Celebrated waist estimates by Gromov and others assert that a map (between, say, Riemannian manifolds) dropping dimension must have a fiber of large volume. We discuss similar questions with volume of a fiber being replaced by its Urysohn \(d\)-width, a certain quantity measuring how effectively a space can be approximated by a \(d\)-dimensional one. For example, a map \(S^4 \to Y^1\) from the unit \(4\)-sphere to a graph always has a fiber of large 1-width, but it might happen that all fibers have small 2-width and are topologically simple.

Let \(\text{Mod}(S_g)\) be the mapping class group of the closed orientable surface \(S_g\) of genus \(g\geq 2\). In this talk, we state necessary and sufficient conditions under which two finite-order mapping classes have representatives in their respective conjugacy classes that commute in \(\text{Mod}(S_g)\). As an application, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate that is liftable under a finite cover. Furthermore, we show that any torsion element in the centralizer of an irreducible finite order mapping class is of order at most \(2\). We also state equivalent conditions for the primitivity of a torsion element of \(\text{Mod}(S_g)\). Finally, we describe a procedure for determining the explicit hyperbolic structures that realize two-generator finite abelian groups of \(\text{Mod}(S_g)\) as isometry groups.

I invented a unifying approach to all important compactifications. I will apply it to create a singular approach to boundaries of hyperbolic spaces and to boundaries of CAT(0)-spaces.

Quantitative Topology studies the relationship between topological invariants of metric spaces and the 'metric complexity' of these invariants. Some tools from topology, such as Sullivan's rational homotopy theory, can readily be made quantitative. With other tools, the relationship between the topology and the metric is less clear In this talk I will discuss how two such tools from topology apply in a quantitative setting: the bar and cobar constructions. Work of K.-T. Chen models the cohomology of loop spaces through the bar construction and I show that this gives quantitative bounds on the cohomology. Dually, work of Adams and Hilton models the homology of a loop space with the cobar construction. However here the quantitative implications are more subtle.

We construct a space \(\mathbb{P}\) for which the canonical homomorphism \(\pi_1(\mathbb{P},p) \rightarrow \check{\pi}_1(\mathbb{P},p)\) from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following properties: (1) \(\mathbb{P}\setminus\{p\}\) is a 2-manifold with connected non-compact boundary; (2) \(\mathbb{P}\) is connected and locally path connected; (3) \(\mathbb{P}\) is strongly homotopically Hausdorff; (4) \(\mathbb{P}\) is homotopically path Hausdorff; (5) \(\mathbb{P}\) is 1-UV\(_0\); (6) \(\mathbb{P}\) admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) \(\pi_1(\mathbb{P},p)\) naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8) \(\pi_1(\mathbb{P},p)\) is locally free.

This will be a very brief, but hopefully gentle, introduction to sheaf cohomology and some of the things it is good for. No prior experience with sheaves will be expected.

Combined work of Eels-Sampson and Hartman asserts the existence of a harmonic diffeomorphism in any homotopy class of maps between a pair of homeomorphic compact hyperbolic surfaces. I will discuss the background theory, present a suitable discretization, and locate discrete harmonic maps by applying a constant step gradient descent method, where convergence is guaranteed by explicit computations in the hyperbolic plane. In particular, we show that the discrete energy functional is strongly convex, a uniform statement not implied by the existing literature. Time permitting, I will discuss a computer implementation that exploits the above viewpoint, and the delicate problem of convergence of the discrete maps to the smooth harmonic map.

Connected sums and boundary connected sums play important roles in the study of closed manifolds and manifolds with boundary, respectively. When working with open manifolds, a third variety of connected sum - the "end sum" - is useful. Each of these operations involves a number of arbitrary choices, making well-definedness of the resulting manifold a significant question. With regards to the end sum operation, we will discuss familiar situations where all goes smoothly (the notion of semistability plays a role here) and others where significant problems arise. Our analysis leads naturally into the subtle and interesting theory of infinitely generated abelian groups.

Since the '70s, the topological Cappell-Shaneson 4-spheres have been considered as possible counterexamples to the differentiable Poincaré conjecture in dimension 4. Over the years, increasingly larger subfamilies of these 4-spheres have been shown to be diffeomorphic to the standard 4-sphere, some by simplifying their Kirby diagram (so far available only for a special class), and some by other means.

By representing a Cappell-Shaneson 4-sphere as the result of a side-pairing of the 4-cube, we obtain a Kirby diagram for the general case and simplify it to two 1-handles and two 2-handles. The diagrams for the special class simplify to the standard 4-sphere in a straightforward way, suggesting our approach may be promising for the general case.

We will show that isospectrality is equivalent to Gassmann equivalence for commensurable covers of a common surface, provided a certain collection of Selberg \(L\)-functions, associated to a regular cover of the base surface, are multiplicatively independent. We show that when the base surface is associated to a cocompact maximal arithmetic lattice in \(PGL(2,{\mathbb R})\), and the cover is principal congruence, the relevant \(L\)-functions are indeed multiplicatively independent. We will show how to apply this result to produce infinitely many infinite families of spectrally rigid surfaces.

If a topological space has a universal cover, then every subgroup of the fundamental group corresponds to a covering space. This correspondence allows one to study the fundamental group using covers spaces and vice versa. For spaces, which do not have a universal cover, the situation is more complicated and requires new techniques. One approach is to consider inverse limits of coverings space, which are a special class of unique path lifting fibrations called covering fibrations. We will discuss when a covering fibration is path connected and how this leads to a splitting of the first homology for many spaces.

We discuss a combinatorial method for finding the topological conformation of DNA bound within a protein complex. We use generalized invariants called virtual colorability and search for possible DNA conformations. We apply this method to generalize the classical knot theory cases in order to determine the topological conformation of DNA bound within a stable protein-DNA complex.

In 2012, V. Fedorchuk, using \(m\)-pairs and \(n\)-partitions, introduced the notion of the \((m,n)\)-dimension of a space. It generalizes covering dimension; Fedorchuk showed that \((m,n)\)-dimension is preserved in inverse limits of compact Hausdorff spaces. We separately have characterized those approximate inverse systems of compact metric spaces whose limits have a specified \((m,n)\)-dimension. Our characterization is in terms of internal properties of the system. Here we are going to give a parallel internal characterization of those inverse systems of compact Hausdorff spaces whose limits have a specified \((m,n)\)-dimension. Fedorchuk's limit theorem will be a corollary to ours.

Geodesic currents are measures that realize a closure of the space of curves on a closed surface. Bonahon introduced geodesic currents in 1986, showed that geometric intersection number extends to geodesic currents and realized hyperbolic length of a curve as intersection number with a distinguished geodesic current. Later, other functions were shown to be intersection numbers, such as negatively curved Riemannian lengths (Otal, 1990) or, recently, word length w.r.t. simple generating sets of a surface group (Erlandsson, 2016). Some other functions aren't intersection numbers, such as word length w.r.t. non-simple generating sets or extremal length. In this talk we answer the following natural question: what are sufficient conditions for a function on curves to be an intersection number?

A 40+ year old question asks if all finitely presented groups have semistable fundamental group at \(\infty\). In 1986 the semistability definition was extended to the class of finitely generated groups and successfully used to show certain classes of finitely presented groups have semistable fundamental group at \(\infty\). Many finitely generated groups have been proposed as candidates to be non-semistable, but up to this point no one has proven that such a group exists. In this talk, we will show that the Lamplighter group does not have semistable fundamental group at \(\infty\). The technique of proof extends to show all finitely generated groups in a significant class of groups fail to have semistable fundamental group at \(\infty\). We discuss prospects for finding a finitely presented group with non-semistable fundamental group at \(\infty\).

A hidden symmetry of a hyperbolic 3-manifold M is a symmetry of a finite-sheeted cover of M that doesn't come from a symmetry of M. When we restrict to hyperbolic knot complements, hidden symmetries seem to be quite rare though it is difficult to show that an infinite set of knot complements doesn't admit hidden symmetries. In this talk, we will examine how the geometric structures of sufficiently twisted knot complements can be exploited to show that such knot complements have no hidden symmetries and very few symmetries. As a result, these knot complements are usually the only knot complements in their respective commensurability classes.

We prove that the space of persistence diagrams on n points (with either the Bottleneck distance or a Wasserstein distance) coarsely embeds into Hilbert space. Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also discuss various non-embeddability results when the number of points is not bounded.

Bestvina defined a \(Z\)-structure on a group \(G\) to generalize the theory of boundaries of CAT(0) and hyperbolic groups. An interesting question to ask in this setting is: given two groups that are quasi-isometric where one group is known to admit a \(Z\)-structure, must the other group also admit a \(Z\)-structure? From previous work, we have a partial answer to this question, but there is a roadblock in giving a complete answer. In this talk, we will discuss this roadblock and provide a modified definition of \(Z\)-structures where we hope to give a more complete answer to this question.

A non-negative numerical quantity \(I\) of a closed oriented manifold \(M\) is said to be functorial if whenever there is a map \(f\colon M \to N\) then \(I(M)\geq |\deg(f)|I(N)\). A prominent example is given by Gromov's simplicial volume. We will discuss vanishing and non-vanishing results of functorial numerical invariants for two classes of manifolds: In one direction, we show that there are non-vanishing functorial numerical invariants on each not virtually trivial circle bundle over an aspherical manifold with hyperbolic fundamental group, generalising thus in all dimensions a result of Brooks and Goldmann for circle bundles over hyperbolic surfaces. In another direction, we show that the simplicial volume of any mapping torus vanishes only in dimensions two and four. Part of this talk is based on joint work with Michelle Bucher.

A basic problem in the study of 3-manifolds is to determine when geometric objects are of 'minimal complexity'. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called 'tautness'.

One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called 'rational homology product'. Most sutured manifolds do not have this form, but do always take the more general form of a 'twisted homology product', which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such?

We explore the case of sutured handlebodies, and see even among the simplest class of these, twisting is required. We give examples that, when restricted to solvable representations, the twisting representation cannot be 'too simple'.

The collection \(\mathrm{Cov}(X)\) of open covers of a topological space \(X\) is a directed set with respect to the relation \(\preceq\) of refinement. This means that if \(\mathcal{U}\), \(\mathcal{V}\in\mathrm{Cov(X)}\), then there exists \(\mathcal{W}\in \mathrm{Cov}(X)\) with \(\mathcal{U}\preceq\mathcal{W}\) and \(\mathcal{V}\preceq\mathcal{W}\). If we are given that \(\mathcal{U}\preceq\mathcal{V}\), then a projection \(p_{\mathcal{U}\mathcal{V}}: \mathcal{V}\to\mathcal{U}\) (not necessarily uniquely defined) is a function such that if \(V\in\mathcal{V}\), then \(V\subset p_{\mathcal{U}\mathcal{V}}(V)\). Such a \(p_{\mathcal{U}\mathcal{V}}\) induces a simplicial map \(p_{\mathcal{U}\mathcal{V}}:|N(\mathcal{V})|\to|N(\mathcal{U})|\), where \(N(\mathcal{V})\), \(N(\mathcal{U})\) are the nerves of \(\mathcal{V}\) and \(\mathcal{U}\), and \(|N(\mathcal{V})|\), \(|N(\mathcal{U})|\) are their polyhedra with the weak topology on each. It is known, however, that although projections \(p_{\mathcal{U}\mathcal{V}}: |N(\mathcal{V})|\to|N(\mathcal{U})|\) and \(p_{\mathcal{U}\mathcal{V}}': |N(\mathcal{V})|\to|N(\mathcal{U})|\) can differ, they are always homotopic.

Since \((\mathrm{Cov}(X),\preceq)\) is a directed set, one might wonder if it is possible to select the projections so that the "Čech system''

\(\mathbf{U}=(|N(\mathcal{U})|,p_{\mathcal{U}\mathcal{V}}, (\mathrm{Cov}(X),\preceq))\)

is an inverse system. This would require that

(†) whenever \(\mathcal{U}\preceq \mathcal{V}\preceq\mathcal{W}\), then \(p_{\mathcal{U}\mathcal{W}}=p_{\mathcal{U}\mathcal{V}}p_{\mathcal{V}\mathcal{W}}.\)

The first author has proved that for any Hausdorff arc-like space \(X\), such a \(\mathbf{U}\) cannot even be an approximate inverse system in the sense of Mardešić and Watanabe, i.e., one in which the commutativity relations in the diagrams (†) are relaxed to certain "gauges of closeness.''

We prove that if \(X\) is a \(\mathrm{T}_1\)-paracompactum that contains a nontrivial component, then none of its Čech systems \(\mathbf{U}\) can be an approximate inverse system. This generalizes the result of Matijević because every Hausdorff arc-like space is also a nontrivial continuum, and hence is a \(\mathrm{T}_1\)-paracompactum containing a nontrivial component.

A construction of Stallings encodes the information of a Heegaard splitting as a continuous map between 2-complexes. We investigate this construction from a more geometric perspective and find that irreducible Heegaard splittings can be encoded as square complexes with certain properties.

When a surface S is non-closed, it can be framed, and the mapping class group acts on the set of framings; the "framed mapping class group'' is the stabilizer of a chosen framing. I will discuss work, joint with Aaron Calderon, which gives a generating set for framed mapping class groups "of holomorphic type''. Remarkably, they are generated by finitely many Dehn twists, despite being infinite-index subgroups. One application is to the study of Abelian differentials: we are able to completely describe the monodromy representations of all strata of differentials in genus at least 5.

The Tits Conjecture (proved by Crisp-Paris) says that the squares of the generators of an Artin group generate the "obvious" right angled Artin subgroup. The Generalized Tits Conjecture says that there should be a larger right angled Artin subgroup generated by high powers of centers of irreducible spherical Artin subgroups. I will explain how the braid group case follows from a result of Koberda, how we can show the conjecture holds for spherical Artin groups not of type \(E_7\) and \(E_8\), and give some applications.

Let \(X\) be an \(n\)-dimensional CW-complex. Then \(X\) is called essential if the classifying map \(X \to B \pi_1(X)\) cannot be deformed to the \((n-1)\)-skeleton of \(B \pi_1(X)\). In the context of LS-category, \(X\) is essential if and only if \(\text{cat}(X) = \text{dim}(X)\); hence, essential is the defining characteristic for having the maximum possible LS-category according to the dimension bound. But more generally, if \(X\) is \((r-1)\)-connected, then we have the bound \(\text{cat}(X) \leq \frac{\text{dim}(X)}{r}\). However, all essential CW-complexes necessarily have nontrivial fundamental groups, so the above perspective can't be used. We generalize essential to a notion called Postnikov essential in an attempt to get a theorem of the form \(X\) is Postnikov essential of type \(r\) if and only if \(\text{cat}(X) = \frac{\text{dim}(X)}{r}\). The backwards direction has been proven with only minor adjustments to the original proof, while the forward direction has currently only been proven in special cases.

In this talk we will begin by describing coarse proximities spaces which are structures in coarse geometry meant to capture the notion of "closeness at infinity". We then describe how to a construct a compact Hausdorff boundary on coarse proximity spaces. Many well known examples of boundaries such as the Higson corona and the Gromov boundary arise as boundaries of coarse proximity spaces. Time permitting we will characterize the covering dimension of boundaries using covers within corresponding coarse proximity spaces.

Let \(N\) be a smooth manifold that is homeomorphic but not diffeomorphic to a hyperbolic manifold \(M\). How much symmetry does \(N\) have? For example, does Isom(\(M\)) act on \(N\) by diffeomorphisms? Farrell-Jones showed that in general the answer is "No"; however, they do not rule out that the orientation-preserving subgroup of Isom(\(M\)) acts on \(N\). In this talk, we will discuss this problem and its relation to Nielsen realization. The main result is a construction of examples of \(N\) such that Isom(\(M\)) does act on \(N\), as well as examples where the largest subgroup of Isom(\(M\)) that acts on \(N\) has arbitrarily large index.

Triangulated Riemann surfaces are compact hyperbolic Riemann surfaces that admit a conformal triangulation by equilateral triangles. They arise naturally in number theory as Riemann surfaces defined over number fields, in probability theory as conjecturally related to Liouville quantum gravity, and in metric geometry as a model to understand arbitrary hyperbolic Riemann surfaces. In many ways, the geometry of triangulated Riemann surfaces mirrors the geometry of arbitrary hyperbolic Riemann surfaces, especially when we consider large genus asymptotics. This suggests that triangulated Riemann surfaces are asymptotically well-distributed in the moduli space of Riemann surfaces. We will describe results in this direction.

A weak tangent of a metric space is the limiting metric space when one zooms in to a metric space. One would wonder how the weak tangents of a metric space is related to the original metric space. In this talk, we explore some of these relationships. We are especially interested in the conformal gauge of weak tangents, i.e. the equivalence classes of weak tangents under quasisymmetry, and their relationships with the conformal gauge of the original class.