We introduce the notion of a geodesic current with corners, a generalization of a geodesic current in which there are singularities (the ``corners'') at which invariance under the geodesic flow can be violated. Recall that the set of closed geodesics is, in the appropriate sense, dense in the space of geodesic currents; the motivation behind currents with corners is to construct a space in which graphs on S play the role of closed curves. Another fruitful perspective is that geodesic currents reside "at infinity'' in the space of currents with corners, in the sense that their (non-existent) corners have been pushed out to infinity. As an application, we count triangulations in a mapping class group orbit with respect to length, and we obtain asymptotics that parallel results of Mirzakhani, Erlandsson-Souto, and Rafi-Souto for curves. This represents joint work with Jayadev Athreya.

The Galvin-Prikry theorem states that Borel subsets of the Baire space are Ramsey. Silver extended this to analytic sets, and Ellentuck gave a topological characterization of Ramsey sets in terms of the property of Baire in the Vietoris topology.

We present work extending these theorems to several classes of countable homogeneous structures. An obstruction to exact analogues of Galvin-Prikry or Ellentuck is the presence of big Ramsey degrees. We will discuss how different properties of the structures affect which analogues have been proved. Presented is work of the speaker for structures with SDAP$^+$+ and joint work with Zucker for binary finitely constrained FAP classes. A feature of the work with Zucker is showing that we can weaken one of Todorcevic’s four axioms guaranteeing a Ramsey space, and still achieve the same conclusion.

Given a set $X$X in the Euclidean plane $\mathbb{R}^2$R2 and a point $p \in X$p∈X, we say $p$p is accessible if there exists an arc $A \subset \mathbb{R}^2$A⊂R2 such that $A \cap X = \{p\}$A∩X={p}. This is an old and vital notion in plane topology and complex analysis, dating back to Schoenflies in the early 1900's.

For a given planar continuum $X$X, in different embeddings of $X$X in $\mathbb{R}^2$R2, the set of points of $X$X which are made accessible may vary. One may ask, then, for a given point $p \in X$p∈X, does there exist an embedding $\varphi$φ of $X$X into $\mathbb{R}^2$R2 for which $\varphi(p)$φ(p) is accessible, or is there some topological obstruction in $X$X which forces $p$p to be inaccessible in every embedding?

In 1972, Nadler and Quinn asked a question in this spirit: For any arc-like continuum $X$X, and any point $p \in X$p∈X, does there exist an embedding $\varphi$φ of $X$X into $\mathbb{R}^2$R2 for which $\varphi(p)$φ(p) is accessible? I will discuss some background for this problem, and describe our recent work in which we give an affirmative answer. This is joint work with Andrea Ammerlaan and Ana Anusic.

This is a joint work with Benjamin Espinoza, Alejandro Illanes, Hayato Imamura and Yoshiyuki Oshima. In this presentation, we discuss some recent results on decomposable continua, in particular, Wilder continua, continuum-wise Wilder continua, closed set-wise Wilder continua, $D$D-continua, $D^{*}$D∗-continua and $D^{**}$D∗∗. First, we introduce some basic definitions and terminology that are used in this talk. Then, we discuss results on the above continua related to Whitney properties and Whitney reversible properties. Also, we deal with product properties as for those continua. Finally, we show the existence of singular decomposable continua using those notion.

Since the space of all cubic polynomials is (complex) two-dimensional and thus too difficult to comprehend, we study a one-dimensional slice of it: the space of all cubic symmetric polynomials of the form $f(z)=z^3+\lambda^2 z$f(z)=z3+λ2z. Thurston has built a topological model for the space of quadratic polynomials $f(z)=z^2+c$f(z)=z2+c by introducing the notion of quadratic invariant laminations. In the spirit of Thurston’s work, we parametrize the space of cubic symmetric laminations and create a model for the space of cubic symmetric polynomials. This is a joint work with Alexander Blokh, Lex Oversteegen, Vladlen Timorin, and Sandeep Vejandla.

In 1982, Leighton proved that any two finite graphs with a common cover admits a finite sheeted common cover. In this talk, I will introduce the combinatorial model X_{m,n} for Baumslag-Solitar group BS(m,n), and classify for which pairs of integers (m,n) the Leighton's theorem can be extended to the orbit space of covering actions on X_{m,n}.

Given a subshift $(X,\sigma)$(X,σ), the mapping class group $\mathcal{M}(\sigma)$M(σ) is the group of self-flow equivalences of $(X,\sigma)$(X,σ), up to isotopy. For a minimal shift, there is an embedding $\textrm{Aut}(X)/\langle \sigma \rangle \xhookrightarrow{} \mathcal{M}(\sigma)$Aut(X)/⟨σ⟩ ​M(σ), where $\textrm{Aut}(X)$Aut(X) is the group of automorphisms.

If $(X,\sigma)$(X,σ) is conjugate to a primitive substitutive shift, then $\mathcal{M}(\sigma)$M(σ) is a finite extension of $\mathbb{Z}$Z, and under mild conditions, this finite group is precisely $\textrm{Aut}(X)/\langle \sigma \rangle$Aut(X)/⟨σ⟩.

We discuss more the general case when $(X,\sigma)$(X,σ) is a minimal subshift of linear complexity, subject to a technical condition, and give some examples.

This is joint work with Scott Schmieding.

Distortion problems, from Banach space geometry, ask about the possibility of distorting the norm of a Banach space in a significant way on all of its subspaces. Big Ramsey degree problems, from combinatorics, are about proving weak analogues of the infinite Ramsey theorem in sets carrying structure. Both topics come back to the seventies and are still not well understood. While their motivations are quite disjoint, both problems share a surprisingly similar flavour.

In a ongoing work with Tristan Bice, Jan Hubička and Matěj Konečný, as a step forward towards the unification of those two topics, we developped an analogue of big Ramsey degrees adapted to the study of metric structures (metric spaces, Banach spaces...). Those metric big Ramsey degrees are compacts metric spaces which are invariants associated to certain monoid actions by isometry, quantifying their default of Ramseyness. We were able to prove the existence of big Ramsey degrees for certain classical metric structures and in some cases, to give an explicit description of them ; it also seems that some classical invariants from topological dynamics can be represented as big Ramsey degrees.

In this talk, I will present this theory, illustrate it on concrete examples (the Urysohn sphere and the Banach space $\ell_\infty$ℓ∞​) and give an overview of its motivations and potential applications (to Banach space theory, Ramsey theory and dynamics).