In this talk, we will introduce the notions of systolic and residual girth growth for finitely generated groups. We will explore the relationship between these types of growth and the usual word growth for finitely generated groups.
Associahedra are polytopes introduced by Stasheff to encode topological semigroups in which associativity holds up to coherent homotopy. These polytopes naturally form a topological operad that gives a resolution of the associative operad. Muro and Tonks recently introduced an operad which encodes $A_\infty$ algebras with homotopy coherent unit.
The material in this talk will be fairly basic. I will cover operads and their algebras, give the construction of the $A_\infty$ operad using the Boardman-Vogt resolution, and of the unital associahedra introduced by Muro and Tonks.
Depending on time and interest of the audience I will define unital $A_\infty$ differential graded algebras and explain how they are precisely the algebras over the cellular chains of the operad constructed by Muro and Tonks.
I will discuss various types of filling functions on topological spaces, stating some results in the area. I will then go onto prove that a finitely presented subgroup of a hyperbolic group of cohomological dimension 2 is hyperbolic. On the way I will prove a stronger result about filling functions of subgroups of hyperbolic groups of cohomological dimension $n$.
In this talk I will try to show how certain asymptotic properties of a random walk on a graph are related to geometric properties of the graph itself. A special focus will be put on spectral properties and isoperimetric inequalities, proving Kesten's criterion for amenability.
We say a group is accessible if the process of iteratively decomposing G as an amalgamated free product or HNN extension over a finite group terminates in a finite number of steps. We will see Dunwoody's proof that FP2 groups are accessible, but that finitely generated groups need not be. If time permits, we will examine generalizations by Bestvina-Feighn, Sela and Louder.
This talk will be an introduction to the weird and wonderful world of Thompson's groups $F$, $T$ and $V$. For example, the group $T$ was the first known finitely presented infinite simple group, $V$ has a finitely presented subgroup with co-NP-complete word problem, and whether or not $F$ is amenable is an infamous open problem.
Many hard problems in Diophantine geometry have analogues over function fields which are less hard. I will give some examples.
With the general election looming upon I will discuss the various different kinds of voting system that one could implement in such an election. I will show that these can give very different answers to the same set of voters. I will then discuss Arrow's Impossibility Theorem which shows that no voting system is compatible with 4 simple axioms which may be desireable.
A result of Jeffrey Brock states that, given a hyperbolic 3-manifold which is a mapping torus over a surface $S$, its volume can be expressed in terms of the distance induced by the monodromy map in the pants graph of $S$. This is an abstract graph whose vertices are pants decompositions of $S$, and edges correspond to some 'elementary alterations' of those.
I will show how this theorem gives an estimate for the volume of hyperbolic complements of closed braids in the solid torus, in terms of braid properties. The core piece of such estimate is a generalization of a result of Masur, Mosher and Schleimer that train track splitting sequences (which I will define in the talk) induce quasi-geodesics in the marking graph.