Abstract:
We review recent developments in the field of ultracold atomic Fermi gases. As the cold atom system evolves from BCS to Bose-Einstein condensation (BEC), the behavior of the thermodynamics, and the particle density profiles evolves smoothly in a way which can be well understood theoretically. In the interesting "unitary" regime, we show that these and other data necessarily requires the introduction of a pseudogap in the fermionic spectrum which exhibits many striking similarities to its counterpart in underdoped high $T_c$ superconductors. We emphasize these similarities, giving an overview of the experimental tools and key issues of common interest in both systems.

Abstract:
We show how recent angle resolved photoemission measurements addressing the Fermi arcs in the cuprates reveal a very natural phenomenological description of the complex superfluid phase. Importantly, this phenomenology is consistent with a previously presented microscopic theory. By distinguishing the order parameter and the excitation gap, we are able to demonstrate how the collapse of the arcs below $T_c$ into well defined nodes is associated with the \emph{smooth} emergence of superconducting coherence. Comparison of this theory with experiment shows good semi-quantitative agreement.

Abstract:
In this comment we demonstrate that the general conjectures of Fong $et~ al$ on their ${\rm YBa_2Cu_3O_7}$ neutron experiments are realized in concrete calculations of the neutron scattering structure factor $S({\bf q} , \omega )$ in this material. We make predictions for similar experiments in ${\rm La_2Sr_{0.15}Cu_{1.85}O_4}$, which yield a different q-structure due to the different fermiology. Our calculations address both the frequency and wave-vector dependence of the data and (1) include realistic Fermi surface effects (2) are compatible with the normal state data and (3) demonstrate the consistency of a $d$-wave order parameter, in ${\rm YBa_2Cu_3O_7}$ although they appear to be also compatible with an alternate symmetry specific to the orthorhombic bi-layer system.

Abstract:
We address those puzzling experiments in underdoped high $T_c$ superconductors which have been associated with normal state "vortices" and show these data can be understood as deriving from preformed pairs with onset temperature $T^* > T_c$. For uncorrelated bosons in small magnetic fields, and arbitrary $T^*/T_c$, we present the exact contribution to \textit{all} transport coefficients. In the overdoped regime our results reduce to those of standard fluctuation theories ($T^*\approx T_c$). Semi-quantitative agreement with Nernst, ac conductivity and diamagnetic measurements is quite reasonable.

Abstract:
We address recent momentum resolved radio frequency (RF) experiments on ultracold trapped Fermi gases of $^{40}$K. We show that momentum resolved RF probes provide measurements of the centrally important fermionic spectral function. They also serve to remove ambiguity plaguing the interpretation of momentum integrated RF experiments by establishing a clear dispersion signature of pairing. We find that the temperature dependence of the spectral function is dramatic at unitarity, and, importantly, smooth from above to below $T_c$ throughout BCS-BEC crossover. This should be tested experimentally, given widespread predictions of first order behavior.

Abstract:
We compute the fraction of closed-channel molecules in trapped atomic Fermi gases, over the entire range of accessible fields and temperatures. We use a two-channel model of BCS--Bose-Einstein condensation (BEC) crossover theory at general temperature $T$, and show that this fraction provides a measure of the $T$ dependent pairing gap. Our calculations, containing no free parameters, are in good quantitative agreement with recent low $T$ measurements in $^6$Li. We present readily testable predictions for the dependencies of the closed-channel fraction on temperature and Fermi momentum.

Abstract:
We present a new many body formalism for BCS-BEC crossover, which represents a modification of the BCS-Leggett ground state to include 4-fermion, and higher correlations. In the BEC regime, we show how our approach contains the \textit{Petrov et al} 4-fermion behavior and associated scattering length $a_{dd}$ at short distances, and secondly reduces to composite-boson Bogoliubov physics at long distances. It reproduces the Lee-Yang term, whose numerical value is also fixed by $a_{dd}$. We have also examined the next term beyond the Lee-Yang correction in a phenomenological fashion, building on cloud size data and collective mode experiments. However, one has to view this phenomenological analysis with some caution since experiments are in a state of flux and are performed close to unitarity.

Abstract:
This article is written as a Lecture given in the 2006 Varenna Summer School on "Ultracold Fermi Gases". Here we present a review of BCS--Bose Einstein condensation (BEC) crossover theory with emphasis on finite temperature effects. We discuss the role of temperature $T$ as it enters a theoretical formalism which is based on the standard BCS-Leggett ground state. We also discuss the role of temperature in the context of experiments ranging from thermometric issues to signatures of superfluidity. Particularly important to this discussion is the novel normal state associated with the crossover regime, intermediate between BCS and BEC. The experimental evidence for this unusual normal state (associated with pre-formed pairs) and its counterpart below $T_c$ (associated with non-condensed pairs) is presented in the context of different experiments. We end with a discussion of finite temperature effects in spin polarized superfluids, where $T$ is found to play a crucial role in both theory and experiment.

Abstract:
We address the physics of equilibration in ultracold atomic gases following a quench of the interaction parameter. We focus on the momentum distribution of the excitations, $n_{\mathbf k}$, and observe that larger ${\mathbf k}$ modes will equilibrate faster, as has been claimed in recent experimental work. We identify three time regimes. At short times $n_{\mathbf k}$ exhibits oscillations; these are damped out at intermediate times where the system appears to be in a false-equilibrium. Finally, at longer times, full equilibration occurs. This false-equilibrium is associated with the necessarily slower relaxation of the condensate which sufficiently high ${\mathbf k}$-states (of the excitation response) will then quasi-adiabatically follow. Our work bears on the recent literature focus on interaction quench experiments. We take issue with the fact that theories to date assume that the oscillatory regime is adequate for addressing experiments.

Abstract:
The nonlinear evolution equation for the scattering amplitude of colour dipole off the heavy nucleus is solved in the double logarithmic approximation. It is found that if the initial parton density in a nucleus is smaller then some critical value, then the scattering amplitude is a function of one scaling variable inside the saturation region, whereas if it is greater then the critical value, then the scaling behaviour breaks down. Dependence of the saturation scale on the number of nucleons is discussed as well.