In this talk we will review some of the main ideas around Hamilton's program for the Ricci flow and see how they fit together to provide a proof of the Poincaré conjecture in dimension 3. We will then analyse this tools in the context of 4-manifolds.

# Past Kinderseminar

Manifolds have been a central object of study for over a century, and the classification of them has been a core theme for the whole of this time. This talk will give an overview of the successes and failures in this effort, with some illustrative examples.

The Banach-Tarski paradox is a celebrated result showing that, using the axiom of choice, it is possible to deconstruct a ball into finitely many pieces that may be rearranged to build two copies of that ball. In this seminar we will sketch the proof of the paradox trying to emphasize the key ideas.

Given a commutative ring A with ideal m, we consider the formal completion of A at m, and we ask when algebraic structures over the completion can be approximated by algebraic structures over the ring A itself. As we will see, Artin's approximation theorem tells us for which types of algebraic structures and which pairs (A,m) we can expect an affirmative answer. We will introduce some local notions from algebraic geometry, including formal and etale neighbourhoods. Then we will discuss some algebraic structures and rings arising in algebraic geometry and satisfying the conditions of the theorem, and show as a corollary how we can lift isomorphisms from formal neighbourhoods to etale neighbourhoods of varieties.

Modularity is a quality function on partitions of a network which aims to identify highly clustered components. Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity q(G) of G is the maximum modularity of a partition of V(G). Knowledge of the maximum modularity of the corresponding random graph is important to determine the statistical significance of a partition in a real network. We provide bounds for the modularity of random regular graphs. Modularity is related to the Hamiltonian of the Potts model from statistical physics. This leads to interest in the modularity of lattices, which we will discuss. This is joint work with Colin McDiarmid.

The curve graph of a surface has a vertex for each curve on the surface and an edge for each pair of disjoint curves. Although it deals with very simple objects, it has connections with questions in low-dimensional topology, and some properties that encourage people to study it. Yet it is more complicated than it may look from its definition: in particular, what happens if we start following a 'diverging' path along this graph? It turns out that the curves we hit get so complicated that eventually give rise to a lamination filling up the surface. This can be understood by drawing some train track-like pictures on the surface. During the talk I will keep away from any issue that I considered too technical.

Cluster algebras are commutative algebras generated by a set S, obtained by an iterated mutation process of an initial seed. They were introduced by S. Fomin and A. Zelevinski in connection with canonical bases in Lie theory. Since then, many connections between cluster algebras and other areas have arisen.

This talk will focus on cluster algebras for which the set S is finite. These are called cluster algebras of finite type and are classified by Dynkin diagrams, in a similar way to many other objects.

Stallings' folding technique lets us factor a map of graphs as a sequence of "folds" (edge identifications) followed by an immersion. We will show how this technique gives an algorithm to express a free-group automorphism as the product of Whitehead automorphisms (and hence Nielsen transformations), as well as proving finite generation for some subgroups of the automorphism group of a free group.

Given any family of varieties over a number field, if we have that the existence of local points everywhere is equivalent to the existence of a global point (for each member of the family), then we say that the family satisfies the Hasse principle. Of more interest, in this talk, is the case when the Hasse principle fails: we will give an overview of the "geography" of the currently known obstructions.