## Mathematical Physics Seminars

2009 - 2010

### Programme

All seminars are held in room M/2.06 on Thursdays at 3:10pm unless otherwise stated. All are welcome.

##### 22 September 2009 - room M/2.30

**Speaker:** Dr Pinhas Grossman (Cardiff).

Title: Operator Algebras Seminar - Subfactor Planar Algebras I

##### 1 October 2009

**Speaker:** Professor Gary Gibbons FRS (Cambridge).

Title: Two Problems in General Relativity

Abstract: I will describe two recent papers

1) Gravitating Opposites Attract.

Robert Beig, Gary W. Gibbons, Richard M. Schoen, . Jul 2009. 13pp.

arXiv:0907.1193 [gr-qc]

Generalizing previous work by two of us, we prove the non-existence of
certain stationary configurations in General Relativity having a spatial
reflection symmetry across a non-compact surface disjoint from the matter
region. Our results cover cases such that of two symmetrically arranged
rotating bodies with anti-aligned spins in $n+1$ ($n \geq 3$) dimensions,
or two symmetrically arranged static bodies with opposite charges in 3+1
dimensions. They also cover certain symmetric configurations in
(3+1)-dimensional gravity coupled to a collection of scalars and abelian
vector fields, such as arise in supergravity and Kaluza-Klein models. We
also treat the bosonic sector of simple supergravity in 4+1 dimensions.

2) Birkhoff's invariant and Thorne's Hoop Conjecture.

G.W. Gibbons, arXiv:0903.1580 [gr-qc]

I propose a sharp form of Thorne's hoop conjecture which relates
Birkhoff's invariant $\beta$ for an outermost apparent horizon to its
$ADM$ mass, $ \beta \le 4 \pi M_{ADM}$. I prove the conjecture in the case
of collapsing null shells and provide further evidence from exact rotating
black hole solutions. Since $\beta$ is bounded below by the length $l$ of
the shortest non-trivial geodesic lying in the apparent horizon, the
conjecture implies $l \le 4 \pi M_{ADM}$. The Penrose conjecture,
$\sqrt{\pi A} \le 4 \pi M_{ADM}$, and Pu's theorem imply this latter
consequence for horizons admitting an antipodal isometry. Quite generally,
Penrose's inequality and Berger's isembolic inequality, $\sqrt{\pi A} \ge
{2\over\sqrt\pi} i$, where $i$ is the injectivity radius, imply $ 4c \le 2
i \le 4 \pi M_{ADM}$, where $c$ is the convexity radius.

##### 6 October 2009 - room M/2.30

**Speaker:** Dr Pinhas Grossman (Cardiff).

Title: Operator Algebras Seminar - Subfactor Planar Algebras II

##### 8 October 2009 - room M/0.34

**Speaker: **Professor Nigel Hitchin FRS (Oxford).

Title: Twisted tangent bundles

The talk concerns the role of the B-field as a mathematical object in the context of generalized geometry. A number of mathematical structures which admit a formulation in generalized terms don't fundamentally involve the symmetries defined by B-fields, but we investigate one, involving holomorphic vector bundles with an extra structure, where the action on the moduli space is nontrivial and relates to spectral curves and varieties. The twisted tangent bundles of the title involve the corresponding structures for local B-field actions, relating to degree three cohomology and the notion of a gerbe.

##### 19 October 2009 - 2pm

Operator Algebras Seminar

**Speaker: **2.00 pm Pinhas Grossman (Cardiff).

Title: "Subfactor Planar Algebras III"

**Speaker**: 4.00 pm Hervé Oyono-Oyono (Clermont)

Title "On non commutative principal torus bundles" (Oyono-Oyono)

NCPT-bundles form a class of continuous fields of noncommutative tori that generalises principal torus bundles. As we shall see, these bundles are quite irregular in any topological sense. This is a special class of C*-bundles , we called K-fibrations, which satisfies a property generalising to the C*-algebras setting the liftingproperty of Serre fibrations. We introduce K-theoretical invariants for this K-fibrations in order to obtain classification results for NCPT -bundles. (joint work with S. Echterhoff and R. Nest)

##### 22 October 2009

**Speaker:** Volker Braun (DIAS, Dublin).

Title: Computing Heterotic String Compactifications

Using Donaldson's algorithm and its generalization, I will present a
numerical computation of Calabi-Yau metrics on K3s and threefolds as
well as Hermitian Yang-Mills connections on holomorphic vector bundles.
We will see how the computation detects slope stability. The metric and
connection defines Laplace operators which can be solved numerically, as
well. The goal is to explicitly find the 4-d effective action.

##### 27 October 2009 - room M/2.30

**Speaker:** Dr Pinhas Grossman (Cardiff).

Title: Operator Algebras Seminar - Subfactor Planar Algebras IV

##### 29 October 2009

**Speaker:** Professor Ulrike Tillmann FRS (Oxford).

Title: Manifolds, cobordism categories, and topological field theory

Following Atiyah, invariants of manifolds can be constructed by
topological field theories. Recall that such a TFT is a functor from
a certain cobordism category to the category of vector spaces.
I will describe the topology of the classifying space of the (enriched)
cobordism category and its higher dimensional analogues. These results
shed new light on classical cobordism theory of Thom. At the same time
they have fed into recent work of Jacob Lurie which classifies TFTs
in any dimension.

##### 3 November 2009 - room M/2.30

**Speaker:**Dr Pinhas Grossman (Cardiff).

Title: Operator Algebras Seminar - Subfactor Planar Algebras V

##### 5 November 2009

**Speaker:** Dr Raf Bocklandt (Newastle)

Title: Calabi Yau algebras and Quiver polyhedra

Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated modules over their centers and their representation spaces are toric varieties. In order to describe these algebras we introduce the notion of a toric order and show that all toric orders 3-dimensional Calabi Yau algebras can be constructed from dimer models on a torus.

Toric orders are examples of a much broader class of algebras: positively graded category algebras with cancellation. For this broader class the CY-3 condition also implies the existence of a weighted quiver polyhedron, which is an extension of dimer models obtained by replacing the torus with any two-dimensional compact orientable orbifold. We discuss which of these quiver polyhedra give rise to Calabi Yau algebras.

##### 10 November 2009 - room M/2.30

**Speaker: **Dr Pinhas Grossman (Cardiff).

Title: Operator Algebras Seminar - Subfactor Planar Algebras VI

##### 12 November 2009

**Speaker:** Dr Gerard Watts (King’s College, London)

Title: Defects in two-dimensional conformal field theory

Conformal Defects can be used to describe both branes in string theory
and interfaces or lines of discontinuity in statistical systems. I shall
review the description and classification of the simplest sort of
defects, topological defects, and explain the physical and mathematical
differences when the topological condition is removed. I shall then
present recent work on exploring the space of defects through their
perturbations, which combines exact, perturbative and numerical methods. (Reference here.)

##### 17 November 2009 - room M/0.40

**Speaker: **Claus Koestler (Aberystwyth).

Title: An application of exchangeability to characters of the group of finite permutations

Thoma's theorem characterizes the extremal characters of the group of finite permutations. Aside of the original result by Thoma from 1964, alternative proofs have been given by Kerov & Vershik in 1981, and more recently by Okounkov in 1997. We provide a new proof based on

exchangeability in the framework of noncommutative probability. Quite surprisingly, our operator algebraic approach reveals that Thoma's theorem can be thought of as a noncommutative de Finetti theorem. This is joint work with Rolf Gohm.

**19 November** 2009

**Speaker:** Simon Brain (SISSA, Trieste and Oxford)

Title: Geometry of the Yang-Mills Gauge Fields

##### 26 November 2009- 5:15pm

**Speaker: **Eugene Lytvynov (Swansea).

Title: Meixner class of non-commutative generalized stochastic processes with freely independent values

**10 December** 2009

**Speaker:** Andrew Bruce (Manchester).

Title: Higher Poisson structures and Higher Kozsul-Schouten brackets

In this talk we recall the basic constructions of higher
Poisson structures as inhomogenous even multivector fields on a
supermanifold that Schouten-Nijenhius self-commute. This
generalises the notion of classical Poisson geometry. We will then
proceed to describe how to generalise the Koszul-Schouten bracket on
differential forms over a Poisson manifold to the higher case. We do
this by constructing a homotopy BV-algebra generated by the Lie
derivative along the higher Poisson structure. The audience are not
expected to be experts in supergeometry or Poisson geometry.

##### 28 January 2010

**Speaker:** Oliver Gray (Bristol).

Title: On the classification of the unitary minimal N=2 superconformal field theories.

Abstract: I will give an introduction to (super) conformal field theories from a more mathematical point of view, explain why the N=2 minimal models are in fact not yet classified, and suggest how a classification might be achieved through a mixture of orbifolding and folklore.

##### 4 February 2010

**Speaker:** Liza Jones (Bristol).

Title: Non-Colliding Brownian Motions, Weyl Chambers and Symmetric Spaces.

Abstract: I will discuss how naturally occuring diffusions possessing the

"non-colliding" property arise in the context of symmetric spaces. This framework has beenparticularly useful in studying and generalizing models from random matrixtheory.

##### 15 February - 5 March 2010

Period on Planar Algebras and Physics

**Speakers:** JulienBichon (Clermont), Stephen Bigelow (UCSB), Richard Burstein (Vanderbilt), Ben Davison (Oxford), Jürgen Fuchs (Karlstad), ShamindraGhosh (KU Leuven), Pinhas Grossman (Cardiff), Ved Gupta (Leuven), AmihayHanany (Imperial), Yang-Hui He (Oxford), PawelKasprzak (Copenhagen), Alastair King (Bath), Mathew Pugh (Cardiff), Alex Quintero-Velez (Glasgow), Sven Raum (Leuven), Peter West (King's College, London)

##### 15 February 2010- 2:30pm

**Speaker:** Yang-Hui He (Oxford).

Title: Quivers, gauge theories and Calabi-Yau singularities

Abstract: We discuss some recent progress on how quiver theories inevitably arise within string theory as world-volume theory of branes in various spacetime dimensions. We discuss intimate relations between the gauge theory, encoded by the quiver representations, the algebraic geometry of affine Calabi-Yau varieties, and auxiliary combinatorics involving dimers, tilings and plethystics.

##### 24 February 2010 - 4pm

**Speaker:** Peter West FRS (King's College, London).

Title: E Theory

Abstract: I will review the current status of string theory and explain the

evidence for a very large Kac -Moody symmetry of an underlying theory

of strings and branes.

##### 26 February 2010

WIMCS Mathematical Physics Colloquium [117KB] (Robert Recorde Room, Swansea University)

**Speakers:** Marco Marino (Geneva) and Richard Thomas (Imperial)

2pm - Thomas. Title: Curves on K3 surfaces and modular forms

Abstract: The Katz-Klemm-Vafa formula is a conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. In genus 0 it reduces to the (proved) Yau-Zaslow formula. I will explain how a correspondence between "stable pairs" and Gromov-Witten theory for toric 3-folds (proved by Maulik-Oblomkov-Okounkov-Pandharipande), some calculations with stable pairs (due to Kawai-Yoshioka) and some deformation theory lead to a proof of the KKV formula.

3:30pm - Marino. Title: Quantum theory and enumerative problems

Abstract: Many problems in mathematics involve counting objects. For example, in combinatorics, one is often interested in counting graphs with certain conditions. Another example is enumerative algebraic geometry, where one counts curves in algebraic varieties. Many of these problems turn out to be deeply related to quantum theories. In this talk I will explore this relation in three important examples: the enumeration of maps, the enumeration of Hurwitz coverings, and the LMO invariant of rational homology spheres. I will show how ideas of quantum theory lead to new insights in these problems, and in particular I will also explain the connection between non-perturbative effects in quantum theory and the asymptotic behavior of these enumerative problems.

##### 5 March 2010 - 4pm

**Speaker:** Amihay Hanany (Imperial).

Title: On the classification of brane tilings

Abstract: Brane tilings give a large class of SCFT's in 3+1 and 2+1 dimensions. In this talk I will discuss several attempts to classify all such models. Statistical properties of these models can be derived using some techniques in crystallography and in number theory.

##### 12 March 2010

**Speaker: **Jeffrey Giansiracusa (Swansea).

Title: Operads and Topological Field Theory

##### 18 March 2010

**Speaker:** Kristian Evans (Swansea).

Title: Variable Order Subordination

##### 15 April 2010

COW Seminar

**Speakers:** Katrin Wendland (Augsburg) and Sven Meinhardt (Oxford)

2.30 Sven Meinhardt (Oxford)

Integrality in Donaldson-Thomas theory

3.00 Tea

4.00 Katrin Wendland (Augsburg)

TQFT & singularities from a conformal field theorist's point of view

Organizers: David E Evans (Cardiff), Miles Reid FRS (Warwick ) and Greg Sankaran (Bath).

##### 22 April 2010

**Speaker:** Boris Zilber (Oxford)

Title: On Model Theory, noncommutative geometry and physics.

Abstract: Studying possible relations between a mathematical structure and its
description in a formal language Model Theory developed a hierarchy
of a 'logical perfection'. On the very top of this hierarchy we discovered
a new class of structures called Zariski geometries. A joint theorem by
Hrushovski and the speaker (1993) indicated that the general Zariski
geometry looks very much like an algebraic variety over an algebraically
closed field, but in general is not reducible to an algebro-geometric
object. Later the present speaker established that a typical Zariski
geometry can be explained in terms of a possibly noncommutative 'co-ordinate'
algebra. Moreover, conversely, many quantum algebras give rise to
Zariski geometries and the correspondence 'Co-ordinate algebra - Zariski
geometry' for a wide class of algebras is of the same type as that
between commutative affine algebras and affine varieties.

General quantum Zariski geometries can be approximated (in a certain
model-theoretic sense) by quantum Zariski geometries at roots of unity. The latter are of a finitary
type, where Dirac calculus has a well-defined meaning. We use this to give a
mathematically rigorous calculation of the Feynman propagator in a few simple cases.

##### 29 April 2010

**Speaker:** Elizabeth Gasparim (Edinburgh)

Title: New isomorphisms of moduli spaces

Abstract: I will describe some new isomorphisms between moduli spaces of rank 2 bundles over surfaces and over threefolds. I will speculate about the possible physics interpretations of those.

##### 14 May 2010 - room M/0.34

**Speaker:** Frances Kirwan FRS (Oxford)

Title: Geometric invariant theory for non-reductive group actions

and jet differentials.

Abstract: Geometric invariant theory (GIT) was developed
by Mumford in the 1960s in order to construct and study
quotients of algebraic varieties by actions of reductive
linear algebraic groups. His main motivation was that many interesting moduli spaces in algebraic geometry
can be constructed in this way. This talk will discuss
a generalisation of GIT to non-reductive group actions,
and if time permits a potential application to global

singularity theory via jet differentials.

##### 17-21 May 2010

**Speaker**: Nigel Higson (Penn State)

Title: The Baum-Connes Conjecture and Group Representations

This includes a Spitalfields Day on 17 May with speakers

Terry Gannon (Alberta) and Nigel Higson

##### 21-25 June 2010

LMS Regional Meeting and Workshop on Operator Algebras and Physics

Principal Speaker: Constantin Teleman (Berkeley) who is giving a series of talks on Two Dimensional Topological Quantum Field Theories and Gauge Theories.

The first talk by Constantin Teleman will be in the LMS regional meeting on the first afternoon, Monday 21 June which is embedded in the five-day workshop.
The other speaker in the regional meeting on the Monday afternoon will be Werner Nahm (Dublin).

Confirmed speakers for the week include
Teodor Banica (Paris), Terry Gannon (Alberta), Johannes Kellendonk (Lyon),
Michael Müger (Nijmegen), Andreas Recknagel (King’s),
Karl-Henning Rehren (Göttingen), Richard Szabo (Heriot-Watt),
Jean-Louis Tu (Metz), Gerard Watts (King’s)

##### 28 June 2010 - 5pm in Wallace Lecture Theatre, Main Building

**Speaker**: Dan-Virgil Voiculescu

##### 28 June - 2 July 2010

Confirmed speakers include: Iakovos Androulidakis (Göttingen), Dorothea Bahns (Göttingen), Moulay-Tahar Benameur (Metz), Sebastiano Carpi (Rome), Kenny De Commer (Rome), Bergfinnur Durhuus (Copenhagen), Robin Hillier (Rome), Stefan Hollands (Cardiff), Tommaso Isola (Rome), Yasuyuki Kawahigashi (Tokyo), Amin Malik (Oslo), Jouko Mickelsson (Helsinki), Denjoe O’Connor (DIAS), Thomas Schucker (Provence), Wojciech Szymanski (Odense), Otgonbayar Uuye (Copenhagen), André Verbeure (K.U. Leuven), Dan Voiculescu (Berkeley), Mihaly Weiner (Rome), Jakob Yngvason (ESI, Vienna), Laszlo Zsido (Rome)