Quantum Chromodynamics (QCD) has rich contents under various
extreme environments. I will discuss recent developments of
hot and dense QCD I have been involved in; a complicated
phase structure of color superconductivity at high baryon
density, an effective description by the idea of the Color
Glass Condensate at high gluon density, and its application
to the very initial stage of heavy-ion collisions. I will
present some ongoing works and address a way to go from my
point of view.

I shall describe a bottom-up approach to constructing a
higher-dimensional theory holographically dual to QCD: AdS/QCD.
Hadronic models built in this way simultaneously satisfy
chiral symmetry constraints together with QCD sum rules
and can be constrained by matching asymptotic behavior of QCD
correlation functions. The simplest model of this type gives a
remarkably good fit to low energy hadronic observables.

In the naive model of the proton, its 1/2 spin is carried by its
quark constituents. However, experiments over the last several decades have shown that the quark spin only contribute a small portion of the proton spin. In this talk, I will discuss the current status of world efforts for solving this "spin crisis", focusing on the theoretical challenges and the perspective from the RHIC spin program.

The Relativistic Heavy-Ion Collider (RHIC) has produced a wide variety of measurements which have led to major strides in our understanding of the structure of strongly interacting matter heated beyond the deconfinement temperature. We focus on a class of observables centered around the perturbative modification of hard jets and jet-like correlations which have been instrumental in resolving the basic picture underlying some of the startling discoveries at RHIC. Jet modification and jet medium interactions will be shown to yield a consistent space-time profile of the expanding bulk matter and in conjunction with bulk fluctuations demonstrate sensitivity to the basic degrees of freedom prevalent in the Quark Gluon Plasma.

According to modern theory and cosmological simulations, the very
first generation of stars that formed in the universe typically
were much more massive than stars forming today. These first stars formed from the material left behind by the big bang, almost exclusively hydrogen and helium. Their resulting evolution and explosive deaths were much different from modern supernovae, with a different central engine and a much more powerful explosion. The resulting nucleosynthesis signatures, the ashes of the explosion, are predicted to show the fingerprint of this peculiar initial condition and evolution. No such fingerprint has ever been found in the observations, however. On the other hand, it will be shown that some not so massive stars with not so powerful explosions seem to be able to explain much of what was observed and considered to be the ashes of the first stars.

I review the progress towards the determination of the QCD phase diagram by lattice simulations, and the difficulties encountered.

We argue that the collinear factorization of the fragmentation functions in high energy hadron and nuclear collisions breaks down at transverse momenta k_T ~ Q_s/g due to high parton densities in the colliding hadrons and/or nuclei. We then argue that gluon recombination, which is basically the merging of two classical fields, should dominate in that k_T regime. We calculate, at next-to-leading order in projectile parton density and to all orders in target parton density, the double-inclusive cross-section for production of a pair of gluons in the scalar J^{PC} = 0^{++} channel. We then generalize our results to AuAu collisions at RHIC energy. Using the low energy theorems of QCD we find the inclusive cross-section for pion meson and baryon production. Finally, we compare our results for baryon to meson ratio with the experimental data from RHIC.

In relativistic heavy ion collisions large numbers of pions are created. It may be possible for these pions to condense into the zero-momentum state, i.e. form a Bose-Einstein condensate. Pions have the special property of being Goldstone bosons of the spontaneously broken chiral symmetry of QCD. The O(N)-symmetric linear sigma model is used as an effective low energy model for QCD to understand the relationship between chiral symmetry breaking and Bose-Einstein condensation. This has recently been studied by Shu and Li and by Andersen. Shu and Li use the framework of the Cornwall-Jackiw-Tomboulis formalism and the 1/N approximation, while Andersen uses the 2PI (Phi derivable) method. I will review their work and attempt to go further in understanding the thermodynamic properties of the sytem.