2021 Topology Festival

The Geometrization Theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means. Specifically, it states that every closed 3-manifold has a decomposition into geometric pieces, and the zoo of these geometric pieces is quite constrained: each is built from one of eight homogeneous 3-dimensional Riemannian model spaces (called the Thurston geometries).

In this talk, we will approach the question of "what does a 3-manifold look like" from the perspective of geometrization. Through animations of simple examples in dimensions 2 and 3 we review what it means to put a (complete, homogeneous) geometric structure on a manifold, and construct an example admitting each of the Thurston geometries.

Using software written in collaboration with Remi Coulon, Sabetta Matsumoto and Henry Segerman, we will explore these manifolds ``from the inside'' - that is, simulating the view one would have inside such a space by raytracing along geodesics. Finally we will touch on how to re-assemble these geometric pieces and understand an "inside view" of general 3-manifolds.

Many interesting groups are generated by torsion elements, for instance, mapping class groups, SL(n,Z) and Homeo+(S1). The word length with respect to this typically infinite generating set is called the torsion length. That is, the torsion length tl(g) of an element g is the smallest k such that g is the product of k torsion elements. The stable torsion length stl(g) is the limit of tl(gn)/n, which measures the growth of the torsion length. I will explain how to use topological methods (and planar surfaces) to compute stl(g) in free products of finite abelian groups. The nature of the method implies that stl(g) is always rational in these free products. This is joint work with Chloe Avery.

A major goal in complex dynamics is to understand dynamical moduli spaces; that is, conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to the complex plane. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. Many tools from complex analysis that pave the way for key breakthroughs in the one-dimensional setting do not carry over to higher dimensions. So instead of considering the whole moduli space, we follow an approach initiated by William Thurston and investigate special subvarieties of moduli space that give rise to dynamical moduli spaces. In this talk, we will explore the topology and geometry of the dynamical moduli spaces that play a prominent role in complex dynamics.

I will discuss the questions that arise when classifying smooth, simply connected manifolds with few non-zero homology groups. This mostly historical talk will begin with work of Milnor, Kervaire, Adams and Wall, and will end with recent joint work with Burklund and Senger. Perhaps surprisingly, the full classification relies on a good structural understanding of the Adams filtration on stable homotopy groups of spheres.

The fine curve graph is a hyperbolic graph which can be used to study homeomorphism groups of surfaces. Although it is inspired by the extremely successful curve graph for mapping class groups, I will assume no prior knowledge about the latter in this talk. After discussing some general properties of fine curve graphs, and how to use it to answer questions about diffeomorphism and homeomorphism groups, I will present recent results which connect the dynamics of torus homeomorphisms to their action on the fine curve graph. This is joint work with Jonathan Bowden, Kathryn Mann, Emmanuel Militon and Richard Webb.

This is joint work with Matei Coiculescu. Sol is one of the strangest of the 8 Thurston geometries. In this talk I will explain an exact criterion for when a geodesic segment in Sol is a distance minimizer. The criterion has to to with a classical Hamiltonian flow on the 2-sphere and also with the arithmetic-geometric mean of Gauss. Our characterization implies that the metric spheres in Sol are topological spheres, smooth away from a union of at most 4 arcs. I'll demonstrate the results with computer plots and animations.

Morita equivalences show up many places - for example, the construction of important trace maps in algebraic topology. I'll talk about a result that uses Mortia equivalence as the connection between maps induced on Hochschild homology by inclusion maps and the Hattori-Stallings trace. My approach to this result is very flexible - as a starting point it is equally accommodating to the topological refinements of these statements. Even more broadly, the underlying insight to this approach implies the familiar formulas for characters of induction and restriction of representations and demonstrates that Euler characteristic is multiplicative on fibrations.

To any topological space one may associate a differential graded coalgebra by considering formal linear combinations of simplices equipped with a boundary map which squares to zero and a compatible coproduct map approximating the diagonal map of the underlying space. This coproduct, also known as the Alexander-Whitney diagonal approximation, is strictly coassociative but symmetric up to an infinite family of higher homotopies. These higher homotopies may be constructed in a natural way resulting in an algebraic structure called an E-infinity coalgebra. In this talk I will describe how the fundamental group of a space is completely determined by a quadratic equation expressed using part of this E-infinity coalgebra "chain level" structure and how this observation fits with a homotopy theory for algebraic structures. The new idea, beyond technical details, is to regard the E-infinity coalgebra of chains under a notion of weak equivalence drawn from the Koszul duality theory for associative structures. This perspective opens up the possibility of removing strong restrictions on the fundamental group from the work of Quillen, Sullivan, Goerss, Mandell (and others) on the problem of classifying homotopy types through algebraic data. This is based on different collaborations with M. Zeinalian, F. Wierstra, and G. Raptis.

An old, fundamental problem is classifying closed n-manifolds admitting a metric of constant curvature. The most mysterious case is constant curvature -1, that is, hyperbolic manifolds, and these divide further into "arithmetic" and "nonarithmetic" manifolds. However, it is not at all evident from the definitions that this distinction has anything to do with the differential geometry of the manifold. Recently, Uri Bader, David Fisher, Nicholas Miller and I gave a geometric characterization of arithmeticity in terms of properly immersed totally geodesic submanifolds, answering a question due independently to Alan Reid and Curtis McMullen. I will give an overview, assuming only basic differential topology, of how (non)arithmeticity and totally geodesic submanifolds are connected, then describe how this allows us to import tools from ergodic theory and homogeneous dynamics inspired by groundbreaking work of Margulis on superrigidity to prove our characterization.

I will discuss a plan started joint with A. Gogolev to find rigidity properties in hyperbolic dynamics. This plan focuses on matching of smooth potentials for different dynamics. I will present a successful instance of this plan in the setting of expanding maps and some ongoing work in the setting of Anosov diffeomorphisms.

Phases of matter are classes of quantum systems that may look different microscopically but share certain common macroscopic properties. An active problem in condensed matter physics is to classify phases of matter. In certain cases, the tools of homotopy theory provide a useful mathematical framework. Specifically, certain classes of phases of matter are expected to form a generalized cohomology theory, a point of view introduced by Kitaev. This can be accessed via quantum field theories as in the work of Freed-Hopkins. Another problem is to get at this connection directly from the point of view of lattice models, which are idealizations of materials commonly used by condensed matter physicists. In this mostly expository talk, I will discuss the latter perspective, with some results on this topic which are part of joint work with Hermele, Moreno, Pflaum, Qi, Spiegel and Wen.

The study of translation surfaces is a rich blend of dynamics and algebraic geometry. One of the most fruitful perspectives has been to look at moduli spaces of translation surfaces known as strata. While there has been spectacular progress in understanding their dynamical and algebro-geometric properties, the topology and geometric group theory of strata is an almost total mystery. In spite of this, there are strong indications that this will develop into an immensely rich topic. The current state of knowledge is such that I can give an essentially complete overview of what is currently known in a single talk, with plenty of time left over to speculate about what might turn out to be true, and why geometric group theorists in particular should be dropping what they are doing to work on this stuff.