The notion of prime decomposition will be defined and illustrated for
manifolds. Two proofs of existence will be given, including Kneser's
classical proof using normal surface theory.

Given a vector space V over a field K, let End(V) denote the algebra of linear endomorphisms of V. If V is finite-dimensional, then it is well-known that the diagonalizable subalgebras of End(V) are characterized by their internal algebraic structure: they are the subalgebras isomorphic to K^n for some natural number n. 

In case V is infinite dimensional, the diagonalizable subalgebras of End(V) cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic.  I will explain how to characterize the diagonalizable subalgebras of End(V) as topological algebras, using a natural topology inherited from End(V).  I will also illustrate how this characterization relates to an infinite-dimensional Wedderburn-Artin theorem that characterizes "topologically semisimple" algebras.

Let $G(n,d)$ be a random $d$-regular graph on $n$ vertices. In 2004 Kim and Vu showed that if $d$ grows faster than $\log n$ as $n$ tends to infinity, then one can define a joint distribution of $G(n,d)$ and two binomial random graphs $G(n,p_1)$ and $G(n,p_2)$ -- both of which have asymptotic expected degree $d$ -- such that with high probability $G(n,d)$ is a supergraph of $G(n,p_1)$ and a subgraph of $G(n,p_2)$. The motivation for such a coupling is to deduce monotone properties (like Hamiltonicity) of $G(n,d)$ from the simpler model $G(n,p)$. We present our work with A. Dudek, A. Frieze and A. Rucinski on the Kim-Vu conjecture and its hypergraph counterpart.

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.