Sep 9

Bill Menasco (UB): Surface Embeddings in $\mathbb{R}^2 \times \mathbb{R}$
Abstract: In this joint work with Margaret Nichols, we consider $\mathbb{R}^3$ as having the product structure $\mathbb{R}^2 \times \mathbb{R}$ and let $\pi : \mathbb{R}^2 \times \mathbb{R} \rightarrow \mathbb{R}^2$ be the natural projection map onto the Euclidean plane. Let $ \epsilon : S_g \rightarrow \mathbb{R}^2 \times \mathbb{R}$ be a smooth embedding of a closed oriented genus $g$ surface such that the set of critical points for the map $\pi \circ \epsilon$ is a piecewise smooth (possibly multicomponent) $1$manifold, $\mathcal{C} \subset S_g$. We say $\mathcal{C}$ is the $\textit{crease set}$ of $\epsilon$ and two embeddings are in the same $\textit{isotopy class}$ if there exists an isotopy between them that has $\mathcal{C}$ being an invariant set. The case where $\pi \circ \epsilon _\mathcal{C}$ restricts to an immersion is readily accessible, since the turning number function of a smooth curve in $\mathbb{R}^2$ supplies us with a natural map of components of $\mathcal{C}$ into $\mathbb{Z}$. The GaussBonnet Theorem beautifully governs the behavior of $\pi \circ \epsilon (\mathcal{C})$, as it implies $\chi(S_g) = 2 \sum_{\gamma \in \mathcal{C}} t( \pi \circ \epsilon (\gamma))$, where $t$ is the turning number function. Focusing on when $S_g \cong S^2$, we give a necessary and sufficient condition for when a disjoint collection of curves $\mathcal{C} \subset S^2$ can be realized as the crease set of an embedding $\epsilon: S^2 \rightarrow \mathbb{R}^2 \times \mathbb{R}$.

Sep 16

Yulan Qing (Fudan University/University of Toronto): Gromov boundary extended
Abstract: Gromov boundary provides a useful compactification for all infinitediameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given basepoint and it has been an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce two topological spaces that are natural analogs of the Gromov boundary for a larger class of metric spaces. First we construct the sublinearly Morse boundaries and show that it is a QIinvariant topological space that can be associated to all finitely generated groups. Furthermore, for many groups, the sublinear boundary can be identified with the Poisson boundaries of the associated group, thus providing a QIinvariant model for Poisson boundaries. This result answers the open problems regarding QIinvariant models of CAT(0) groups and the mapping class group. Lastly, for a subset of the metric spaces we define a compactification of the sublinearly Morse boundary and show that in these cases they are naturally identified with the Bowditch boundary. This is a series of joint work with Kasra Rafi and Giulio Tiozzo.

Sep 23

Abdul Zalloum (University of Toronto): Hyperbolic models for CAT(0) spaces
Abstract: Two of the most wellstudied topics in geometric group theory are CAT(0) cube complexes and mapping class groups. This is in part because they both admit powerful combinatoriallike structures that encode their (coarse) geometry: hyperplanes for the former and curve graphs for the latter. In recent years, analogies between the two theories have become more apparent. For instance: there are counterparts of curve graphs for CAT(0) cube complexes and rigidity theorems for such counterparts that mirror the surface setting, and both can be studied using the machinery of hierarchical hyperbolicity. However, the considerably larger class of CAT(0) spaces is left out of this analogy, as the lack of a combinatoriallike structure presents a difficulty in importing techniques from those areas. In this talk, I will speak about recent work with Petyt and Spriano where we bring CAT(0) spaces into the picture by developing analogues of hyperplanes and curve graphs for them. The talk will be accessible to everyone, and all the aforementioned terms will be defined.

Oct 14

Bojun Zhao (UB): Left orderability and taut foliations with onesided branching
Abstract: For a closed orientable irreducible 3manifold $M$ that admits a coorientable taut foliation with onesided branching, we show that $\pi_1(M)$ is left orderable.

Oct 21

José Román Aranda Cuevas (Binghamton University): Bounds for Linvariants of knotted surfaces
Abstract: Take two 3dimensional handlebodies with the same boundary surface. One can tell them apart by studying the curves on the boundary surface bounding disks on each handlebody. Hempel studied Heegaard splittings of closed 3manifolds by comparing these disk sets in the curve complex. For trisections of 4manifolds, one can measure the length of loops in some complex passing through the disk set of each 3dimensional handlebody. Kirby and Thompson used cut systems this way to define the Linvariant of a trisection of a closed 4manifold. Other authors extended this definition for relative trisections and bridge trisections. Naturally, L is hard to compute. We will discuss lower bounds for $(b,c)$bridge trisections of closed surfaces. This is joint work with Taylor, Pongtanapaisan, and Zhang.
