The truncated conformal space approach is a very useful tool in the numerical study of renormalisation group flows. I will discuss how one can study the effect of truncation in a standard RG manner and its relevance for studying perturbative fixed points.

(This talk is an exceptional colloquium of the Department of Mathematics.) A real-valued function, F, on an interval (a,b) is called matrix monotone if F(A) is less than F(B) whenever A and B are finite matrices of the same order with eigenvalues in (a,b) and A less than B. In 1934, Loewner proved the remarkable theorem that F is matrix monotone if and only if F is real analytic with continuations to the upper and lower half planes so that Im F is positive in the upper half plane. This deep theorem has evoked enormous interest over the years and a number of alternate proofs. There is a lovely 1954 proof that seems to have been lost in that the proof is not mentioned in various books and review article presentations of the subject, and I have found no references to the proof since 1960. The proof uses continued fractions. I'll provide background on the subject and then discuss the lost proof and a variant of that proof which I've found, which even avoids the need for estimates, and proves a stronger theorem.

The linearised curvatures of de Wit and Freedman for totally symmetric gauge fields of any spin are shown to give rise to geometric equations in which both fields and gauge parameters are not constrained a priori. These equations, which are non local, reduce to the traditional, local Fronsdal form by a suitable gauge fixing, and on-shell degrees of freedom are just the physical ones. Developments and generalisations of these results are discussed.

I will introduce the Gaudin Model. This model is described by a system of commuting Hamiltonians. I will explain how the eigenvalue equations for these Hamiltonians arise as the critical level limit of the Knizhnik-Zamolodchikov equations. In particular, some eigenvectors can be built from H3 correlators. Then I will use the H3-Liouville relation to relate these correlators to Liouville theory correlators. The critical level limit is interpreted in Liouville theory as a geometrical limit. This leads to the construction of Gaudin eigenvalues from the accessory parameters which arise in the uniformization of certain Riemann surfaces.

I will give an elementary and informal introduction to an efficient method of solving the Killing spinor equations. The method is based on the description of spinors in terms of forms and can be used to classify the supersymmetric solutions in e.g. 11D and IIB supergravity.

I am going to review the status of loop quantum gravity as it is seen from the point of view of a Lorentz covariant approach to loop quantization. I'll start from a brief review of the standard loop approach based on the SU(2) gauge group, its main results and problems. Then I'll present another approach which is based on a canonical formulation of general relativity which is explicitly covariant under the local Lorentz transformations. It allows to overcome several problems of the standard loop quantization and at the same time shows that the latter breaks the diffeomorphism invariance and is not a correct way for quantizing gravity. In the covariant framework I'll derive a new spectrum of the area operator, both for spacelike and timelike surfaces, and show that its predictions agree with the spin foam quantization.

I present results on non-perturbative effects in the c=1 string theory. First, I describe a geometric picture found in the CFT framework which gives an interpretation of D-branes in non-critical strings in terms of a complex curve associated with any closed string background. I show that its c=1 limit is degenerate and the degeneracy can be removed by considering a condensation of tachyon modes. Using the matrix model description, I calculate the leading as well as the subleading non-perturbative corrections to the string partition function. We find them by using the Toda integrable structure and from the realization of 2D string theory in terms of free fermions. Both methods give the same result which is also interpreted through correlation functions of a bosonic field. The leading corrections can be interpreted in terms of localized D-branes, whereas the sub-leading ones do not have a simple D-brane description.

(Note: this talk will have a great overlap with the April 5, 2004 talk by the same speaker.) The talk will give a fairly basic introduction to the evidence why a special class of Kac-Moody algebras, called very-extended algebras, might play a role in the formulation of supergravity theories. This evidence will contain a natural understanding of the field contents of some (super)gravity theories as well as the known dualities that usually are expected to arrive only after dimensional reduction.

This talk is part of the joint Triangular Seminars.

This part is part of the Triangular Seminars.

Chosen aspects of twisted brane geometry in the Wess-Zumino-Witten models of type A_2n shall be discussed, both classical and stringy, in reference to a class of coideal subalgebras of Drinfel'd-Jimbo quantum groups known as twisted orthogonal quantum groups. An explicit relation between the two families of algebras, together with a realisation of the latter as (twisted) Reflection Equation Algebras shall be invoked to emphasise the role played by them in a compact algebraic description of quantum twisted branes on SU(2n+1) in the framework of R-matrix Reflection Equations and associated quantum group geometries.

I will explain how to do concrete computations in bosonic closed string field theory. For this, I will show how to numerically describe the geometry of the four-point contact interaction, by computing the boundary of the relevant region of the moduli space of the four-punctured spheres, and by computing everywhere in this region the local coordinates around each punctures, in terms of a Strebel quadratic differential and mapping radii. I will then explain how these results are used and checked in the recent paper of Yang and Zwiebach by considering marginal fields in closed bosonic string field theory.

I explain how arbitrary correlators of string theory in the Euclidean AdS3 can be computed from the correlators of Liouville theory. This makes use of the KZ-BPZ correspondence, which relates on the one hand the Knizhnik-Zamolodchikov equations reflecting the group symmetry in AdS3, on the other hand the Belavin-Polyakov-Zamolodchikov equations reflecting the presence of degenerate fields in the Liouville correlators.