Unless noted, all seminars are on Friday at 4pm, in Mathematics 122. 201617 seminar listing can be found here.
Feb 2  Yulan Qing (Toronto): Convexity of balls in the outer space
Abstract: In this talk we answer questions regarding the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop α, the length of α along a balanced folding path is not larger than the maximum of its lengths at the end points. This implies that outgoing balls are weakly convex. We then show that these results are sharp by providing several counterexamples.


Feb 23  David Cohen (Chicago): Strongly aperiodic subshifts of finite type on oneended hyperbolic groups
Abstract: We discuss the ways in which the geometry of a group $G$ constrains the possible behavior of symbolic dynamical systems over $G$. In particular, we explain our results with Chaim GoodmanStrauss and Yoav Rieck on SFTs over hyperbolic groups.

Mar 16  Abdalrazzaq Zaollum (Buffalo): Contracting geodesics in CAT(0) groups
Abstract: The geodesicity condition for Gromov delta hyperbolic groups is a coarse local condition, in other words, in order to check whether a given edge path is a geodesic you need only to check that subsegments of length $m$, where $m$ depends on delta, are geodesics. The formal way of stating the above is to say that hyperbolic groups admit a regular language that reads all geodesics in the group. This seemingly simple observation has a lot of interesting consequences. For example, one can use it to show that hyperbolic groups have a rational growth function. Another consequence is the fact that the boundary of a hyperbolic group is a subshift of finite type (over $\mathbb{Z}$). Charney and Sultan introduced the notion of contracting boundaries for CAT(0) spaces, in short, these are the collection of all "hyperbolic directions" in a CAT(0) space. This talk will be about our theorem with Josh Eike proving that contracting geodesics in CAT(0) groups are realized by a regular language.

Apr 20  Bill Menasco (Buffalo): Distance and intersection number in the curve complex
Abstract: Let $S_g$ be a closed oriented surface of genus $g \geq 2$ and $\mathcal{C}^1(S_g)$ be its curve complex—vertices are homotopy classes of essential simple closed curves with two vertices sharing an edge if they have disjoint representatives. It is known that $\mathcal{C}(S_g)$ is path connected , and the distance, $d(\alpha , \beta)$, between two vertices $\alpha , \beta \in \mathcal{C}^1(S)$ is just the minimal count of the number of edges in an edgepath between $\alpha$ and $\beta$. One can also consider, $ i(\alpha , \beta)$, the minimal intersection between curve representatives of $\alpha$ and $\beta$. This talk discusses how $i(\alpha , \beta)$ will grow as $d(\alpha, \beta)$ grows. This is joint work with Dan Margalit.

May 4  Angelica Deibel (Brandeis): Random Coxeter groups
Abstract: Random rightangled Coxeter groups have been studied extensively using methods and results from random graph theory. Some of these methods can be extended to study random Coxeter groups in general. In this talk, I will introduce random Coxeter groups and give several results.

May 11  Bill Menasco (Buffalo): Distance and intersection number in the curve complex (Continued)
Abstract: Let $S_g$ be a closed oriented surface of genus $g \geq 2$ and $\mathcal{C}^1(S_g)$ be its curve complex—vertices are homotopy classes of essential simple closed curves with two vertices sharing an edge if they have disjoint representatives. It is known that $\mathcal{C}(S_g)$ is path connected , and the distance, $d(\alpha , \beta)$, between two vertices $\alpha , \beta \in \mathcal{C}^1(S)$ is just the minimal count of the number of edges in an edgepath between $\alpha$ and $\beta$. One can also consider, $ i(\alpha , \beta)$, the minimal intersection between curve representatives of $\alpha$ and $\beta$. This talk discusses how $i(\alpha , \beta)$ will grow as $d(\alpha, \beta)$ grows. This is joint work with Dan Margalit.

Fall 2017
Sep 8  Piotr Hajac (Polish Academy of Sciences): There and back again: from the BorsukUlam theorem to quantum spaces
Abstract: Assuming that both temperature and pressure are continuous functions, we can conclude that there are always two antipodal points on Earth with exactly the same pressure and temperature. This is the twodimensional version of the celebrated BorsukUlam Theorem which states that for any continuous map from the ndimensional sphere to ndimensional real Euclidean space there is always a pair of antipodal points on the sphere that are identified by the map. Our quest to unravel topological mysteries in the Middle Earth of quantum spaces will begin with gentle preparations in the Shire of elementary topology. Then, after riding swiftly through the Rohan of $C^\ast$algebras and GelfandNaimark Theorems, and carefully avoiding the Mordor of incomprehensible technicalities, we shall arrive in the Gondor of compact quantum groups acting freely on unital $C^\ast$algebras. It is therein that the generalized BorsukUlamtype statements dwell waiting to be proven or disproven. To end with, we will pay tribute to the ancient quantum group $SUq(2)$, and show how the proven nontrivializability of Pflaum's $SUq(2)$principal instanton bundle is a special case of two different noncommutative BorsukUlamtype conjectures. Time permitting, we shall also explain a general method to extend the nontrivializability result from the Pflaum quantum instanton bundle to an arbitrary finitely iterated equivariant join of $SUq(2)$ with itself. The latter is a quantum sphere with a free $SUq(2)$action whose space of orbits defines a quantum quaternionic projective space. (Based on joint work with Paul F. Baum, Ludwik Dabrowski, Tomasz Maszczyk and Sergey Neshveyev.)


Sep 22  Devin Murray (Brandeis): Groups, rigidity, and the Morse boundary
Abstract: In geometric group theory a finitely presented group is often
identified with its Cayley graph. Unfortunately, the Cayley graph
depends on the generating set of the group. When the generating set is
changed the homeomorphism type of the graph can change quite a lot.
However, there is a metric equivalence relation called a quasiisometry
(QI) and the Cayley graph *is* well defined up to quasiisometries.
Unfortunately quasiisometries are a very weak geometric equivalence
(in general there may be no continuous bijections between quasi
isometric spaces!). In this context a very natural question to ask is,
if two groups are quasiisometric how similar are they as groups? For
some classes of groups the answer is quite surprising! For example, if
a group $G$ is QI to $\pi_1(S)$ where $S$ is a surface group, then $G$ is
(virtually) isomorphic to a surface group! A class of groups which
exhibits this property is called quasiisometrically rigid.
Many classes of groups are quasiisometrically rigid, free abelian
groups, finite volume discrete subgroups of a noncompact Lie group,
surface groups, etc. Often, some form of negative curvature plays an
important role in proving rigidity style theorems.
The Morse property is a natural generalization of the behavior of
geodesics in negatively curved manifolds. The Morse boundary is a
quasiisometry invariant for finitely generated groups that serves a
similar role that the boundary of hyperbolic nspace serves for
hyperbolic manifold groups. I will introduce all of the objects in
question and talk about some results for the morse boundary that make
it a promising tool to study the quasiisometric rigidity phenomena of
some classes of groups, in particular some CAT(0) groups, Mapping Class
groups, some artin/coxeter groups etc.
This is joint work with Ruth Charney and Matt Cordes.

Back to homepage