We will announce two theorems. The first theorem will classify all topological types of degenerate fibers appearing in one-parameter families of Riemann surfaces, in terms of ``pseudoperiodic'' surface homeomorphisms. The second theorem will give a complete set of conjugacy invariants for the mapping classes of such homeomorphisms. This latter result implies that Nielsen's set of invariants [{\it Surface transformation classes of algebraically finite type}, Collected Papers 2, Birkh\"auser (1986)] is not complete.

We discuss an interpretation of the projection postulate that implies collapse of the wavefunction along the lightcone.

Electronic structures of 13-atom Rh clusters with three possible high-symmetry geometries are studied using the discrete-variational local-spin-density-functional method. The ground state is found to be the icosahedral structure, and a total magnetic moment of 15$\mu_B$ is obtained for the cluster. This value is anomalously smaller than those for clusters with lower symmetries, but in agreement with recent experiments. The magnetic interactions between the central and surface atoms of the cluster are {\it not} fully ferromagnetic, and a small amount of antiferromagnetic interactions is found to be mixed in. An energy parameter is introduced to explain the anomalous magnetic properties, which is found to be also useful for judging whether some techniques can or must be used in the local-spin-density-functional calculations.

A formalism is presented for computing and organizing actions for autonomous agents in dynamic environments. We introduce the notion of teleo-reactive (T-R) programs whose execution entails the construction of circuitry for the continuous computation of the parameters and conditions on which agent action is based. In addition to continuous feedback, T-R programs support parameter binding and recursion. A primary difference between T-R programs and many other circuit-based systems is that the circuitry of T-R programs is more compact; it is constructed at run time and thus does not have to anticipate all the contingencies that might arise over all possible runs. In addition, T-R programs are intuitive and easy to write and are written in a form that is compatible with automatic planning and learning methods. We briefly describe some experimental applications of T-R programs in the control of simulated and actual mobile robots.

The Kazakov-Migdal model, if considered as a functional of external fields, can be always represented as an expansion over characters of $GL$ group. The integration over "matter fields" can be interpreted as going over the {\it model} (the space of all highest weight representations) of $GL$. In the case of compact unitary groups the integrals should be substituted by {\it discrete} sums over weight lattice. The $D=0$ version of the model is the Generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with partition function of the $2d$ Yang-Mills theory with the target space of genus $g=0$ and $m=0,1,2$ holes. This particular quantity is always a bilinear combination of characters and appears to be a Toda-lattice $\tau$-function. (This is generalization of the classical statement that individual $GL$ characters are always singular KP $\tau$-functions.) The corresponding element of the Universal Grassmannian is very simple and somewhat similar to the one, arising in investigations of the $c=1$ string models. However, under certain circumstances the formal sum over representations should be evaluated by steepest descent method and this procedure leads to some more complicated elements of Grassmannian. This "Kontsevich phase" as opposed to the simple "character phase" deserves further investigation.

In previous papers we have pointed out that there exists a QCD analog of the phenomenological concept of the so called Fermi motion for the heavy quark inside a hadron. Here we show in a more detailed way how this comes about and we analyze the limitations of this concept. Non-perturbative as well as perturbative aspects are included. We emphasize both the similarities and the differences to the well-known treatment of deep inelastic lepton-nucleon scattering. We derive a model-independent {\em lower} bound on the kinetic energy of the heavy quark inside the hadron.

Isodoublet color-octet scalar bosons appear in the low-energy limit of a natural extension of the Standard Model in which the electroweak symmetry is broken by a $t\bar t$ condensate. We briefly discuss the model and show that radiative corrections (involving these scalars) to the branching ratio $R_b=\Gamma (Z\rightarrow b\bar b)/\Gamma (Z\rightarrow {\rm hadrons})$ are negative and thus place a stringent lower bound on the masses of the colored scalars. This turns out to be $\sim 400$ GeV with $m_t=150$ GeV and increases quadratically with $m_t$. It is emphasized that $R_b$ is a parameter that is well-determined experimentally and that theoretical estimates are relatively free from uncertainties emanating from hadronic corrections and are comparatively insensitive to the value of the top mass.

Representations of braid group obtained from rational conformal field theories can be used to obtain explicit representations of Temperley-Lieb-Jones algebras. The method is described in detail for SU(2)$_k$ Wess - Zumino conformal field theories and its generalization to an arbitrary rational conformal field theory outlined. Explicit definition of an associated linear trace operation in terms of a certain matrix element in the space of conformal blocks of such a conformal theory is presented. Further for every primary field of a rational conformal field theory, there is a subfactor of hyperfinite II$_1$ factor with trivial relative commutant. The index of the subfactor is given in terms of identity - identity element of certain duality matrix for conformal blocks of four-point correlators. Jones formula for index ( $<$ 4 ) for subfactors corresponds to spin ${\frac{1}{2}}$ representation of SU(2)$_k$ Wess-Zumino conformal field theory. Definition of the trace operation also provides a method of obtaining link invariants explicitly.

The effective operator tr(G^3) is the only dimension-6 gluonic operator that cannot be related to four-quark operators. A peculiar property of this operator is that it does not contribute to two-jet production at hadron colliders, at the level of one operator insertion and leading-order in alpha_s; therefore we study its effects on three jet events. To calculate the helicity amplitudes induced by this operator we make extensive use of collinear factorization. We propose several ways of detecting the tr(G^3) signal, one of which exploits its non-trivial behavior under azimuthal rotations of two almost collinear jets.

We first review the properties of the conventional $\tau$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve however the concept of operator-valued $\tau$-function nor the one associated with non-Cartanian (level $k\ne1$) algebras. The present study could be useful to understand better $q$-free fields and their relation to ordinary free fields.