Past Kinderseminar

Two polyhedra are said to be scissors congruent if they can be subdivided into the same finite number of polyhedra such that each piece in the first polyhedron is congruent to one in the second. In 1900, Hilbert asked if there exist tetrahedra of the same volume which are not scissors congruent. I will give a history of this problem and its proofs, including an incorrect 'proof' by Bricard from 1896 which was only rectified in 2007.

The problem of computing the Galois group of an irreducible, rational polynomial has been studied for many years. I will discuss the methods developed over the years to approach this problem, and give some examples of them in practice. These methods mainly involve constructing and factorising resolvent polynomials, and thereby determining better upper bounds for the conjugacy class of the Galois group within the symmetric group, i.e. describe its action on the roots of the polynomial explicitly. I will describe how using approximations to the zeros of the polynomial allows us to construct resolvents, and in particular, how using p-adic approximations can be advantageous over numerical approximations, and how this can yield a direct and systematic method of determining the Galois group.

We will explore the many guises under which groups of 2x2 matrices appear, such as isometries of the hyperbolic plane, mapping class groups and the modular group. Along the way we will learn some interesting and perhaps surprising facts.

In 1924, James W. Alexander constructed a 2-sphere in R3 that is not ambiently homeomorphic to the standard 2-sphere, which demonstrated the failure of the Schoenflies theorem in higher dimensions. I will describe the construction of the Alexander horned sphere and the Antoine necklace and describe some of their properties.

Since the symmetric group is a finite group it’s representation theory is not too complex, however in this special case we can realise these representations in a particular nice combinatorial way using young tableaux and young symmetrizers. I will introduce these ideas and use them to describe the representation theory of Sn over the complex numbers.

I will talk about a remarkable theorem by Paulin, which says
that if a one-ended hyperbolic group has infinite outer automorphism
group, then it splits over a two-ended subgroup. In particular, this
gives a condition which ensures a hyperbolic group doesn't have property
(T).

The representation dimension of an algebra was introduced in the early 70's by M. Auslander, with the goal of measuring how far an algebra is from having finite number of finitely generated indecomposable modules (up to isomorphism). This invariant is not well understood. For instance, it was not until 2002 that O. Iyama proved that every algebra has finite representation dimension. This was done by constructing special quasihereditary algebras. In this talk I will give an introduction to this topic and I shall briefly explain Iyama's construction.

I will give a historical overview of some of the theorems proved by the
Ancient Greeks, which are now taken for granted but were, and are,
landmarks in the history of mathematics. Particular attention will be
given to the calculation of areas, including theorems of Hippocrates,
Euclid and Archimedes.

Many people talk about properties that you would expect of a group. When they say this they are considering random groups, I will define what it means to pick a random group in one of many models and will give some properties that these groups will have with overwhelming probability. I will look at the proof of some of these results although the talk will mainly avoid proving things rigorously.

This'll be a nice and slow paced introduction to topological quantum field theory in general, and 1-2-3 dimensional theories in particular. If time permits I will explain the spin version of these and their connection to physics. There will be lots of pictures. 

In the classification of surfaces, K3 surfaces hold a place not dissimilar to that of elliptic curves within the classification of curves by genus. In recent years there has been a lot of activity on the problem of rational points on K3 surfaces. I will discuss the problem of finding the Picard group of a K3 surface, and how this relates to finding counterexamples to the Hasse principle on K3 surfaces.

Expansion, rank gradient and virtual splitting are all concepts of great interest in asymptotic group theory. We discuss a result of Marc Lackenby which demonstrates a surprising relationship between then, and give examples exhibiting different combinations of asymptotic behaviour.

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.

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.

Expansion, Random Walks and Sieving in $SL_2 (\mathbb{F}_p[t])$

We pose the question of how to characterize "generic" elements of finitely generated groups. We set the scene by discussing recent results for linear groups in characteristic zero. To conclude we describe some new work in positive characteristic.

Soluble groups, and other classes of groups that can be built from simpler groups, are useful test cases for studying group properties. I will talk about techniques for building profinite groups from simpler ones, and how  to use these to investigate the cohomology of such groups and recover information about the group structure.

In this talk we will introduce the (bounded) derived category of coherent sheaves on a smooth projective variety X, and explain how the geometry of X endows this category with a very rigid structure. In particular we will give an overview of a theorem of Orlov which states that any sufficiently ‘nice’ functor between such categories must be Fourier-Mukai.

In particular, some nice things about branch groups, whose subgroup structure  "sees" all actions on rooted trees.

In particular, some nice things about branch groups, whose subgroup structure "sees" all actions on rooted trees.

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.

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.

A word w has finite width n in a group G if each element in the subgroup generated by the w-values in G can be written as the product of at most n w-values. A group G is called verbally elliptic if every word has finite width in G. In this talk I will present a proof for the fact that every finitely generated virtually nilpotent group is verbally elliptic.

"Show that there is a function $f$ such that for any sequence $(x_1, x_2, \dots)$ we have $x_n = f(x_{n + 1}, x_{n + 2}, \dots)$ for all but finitely many $n$."

Fred Galvin. Problem 5348. The American Mathematical Monthly, 72(10):p. 1135, 1965.\\

This quote is one of the earliest examples of an infinite hat problem, although it's not phrased this way. A hat problem is a non-empty set of colours together with a directed graph, where the nodes correspond to "agents" or "players" and the edges determine what the players "see". The goal is to find a collective strategy for the players which ensures that no matter what "hats" (= colours) are placed on their heads, they will ensure that a "sufficient" amount guess correctly.\\

In this talk I will discuss hat problems on countable sets and show that in a non-transitive setting, the relationship between existence of infinitely-correct strategies and Ramsey properties of the graph breakdown, in the particular case of the parity game. I will then introduce some small cardinals (uncountable cardinals no larger than continuum) that will be useful in analysing the parity game. Finally, I will present some new results on the sigma-ideal of meagre sets of reals that arise from this analysis.

Riley and Dison's hydra groups are a family of group and subgroup pairs $(G_k, H_k)$ for which the subgroup $H_k$ has distortion like the $k$-th Ackermann function. One wants to know if finite quotients can distinguish elements that are not in $H_k$, as a positive answer would allow you to construct a hands-on family of finitely presented, residually finite groups with arbitrarily large Dehn functions. I'll explain why we get a negative answer.

In the late 70s -- early 80s Makanin came up with a very simple, but very powerful idea to approach solving equations in free groups. This simplicity makes Makanin-like procedures ubiquitous in mathematics: in dynamical systems, geometric group theory, 3-dimensional topology etc. In this talk I will explain loosely how Makanin's algorithm works.

The Haagerup property is a group theoretic property which is a strong converse of Kazhdan's property (T). It implies the Baum-Connes conjecture and has connections with amenability, C*-algebras, representation theory and so on. It is thus not surprising that quite some effort was made to investigate how the Haagerup property behaves under the formation of free products, direct products, direct limits,... In joint work with Y.Antolin, we investigated the behaviour of the Haagerup property under graph products. In this talk we introduce the concept of a graph product, we give a gentle introduction to the Haagerup property and we discuss its behaviour under graph products.

In this talk we aim to introduce the key ideas of homotopy type theory and show how it draws on and has applications to the areas of logic, higher category theory, and homotopy theory. We will discuss how types can be viewed both as propositions (statements about mathematics) as well as spaces (mathematical objects themselves). In particular we will define identity types and explore their groupoid-like structure; we will also discuss the notion of equivalence of types, introduce the Univalence Axiom, and consider some of its implications. Time permitting, we will discuss inductive types and show how they can be used to define types corresponding to specific topological spaces (e.g. spheres or more generally CW complexes).\\

This talk will assume no prior knowledge of type theory; however, some very basic background in category theory (e.g. the definition of a category) and homotopy theory (e.g. the definition of a homotopy) will be assumed.

An important moral truth about the mapping class group of a closed orientable surface is the following: a generic mapping class has no power fixing a finite family of simple closed curves on the surface. Such "generic" elements are called pseudo-Anosov. In this talk I will discuss one instantiation of this principle, namely that the probability of a simple random walk on the mapping class group returning a non-pseudo Anosov element decays exponentially quickly.

For any infinite group with a distinguished family of normal subgroups of finite index -- congruence subgroups-- one can ask whether every finite index subgroup contains a congruence subgroup. A classical example of this is the positive solution for $SL(n,\mathbb{Z})$ where $n\geq 3$, by Mennicke and Bass, Lazard and Serre. \\

Groups acting on infinite rooted trees are a natural setting in which to ask this question. In particular, branch groups have a sufficiently nice subgroup structure to yield interesting results in this area. In the talk, I will introduce this family of groups and the congruence subgroup problem in this context and will present some recent results.

I will discuss free-by-cyclic groups and cases where they can and cannot act on CAT(0) spaces. I will specifically go into a construction building CAT(0) 2-complexes on which free of rank 2-by-cyclic act. This is joint work with Martin Bridson and Martin Lustig.

The modularity theorem saying that all (semistable) elliptic curves are modular was one of the two crucial parts in the proof of Fermat's last theorem. In this talk I will explain what elliptic curves being 'modular' means and how an alternative definition can be given in terms of Galois representations. I will then state some of the conjectures of the Langlands program which in some sense generalise the modularity theorem.

The classical small cancellation theory goes back to the 1950's and 1960's when the geometry of 2-complexes with a unique 0-cell was studied, i.e. the standard 2-complex of a finite presentation. D.T. Wise generalizes the Small Cancellation Theory to 2-complexes with arbitray 0-cells showing that certain classes of Small Cancellation Groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and hence have codimesion 1-subgroups. To be more precise I will introduce "his" version of small Cancellation Theory and go roughly through the main ideas of his construction of the cube complex using Sageeve's famous construction. I'll try to make the ideas intuitively clear by using many pictures. The goal is to show that B(4)-T(4) and B(6)-C(7) groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and if there is time to explain the difficulty of the B(6) case. The talk should be self contained. So don't worry if you have never had heard about "Small Cancellation".

We will introduce some necessary basic notions regarding formal languages, before proceeding to give the classification of groups whose word problem is context-free as the virtually free groups (due to Muller and Schupp (1983) together with Dunwoody's accessibility of finitely presented groups (1985) for full generality). Emphasis will be on the group theoretic aspects of the proof, such as Stalling's theorem on ends of groups, accessibility, and geometry of the Cayley graph (rather than emphasizing details of formal languages).

A Kähler group is a finitely presented group that can be realized as fundamental group of a compact Kähler manifold. It is known that every finitely presented group can be realized as fundamental group of a compact real and even symplectic manifold of dimension greater equal than 4 and of a complex manifold of complex dimension greater equal than 2. In contrast, the question which groups are Kähler groups is surprisingly harder and there are large classes of examples for both, Kähler, and non-Kähler groups. This talk will give a brief introduction to the theory of Kähler manifolds and then discuss some basic examples and properties of Kähler groups. It is aimed at a general audience and no prior knowledge of the field will be required.