss the properties that make the geodesic flow pseudo-Anosov and construct the bifoliated plane associated to this flow.

The basic objects of study in stable homotopy theory are spectra, a simultaneous generalization of generalized cohomology theories and infinite loop spaces. Despite this topological origin, the category of spectra admits enough algebraic structure to make sense of rings and modules therein. In this talk, I'll set the stage for spectra to appear later this weekend by elucidating their relation to more familiar notions and giving several examples.

Ring spectra are homotopy-theoretic generalizations of rings or dgas. Algebraic K-theory, a spectrum-valued invariant of rings, can be naturally extended to take ring spectra as input. The theory of algebraic K-theory of ring spectra has seen tremendous development over the past 30 years, in both computational and theoretical aspects, culminating in the disproof of the telescope conjecture. In this talk, I will start by introducing basic notions of stable homotopy theory: the category of spectra and its properties. Then, I will briefly explain about algebraic K-theory and the trace method, which is how we compute algebraic K-theories.

Periodic Floer Homology (PFH), originally introduced by Hutchings, is a Floer theory developed to study area-preserving maps on closed surfaces. In this talk, I will introduce PFH and its spectral invariants, which have powerful applications in the study of surface dynamics and the structure of transformation groups. I will focus on the case of the $2$-sphere, explaining the computation of its PFH and how the spectral invariants give rise to new homogenized quasimorphisms on its group of Hamiltonian diffeomorphisms.

Computation of the stable homotopy groups of spheres is a long-standing open problem in algebraic topology. I will introduce chromatic homotopy theory -- a powerful framework for organizing computations and describing large scale patterns in stable homotopy groups. It splits the problem into chromatic levels, and each level can be studied using the theory of formal group laws and their deformations. I will discuss the history of this field and some recent developments.

The telescope conjecture is an attempt to describe telescopes, which are certain fundamental building blocks of the stable homotopy category. It predicts that these telescopes have homotopy groups that are finite dimensional and describable in terms of arithmetic geometry information. I will explain the disproof of it in joint work with Burklund, Hahn, and Schlank, for heights at least 2. The idea is to use certain algebraic K-theory spectra and trace methods to be able to detect the discrepancy between the actual telescopes and what is predicted by the telescope conjecture. If time permits, I will indicate future directions and open problems.

The construction of homotopy-theoretic invariants from geometric topology often passes through techniques from Morse theory and Floer theory, using the machinery of flow categories. In this talk I'll discuss a structural interpretation of these constructions with a stable homotopy-theoretic flavor, based on Abouzaid-Blumberg's proof of the Arnold conjecture. This is a report on joint work with Erkao Bao.

In his 1990 ICM address, Goodwillie conjectured a very close relationship between algebraic K-theory and topological cyclic homology (TC) of ring spectra -- that is, they have identical Taylor series (in the sense of functor calculus). The strongest form of this conjecture is now a celebrated theorem of Dundas, Goodwillie, and McCarthy. In closely related work, Lindenstrauss and McCarthy characterized these Taylor series in terms of "topological Witt vectors" — an invariant which naturally arises in the study of TC. In this talk, I will introduce a THH-like invariant that is designed to recover the Taylor tower of TC, and whose construction makes use of Kaledin's 'cyclic trace theories.' I will discuss some applications of this invariant, and, time permitting, describe connections to the work of Lindenstrauss and McCarthy.

In 1980, McDuff showed that the group of diffeomorphisms of the open n-dimensional disc preserving a given volume form is perfect whenever n is at least 3, and asked what happens when n = 2. I will explain recent joint work showing that in the two-dimensional case, this group is in fact not perfect as long as the area form has finite area. The proof will use some new quasimorphisms on the group of area-preserving homeomorphisms of the sphere, of independent interest; these quasimorphisms are constructed using ideas from Heegaard Floer cohomology.

We compute the automorphism group of the intersection graph of many large-type Artin groups. This graph is an analogue of the curve graph for mapping class groups, adapted to the setting of Artin groups. As an application, we derive a range of rigidity and classification results for these groups, including the computation of outer automorphism groups, commensurability classification, quasi-isometric rigidity, and other related results.

This is joint work with Jingyin Huang and Nicolas Vaskou.

Artin groups simultaneously generalise braid groups, free groups, and free abelian groups. They are closely related to Coxeter groups, and like these, can be defined in terms of labelled simplicial graphs. Despite their seemingly straightforward combinatorial nature, hardly anything is known about arbitrary Artin groups, and many outstanding questions remain unanswered for large subclasses of these groups.

In this talk we will explain how to view an Artin group as a quotient of a cubulated group in a way that encodes many aspects of its geometry. We will then explain how this viewpoint – cubical small-cancellation theory – allows us to understand in many cases important properties of the group, such as asphericity and cubulability.

Up to quasi-isometry, finitely generated groups admit a canonical left-invariant metric, making large-scale geometric invariants into group-theoretic invariants.  Are there other, non-finitely generated abstract groups with this property?  In this talk, we exhibit such examples by "going against nature"—stripping the topology from several large, rich families of topological transformation groups (e.g., homeomorphism groups of manifolds) and showing that they nevertheless admit canonical large-scale geometries as abstract groups.

This talk is about the interaction of foliations, 3-manifold topology and dynamics.  Pseudo-Anosov flows on 3-manifolds are dynamical systems generalizing the behavior of geodesic flow on the unit tangent bundle of a hyperbolic surface. Like geodesic flows, they come with two transverse, invariant 2-dimensional foliations (possibly with some prong singularities) which meet along the 1-dimensional foliation by orbits. Because of various surgery techniques, there are many examples known, and their "topological" classification is an interesting and important problem both in low-dimensional geometric topology and dynamics. I will describe some of this framework, and then some joint work with T. Barthelmé, S. Frankel, S. Fenley and C. Bonatti, on describing the structure and classification of such flows and their associated foliations.  

We consider the problem of generalizing the result of the speaker and Vladimir Markovic, building nearly geodesic surfaces in a closed hyperbolic 3-manifold M out of ``good pants’’ in M that are identified along their cuffs. The general setting is a semisimple Lie group G, an sl_2(R) subgroup H, and a cocompact lattice Gamma < G. The first problem is how to define good pants groups and their feet, such that a good matching of these pants produces a surface subgroup of Gamma that is ``closely aligned’’ with H. We’ll present several solutions to this problem coming from joint work with François Labourie, Shahar Mozes, and Zhenghao Rao. We’ll also discuss the problems that can arise with matching the pants and how they may be solved.