Abstract:
An explicit computation of the so-called string-theoretic E-function of a normal complex variety X with at most log-terminal singularities can be achieved by constructing one snc-desingularization of X, accompanied with the intersection graph of the exceptional prime divisors, and with the precise knowledge of their structure. In the present paper, it is shown that this is feasible for the case in which X is the underlying space of a class of absolutely isolated singularities (including both usual A_{n}-singularities and Fermat singularities of arbitrary dimension). As byproduct of the exact evaluation of e_{str}(X), for this class of singularities, one gets (in contrast to the expectations of V1!) counterexamples to a conjecture of Batyrev concerning the boundedness of the string-theoretic index. Finally, the string-theoretic Euler number is also computed for global complete intersections in P^{N} with prescribed singularities of the above type.

Abstract:
We examine, in the context of certain string compactifications resulting in five dimensional brane worlds the mechanisms of (self) tuning of the cosmological constant and the recovery of standard cosmological evolution. We show that self tuning can occur only as long as supersymmetry is unbroken (unless additional assumptions are made) and that the adjustment of the cosmological constant to zero after supersymmetry breaking and the recovery of standard evolution are the same problem verifying previously made statements in the context of general i.e. not necessarily string theoretic brane worlds. We emphasize, however, that contrary to general brane worlds where the above adjustment requires a fine tuning, stringy brane worlds contain an additional integration constant due to the presence of the compact space thus allowing the adjustment to be done only with integration constants.

Abstract:
The string-theoretic E-functions E_{str}(X;u,v) of normal complex varieties X having at most log-terminal singularities are defined by means of snc-resolutions. We give a direct computation of them in the case in which X is the underlying space of the 3-dimensional A-D-E singularities by making use of a canonical resolution process. Moreover, we compute the string-theoretic Euler number for several compact complex threefolds with prescribed A-D-E singularities.

Abstract:
The study of brane-antibrane configurations in string theory leads to the understanding of supersymmetric D$p$-branes as the bound states of higher dimensional branes. Configurations of pairs brane-antibrane do admit in a natural way their description in terms of K-theory. We analyze configurations of brane-antibrane at fixed point orbifold singularities in terms of equivariant K-theory as recently suggested by Witten. Type I and IIB fivebranes and small instantons on ALE singularities are described in K-theoretic terms and their relation to Kronheimer-Nakajima construction of instantons is also provided. Finally the D-brane charge formula is reexamined in this context.

Abstract:
This paper is an exposition of the new subject of String Topology. We present an introduction to this exciting new area, as well as a survey of some of the latest developments, and our views about future directions of research. We begin with reviewing the seminal paper of Chas and Sullivan, which started String Topology by introducing a BV-algebra structure on the homology of a loop space of a manifold, then discuss the homotopy theoretic approach to String Topology, using the Thom-Pontrjagin construction, the cacti operad, and fat graphs. We review quantum field theories and indicate how string topology fits into the general picture. Other topics include an open-closed version of string topology, a Morse theoretic interpretation, relation to Gromov-Witten invariants, and "brane'' topology, which deals with sphere spaces. The paper is a joint account of the lecture series given by each of us at the 2003 Summer School on String Topology and Hochschild Homology in Almeria, Spain.

Abstract:
We use a method of linearization to study the emergence of the future cosmological singularity characterized by finite value of the cosmological radius. We uncover such singularities that keep Hubble parameter finite while making all higher derivatives of the scale factor (starting out from the $\ddot a$) diverge as the cosmological singularity is approached. Since such singularities has been obtained before in the brane world model we name them the "brane-like" singularities. These singularities can occur during the expanding phase in usual Friedmann universe filled with both a self-acting, minimally coupled scalar field and a homogeneous tachyon field. We discover a new type of finite-time, future singularity which is different from type I-IV cosmological singularities in that it has the scale factor, pressure and density finite and nonzero. The generalization of $w$-singularity is obtained as well.

Abstract:
Brane-world singularities are analysed, emphasizing the case of supergravity in singular spaces where the singularity puzzle is naturally resolved. These naked singularities are either time-like or null, corresponding to the finite or infinite amount of conformal time that massless particles take in order to reach them. Quantum mechanically we show that the brane-world naked singularities are inconsistent. Indeed we find that time-like singularities are not wave-regular, so the time-evolution of wave packets is not uniquely defined in their vicinity, while null singularities absorb incoming radiation. Finally we stress that for supergravity in singular spaces there is a topological obstruction, whereby naked singularities are necessarily screened off by the second boundary brane.

Abstract:
We discuss K-theoretic matching of D-brane charges in the string duality between type I on the 4-torus and type IIA on a K3 surface. This case is more complex than the familiar case of IIA/IIB duality, which is already well understood, but it turns out that replacing K3 by its orbifold blow-down seems largely to resolve the apparent problems with the theory. In particular, this allows for precise matching of 2-torsion brane charges.

Abstract:
By utilizing non-standard slicings of 5-dimensional Schwarzschild and Schwarzschild-AdS manifolds based on isotropic coordinates, we generate static and spherically symmetric braneworld spacetimes containing shell-like naked null singularities. For planar slicings, we find that the brane-matter sourcing the solution is a perfect fluid with an exotic equation of state and a pressure singularity where the brane crosses the bulk horizon. From a relativistic point of view, such a singularity is required to maintain matter infinitesimally above the surface of a black hole. From the point of view of the AdS/CFT conjecture, the singular horizon can be seen as one possible quantum correction to a classical black hole geometry. Various generalizations of planar slicings are also considered for a Ricci-flat bulk, and we find that singular horizons and exotic matter distributions are common features.

Abstract:
Brane Box Models of intersecting NS and D5 branes are mapped to D3 branes at C^3/Gamma orbifold singularities and vise versa, in a setup which gives rise to N=1 supersymmetric gauge theories in four dimensions. The Brane Box Models are constructed on a two-torus. The map is interpreted as T-duality along the two directions of the torus. Some Brane Box Models contain NS fivebranes winding around (p,q) cycles in the torus, and our method provides the geometric T-dual to such objects. An amusing aspect of the mapping is that T-dual configurations are calculated using D=4 N=1 field theory data. The mapping to the singularity picture allows the geometrical interpretation of all the marginal couplings in finite field theories. This identification is further confirmed using the AdS/CFT correspondence for orbifold theories. The AdS massless fields coupling to the marginal operators in the boundary appear as stringy twisted sectors of S^5/Gamma. The mapping for theories which are non-finite requires the introduction of fractional D3 branes in the singularity picture.