Forthcoming events in this series


Tue, 17 Feb 2015

11:00 - 12:30
N3.12

Groups acting on R(ooted) trees

Alejandra Garrido
(Oxford)
Abstract

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

Wed, 11 Feb 2015

11:00 - 12:30
N3.12

The Poincaré conjecture in dimensions 3 and 4.

Alejandro Betancourt
(Oxford)
Abstract

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.

Wed, 04 Feb 2015
11:30
N3.12

A brief history of manifold classification

Gareth Wilkes
(Oxford University)
Abstract

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.

Wed, 03 Dec 2014
12:30
N3.12

The Banach-Tarski paradox

Federico Vigolo
(Oxford University)
Abstract

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.
 

Wed, 26 Nov 2014
12:30
N3.12

The Artin approximation theorem in algebraic geometry

Emily Cliff
(Oxford University)
Abstract

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.

Wed, 19 Nov 2014
12:30
N3.12

Modularity of networks

Fiona Skerman
(Oxford University)
Abstract

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.

Wed, 12 Nov 2014
12:30
N3.12

The boundary of the curve complex: a journey by train

Antonio De Capua
(Oxford University)
Abstract

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.

Wed, 05 Nov 2014
12:30
N3.12

Cluster algebras of finite type

Teresa Conde
(Oxford University)
Abstract

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.

 
Wed, 29 Oct 2014
12:30
N3.12

Folding free-group automorphisms

Giles Gardam
(Oxford University)
Abstract

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.

 
Wed, 22 Oct 2014
12:30
N3.12

How badly can the Hasse principle fail?

Francesca Balestrieri
(Oxford University)
Abstract

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.

Wed, 18 Jun 2014
11:00
N3.12

Verbal Width in Virtually Nilpotent Groups

Constantin Gresens
Abstract

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.

Wed, 11 Jun 2014
10:30
N3.12

Hat problems and small cardinals

Robert Leek
Abstract

"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.

Wed, 04 Jun 2014
10:30
N3.12

Separability in the Hydra Groups

Kristen Pueschel
(Cornell)
Abstract

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.

Wed, 28 May 2014
10:30
N3.12

Makanin's algorithm

Ilya Kazachkov
Abstract

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.

Wed, 21 May 2014
10:30
N3.12

The behaviour of the Haagerup property under graph products

Dennis Dreesen
(Southampton)
Abstract

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.

Wed, 14 May 2014
10:30
N3.12

An introduction to homotopy type theory and Univalent Foundations

Emily Cliff
Abstract

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.

Wed, 07 May 2014
10:30
N3.12

Random Walks on Mapping Class Groups

Henry Bradford
Abstract

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.

Wed, 30 Apr 2014
10:30
N3.12

On the congruence subgroup problem for branch groups

Alejandra Garrido Angulo
Abstract

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.

Wed, 12 Mar 2014
10:30
N3.12

CAT(0) structures for free-by-cyclic groups

Robert Kropholler
Abstract

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.

Wed, 05 Mar 2014
10:30
N3.12

Modularity and Galois Representations

Benjamin Green
Abstract

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.

Wed, 19 Feb 2014
10:30
N3.12

Wise Small Cancellation Theory

Lukas Buggisch
Abstract

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".

Wed, 12 Feb 2014
10:30
N3.12

Groups whose word problem is context-free

Giles Gardam
Abstract

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).

Wed, 05 Feb 2014
10:30
N3.12

An introduction to Kähler groups

Claudio Llosa Isenrich
Abstract

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.