## Mathematical Physics Seminars

2008 - 2009

### Programme

#### 9 October 2008

**Speaker: **Sophie Jackson (Reader in Biophysical Chemistry, University of Cambridge)

Title: How do knotted proteins fold?

Abstract: For a long time, it was thought that it was impossible for a polypeptide (protein) chain to both knot and fold into a specific three-dimensional structure. However, since their discovery earlier this decade, protein structures which contain topological knots in their polypeptide chains have now been identified for almost 400 proteins. These unusual proteins represent a novel challenge for both the experimental and computational protein folding communities, and raise a number of important questions such as how do such structures fold and what is the role of the knot in the protein structure and function? The talk will be broken up into four main sections. First, I will give some background on protein structure in general, including the structures of knotted proteins and their knotted topologies. This will be followed by a brief introduction to the protein folding problem and what has been learnt from 20 years of study of small model systems. The experimental program of work and results that my own group have obtained in the last five years on the folding pathways of three knotted protein structures will be described and, finally, recent computational efforts at simulating the folding pathways of knotted proteins will be discussed.

##### 16 October 2008

**Speaker: **John DS Jones (Professor of Mathematics, Warwick University)

Title: String Topology

##### 6 November 2008WIMCS Colloquium - 12:30pm

**Speaker: **Carloz Nunez (Reader in Physics, Swansea University)

Title: The Duality between Gauge Fields and Strings

**Speaker: ** Man-Duen Choi (Professor of Mathematics, University of Toronto)

Title: The magic of non-commutative computations

**Speaker: ** Denjoe O'Connor (School of Theoretical Physics, Dublin
Institute for Advanced Studies)

Title: Geometry in Transition

##### 20 November 2008

**Speaker: **Balázs Szendroi (Faculty Lecturer, Mathematics Institute, University of Oxford
and
Martin Powell Fellow, St Peter’s College, Oxford)

Title: Deformed partition functions and Poincare polynomials of moduli spaces

I will discuss the topological string partition function of some local Calabi--Yau threefolds, and some recently introduced deformations thereof, giving a tentative interpretation of the latter as Poincare polynomials of certain highly singular moduli spaces associated to the threefolds.

##### 11 December 2008

**Speaker:** Geoffrey L Sewell (Emeritus Professor in Mathematical Physics Department of Physics,
Queen Mary College, London)

Title: Relativistic statistical thermodynamics of moving bodies

I provide an operator algebraic solution of the long standing question of temperature transformations under Lorentz and Galilei boosts. The key ingredients of my treatment are (a) the Tomita-Takesaki modular theory and (b) the connection between the Kubo-Martin-Schwinger conditions and the Zeroth law of Thermodynamics. On this basis, I prove that, in both the relativistic and nonrelativistic settings, a state cannot satisfy the thermal equilibrium conditions for different inertial frames that are in uniform motion relative to one another. This implies that the concept of temperature is restricted to states of bodies in their rest frames and thus that there is no law of temperature transformation under either Lorentz or Galilei boosts.

##### 13 and 14 January 2009 WIMCS Mathematical Physics Lectures

**Speaker:** Professor Tim Porter (Bangor, Wales).

Title: Categorification and bundles

The aim of the talks will be to illustrate some of the aspects of categorification in the case of bundles. This will lead from bundles and vector bundles via sheaves to stacks, gerbes, 2-bundles and eventually to 2-vector bundles. On the way I hope to indicate some of the links with higher category theory and non-Abelian cohomology.

##### 22 January 2009- 1pm Operator Algebras Seminar

**Speakers: **Dr Simon Wassermann (Glasgow), Dr Wilhelm Winter (Nottingham) and Dr Rolf Gohm (Aberystwyth).

1.00 Wilhelm Winter (Nottingham)

Title: The classification of C*-algebras associated to minimal uniquely ergodic dynamical systems.

2.15 Rolf Gohm (Aberystwyth)

Title: Non-commutative Markov Chains and Multi-analytic Operators

3.30 Simon Wassermann (Glasgow)

Title: Masas in UHF algebras

##### 29 January 2009

**Speaker:** Professor Piotr Chrusciel (Oxford)

Title: Black holes: an introduction

After a crash course on general relativity and the Einstein equations, I will review the current experimental and theoretical understanding of black holes.

##### 12 February 2009

**Speaker:** Dr Joost Slingerland (Dublin Institute for Advanced Studies)

Title: Phase transitions and domain walls in 2+1 dimensional topological field theory

Recently there has been much interest in 2+1 dimensional physical
systems with "topological order". At low energies, the phases of such
systems can be described by topological field theory, in particular
their excitations may have nontrivial braiding and fusion interactions
described by a suitable braided tensor category. Using a knowledge of
just this fusion and braiding as a starting point, one may ask whether
it is still possible to make useful statements about phase transitions
that may occur. I will argue that this is indeed the case for phase
transitions caused by an analogue of bose condensation, and indicate
how one may obtain the spectrum, fusion and braiding of the condensed
phase. This leads to some interesting conjectures in topological field
theory, including conjectured analogues of constructions known from
Conformal Field Theory, notably the coset construction.

##### 26 February 2009 WIMCS Colloquium

**Speakers: **Professor Ken Brown (Glasgow) and
Professor Richard Szabo (Edinburgh)

3:00pm (Professor Szabo): Instantons and enumerative geometry

4:30pm (Professor Brown): Small infinite non-commutative groups

The facetious title is intended partly for disguise, partly for
motivation - a more "academic" title might be "Classification of affine
prime Hopf algebras of Gelfand-Kirillov dimension one". But the talk will
not assume prior knowledge of any of the terms in the posh title. Rather,
I will aim to explain the above terms, motivate the problem, and explain
what is known.

##### 5 March 2009

**Speaker: **Dr Akihiro Ishibashi (KEK)

Title: A uniqueness theorem for charged rotating black holes in five-dimensional minimal supergravity

We show that a charged rotating black hole in five-dimensional
Einstein-Maxwell-Chern-Simons theory is uniquely characterized
by the mass, charge, and two independent angular momenta, under
the assumptions of the existence of two commuting axial isometries
and spherical topology of horizon cross-sections. Therefore, such
a black hole must be described by the Chong-Cveti\v{c}-L\"u-Pope metric.

##### 12 March 2009

**Speaker:** Dr Ingo Runkel (King's College, London)

Title: Algebraic structures in conformal field theory

It turned out to be fruitful to isolate questions in CFT which can be formulated in a purely categorical fashion. The way left and right moving degrees of freedom can be combined to a consistent theory is an example of this, the relevant structure being a commutative symmetric Frobenius algebra. This is true independently of whether CFT is formulated via sewing of surfaces or nets of operator algebras. Another example is modular invariance, which has a surprising alternative formulation as a certain maximality condition.

##### 17 March 2009 - 2pm Operator Algebras Seminar

**Speakers:** Professor George A Elliott (Copenhagen and Toronto) and Dr Michael Dritschel (Newcastle)

Title (Professor Elliott): A brief survey of classification theory

The theory of operator algebras has shown that classification
of mathematical objects first of all cannot be expected to be
achieved in terms of the simplest possible labels, such as
numbers, but one can often hope to achieve it in terms of a
more complicated parameter, such that the values of the
parameter corresponding to two isomorphic objects are no longer
necessarily equal---this may not even make sense!---but are

isomorphic. (In other words, the parameter values belong to
their own category---either an abstract category---very general
results of this kind can be proved---or even, as can be proved
in important cases, an interesting concrete category.
Sometimes the concrete invariant can be defined directly, but sometimes, it seems, it can only be seen as a concrete
manifestation of an abstract one---in other words, by showing
that a certain abstract category is equivalent to a certain
concrete one.)

Title (Dr Drirschel): Completely bounded kernels

Given a set $X$ and two $C^*$-algebras $A$ and $B$, a kernel $k$ is defined as a function from $X\times X$ to $L(A, B)$, the bounded linear maps from $A$ to $B$. The kernel $k$ is positive if for all finite sets $F = \{(x_j , a_j)\} \subset X \times A$, the matrix \begin{equation*} (k(x_i, x_j)[a_ia^*_j ])_{F\times F} \qquad\qquad(*) \end{equation*} is nonnegative. If the same is true whenever we replace $X\times A$ by $X\times M_n(A)$ and $k$ by $k\otimes 1_n$ for any $n\in \mathbb N$, then $k$ is said to be completely positive (the two concepts coincide when $A = B = \mathbb C$). Completely positive kernels have several equivalent characterisations, including the existence of a so-called Kolmogorov decomposition. Constantinescu and Gheondea, generalising results of Laurent Schwartz, considered kernels $k$ where the matrix in ($*$) is merely selfadjoint with $L(A, B) = B(H)$, $H$ a Hilbert space, and found necessary and sufficient conditions for the decomposability of such kernels as the difference of (completely) positive kernels. A result of Haagerup implies that when $A$ and $B$ are von Neumann algebras such decompositions in terms of completely positive kernels will fail if $B$ is not injective. In this talk we discuss decomposability of self adjoint kernels as differences of completely positive kernels when $A$ and $B$ are $C^*$-algebras, characterising decomposable kernels. We also discuss the case when the matrix in ($*$) is a only a completely bounded map, giving an analogue of the Wittstock decomposition for such kernels.

##### 14 May 2009

**Speaker:** Dr Dorothy Buck (Imperial)

Title: DNA Knots and How They Arise

DNA molecules often have a circular, or topologically
constrained, central axis. The topology of this axis can influence which
proteins interact with the underlying DNA. Subsequently, there are
proteins, topoisomerases, whose primary function is to change the DNA axis
topology. Additionally, there are protein families that change the axis
topology as a by-product of their interaction with DNA.

This informal talk will describe typical DNA conformations, and the families of
proteins that change these. We will present one example illustrating how 3-manifold
topology has been useful in understanding certain DNA-protein
interactions, and discuss the most common topological techniques used to consider these biological questions.

(No prior biological knowledge necessary)

##### 21 May 2009

**Speaker:**Dr Danny Stephenson (Glasgow)

Title: The basic bundle gerbe on unitary groups, revisited

Let G be the unitary group U(n) or more generally one of the groups
U_p(H) consisting of unitary operators on an infinite dimensional
Hilbert space H which differ from the identity by an element of the
Schatten ideal L^p. For these groups G, the degree three integer
cohomology group H^3(G,Z) of G is canonically isomorphic to the integers
Z. The generator of H^3(G,Z) = Z can be realized geometrically as the
`basic bundle gerbe'. Building on work of Meinrenken and Mickelsson we
will give a construction of this basic bundle gerbe. We will explain how
the holomorphic functional calculus can be used to describe the geometry
of this gerbe. This is joint work with Michael Murray.