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# Pure Mathematics

All seminars will take place in the McCrea Building, Room 219, on Tuesdays at 2 pm, unless stated otherwise. Tea will be served after the seminar at 3pm in Room 237 of the McCrea Building. All are welcome!

## Autumn 2012

2nd October Lukasz Grabowski (Imperial College): Spectral properties of group rings of semidirect products (abstract)

9th October Owen Cotton-Barratt (University of Oxford): Scale invariance in geometry and topology (abstract)

16th October Jörg Brüdern (Universität Göttingen): Anisotropic diagonal forms

23rd October Lilian Matthiesen (University of Bristol): Rational points on conic bundle surfaces via additive combinatorics

30th October David Singerman (University of Southampton): The geometry of Galois' last theorem (abstract)

6th November Steven Noble (Brunel University): The Merino-Welsh conjecture: an inequality for Tutte polynomials (abstract)

7th November: Sergei Chmutov(Ohio State University): Beraha numbers and graph polynomials

Please note that the seminar by Sergei Chmutov will be Wednesday, 3pm–4pm, in ABLT2.

13th November Gareth Jones (University of Southampton): Bipartite graph embeddings, Riemann surfaces and Galois groups (abstract)

20th November Oleg Pikhurko (University of Warwick): On possible Turán densities (abstract)

27th November Andrei Yafaev (UCL): Some applications of Model Theory to Diophantine Geometry (abstract)

4th December Dan Loughran (Bristol): Rational points of bounded height and the Weil restriction (abstract)

13th December Dorothy Buck (Imperial): The Topology DNA (abstract)

Abstracts:

Lukasz Grabowski. Consider a discrete group G and its regular representation on theHilbert space l2(G). We want to understand spectral properties (i.e. kernels, eigenspaces, etc.) of elements of the group ring of G. In the recent years it has been observed that if G is a semidirect product, we can often get a very good description of such properties. This has led to various counterexamples to the Atiyah conjecture, examples of interesting spectral measures (in particular to manifolds with trivial Novikov-Schubin invariants), and to relating random walks on semidirect products with random Schroedinger operators. I will start by giving some motivations for the study of spectral invariants; then I'll explain why are the semidirect products amenable to computations, and I'll indicate the ways to obtain the examples mentioned above.

Owen Cotton-Barratt. The traditional scope of geometry and topology has been idealised objects. Despite no perfect examples of such ideal objects existing in the external world, we often feel our intuitions in these subjects do relate to it. We will explore this discrepancy, and develop some theory to explain it. We take some inspiration from coarse geometry (which cannot see finite structures) to generalise the ideas behind persistent homology, a recent tool for detecting scale-dependent structures in finite metric spaces.

Jörg Brüdern. This talk addresses the classical question whether a diagonal form admits non-trivial zeros. We shall discuss this problem in the obviously related cases where the underlying field is finite, or p-adic, or the rational numbers. In all these cases, the familiar conjecture of Artin has been confirmed long ago, and our primary concern is to go well beyond this. It turns out that a complete classification of the anisotropic forms is possible if the number of variables is only half as large as is needed for Artin's conjectures.

Lilian Matthiesen. Methods of Green and Tao can be used to prove the Hasse principle and weak approximation for some special intersections of quadrics defined over the rational numbers. This implies that the Brauer-Manin obstruction controls weak approximation on conic bundles with an arbitrary number of degenerate fibres, all defined over Q. This is joint work with Tim Browning and Alexei Skorobogatov.

David Singerman. We are referring to the Theorem in Galois' last letter, written the night before his fatal duel. PSL(2,p), (p prime) acts transitively on the p+1 points of the projective line. In this letter Galois showed that if p is not equal to 2, 3, 5, 7, 11, then PSL(2,p) does not act transitively on fewer points. The cases of most interest are = 7 and 11. Now PSL(2,7) is the automorphism group of the Fano plane, the projective plane with 7 points and also the automorphism group of the Klein quartic, a famous Riemann surface of genus 3 with its well known map of type {3,7}. PSL(2,11) is the automorphism group of a biplane with 11 points. Is there a corresponding Riemann surface? Yes! There is a Riemann surface of genus 70 and a corresponding map of type {5,11}. This surface is composed of Buckyballs and is known as the Buckyball curve. This curve is a direct cousin of the Klein quartic.

Steve NobleWhile investigating convexity properties of the Tutte polynomial, Criel Merino and Dominic Welsh conjectured that in any 2-connected loopless, bridgeless graph, the larger of the number of acyclic orientations and the number of totally cyclic orientations is at least the number of spanning trees of the graph. Each of these invariants is an evaluation of the Tutte polynomial. We will discuss the background to this conjecture and explain why various "obvious" approaches do not work. We show that a stronger version of it holds for series-parallel networks. This is joint work with many co-authors including, for the most recent result, Gordon Royle.

Sergei Chmutov. The n-th Beraha number is defined as B_n =2+2cos(2\pi/n). According  to W. Tutte the Beraha numbers are tightly related to chromatic polynomials of graphs. It is known that a non-integer Beraha number can never be a root of the  chromatic polynomial of any graph. Nevertheless, conjecturally the roots of the chromatic polynomial tend to accumulate around the Beraha numbers. In the talk I first briefly review the Beraha numbers and then I turn to the Tutte  polynomial which specializes to the chromatic polynomial. After that I plan to discuss applications of the Tutte polynomial in knot theory motivated by topology.

Gareth Jones. I shall explain how surface embeddings of bipartite graphs (called dessins d'enfants by Grothendieck) give a link between compact Riemann surfaces and algebraic number fields, and how they provide a faithful representation of the absolute Galois group $\Gamma={\rm Gal}\,\overline{\mathbb Q}/{\mathbb Q}$, an important profinite group. I shall also explain how results of Hall on finite solvable groups and of Huppert and Wielandt on products of cyclic groups allow the classification of the most symmetric dessins, namely the regular embeddings of complete bipartite graphs $K_{n,n}$, and a description of how $\Gamma$ acts on them.

Oleg Pikhurko. Let $k \ge 3$ and $\cal F$ be a family of $k$-graphs, i.e. $k$-uniform set systems. The Turán function $ex(n,F)$ is the maximum number of edges in a $\cal F$-free $k$-graph on $n$ vertices. The Turán density $\pi(\cal F)$ is the limit of $ex(n,F)/\binom{n}{k}$ as $n$ tends to infinity.

We disprove the conjecture of Chung and Graham that $\pi(\cal F)$ is rational for every finite family $\cal F$. The conjecture was independently disproved by Baber and Talbot. Also, we show that the set of possible Turán densities has cardinality of the continuum.

Andrei Yafaev. Very recently Jonathan Pila came up with a new and very promising approach to the Andre-Oort conjecture involving the ideas from Model Theory (more specifically, the theory of o-minimality). This approach had already been sucessfully applied to many special cases. In this talk I will explain the approach, focussing on the simplest case of the Andre-Oort conjecture, namely that of products of two modular curves.

Dan Loughran. If one is interested in studying diophantine equations over number fields, there is a clever trick due to Weil where one may move the problem from the number field setting to the usual field of rational numbers by performing a "restriction of scalars". In this talk, we consider the problem of how the height of a solution (a measure of the complexity of a solution) changes under this process, and in particular how the number of solutions of bounded height changes.

Dorothy Buck. The central axis of the famous DNA double helix is typically topologically constrained or circular, and can become knotted or linked during important cellular reactions. The shape of this axis can influence which proteins interact with the underlying DNA.

I will give an overview of some of the methods from 3-manifold topology that are used to model both these DNA molecules and a variety of DNA-protein reactions. We'll conclude with a few examples showing how the answers from these models aid biologists.

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