Abstract:
We have identified two useful exact properties of the perturbative expansion for the case of a two-dimensional electron liquid with Rashba or Dresselhaus spin-orbit interaction and in the absence of magnetic field. The results allow us to draw interesting conclusions regarding the dependence of the exchange and correlation energy and of the quasiparticle properties on the strength of the spin-orbit coupling which are valid to all orders in the electron-electron interaction.

Abstract:
We discuss by analytic means the theory of the high-density limit of the unpolarized two-dimensional electron liquid in the presence of Rashba or Dresselhaus spin-orbit coupling. A generalization of the ring-diagram expansion is performed. We find that in this regime the spin-orbit coupling leads to small changes of the exchange and correlation energy contributions, while modifying also, via repopulation of the momentum states, the noninteracting energy. As a result, the leading corrections to the chirality and total energy of the system stem from the Hartree-Fock contributions. The final results are found to be vanishing to lowest order in the spin-orbit coupling, in agreement with a general property valid to every order in the electron-electron interaction. We also show that recent quantum Monte Carlo data in the presence of Rashba spin-orbit coupling are well understood by neglecting corrections to the exchange-correlation energy, even at low density values.

Abstract:
The screened electron-electron interaction $W_{\sigma, \sigma'}$ and the electron self-energy in an infinitesimally polarized electron gas are derived by extending the approach of Kukkonen and Overhauser. Various quantities in the expression for $W_{\sigma, \sigma'}$ are identified in terms of the relevant response functions of the electron gas. The self-energy is obtained from $W_{\sigma, \sigma'}$ by making use of the GW method which in this case represents a consistent approximation. Contact with previous calculations is made.

Abstract:
We show that a result equivalent to Overhauser's famous Hartree-Fock instability theorem can be established for the case of a two-dimensional electron gas in the presence of Rashba spin-obit coupling. In this case it is the spatially homogeneous paramagnetic chiral ground state that is shown to be differentially unstable with respect to a certain class of distortions of the spin-density-wave and charge-density-wave type. The result holds for all densities. Basic properties of these inhomogeneous states are analyzed.

Abstract:
We show that the pitfalls encountered in earlier calculations of the RKKY range function for a non interacting one dimensional electron gas at zero temperature can be unraveled and successfully dealt with through a proper handling of the impurity potential.

Abstract:
Tian initiated the study of incomplete K\"ahler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle $2\pi(1-\alpha)$ for $\alpha\in (0, 1)$. In this paper we study how the existence of such K\"ahler-Einstein metrics depends on $\alpha$. We show that in the negative scalar curvature case, if such K\"ahler-Einstein metrics exist for all small cone-angles then they exist for every $\alpha\in(\frac{n+1}{n+2}, 1)$, where $n$ is the dimension. We also give a characterization of the pairs that admit negatively curved cone-edge K\"ahler-Einstein metrics with cone angle close to $2\pi$. Again if these metrics exist for all cone-angles close to $2\pi$, then they exist in a uniform interval of angles depending on the dimension only. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed.

Abstract:
The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We estimate the number of ends of such manifolds in terms of their volume. We give effective bounds on the number of complex hyperbolic manifolds with given upper bounds on the volume. Moreover, we give two sided bounds on their Picard numbers in terms of the volume and the number of cusps.

Abstract:
Let $X$ be a smooth variety and let $L$ be an ample line bundle on $X$. If $\pi^{alg}_{1}(X)$ is large, we show that the Seshadri constant $\epsilon(p^{*}L)$ can be made arbitrarily large by passing to a finite \'etale cover $p:X'\rightarrow X$. This result answers affirmatively a conjecture of J.-M. Hwang. Moreover, we prove an analogous result when $\pi_{1}(X)$ is large and residually finite. Finally, under the same topological assumptions, we appropriately generalize these results to the case of big and nef line bundles. More precisely, given a big and nef line bundle $L$ on $X$ and a positive number $N>0$, we show that there exists a finite \'etale cover $p: X'\rightarrow X$ such that the Seshadri constant $\epsilon(p^{*}L; x)\geq N$ for any $x\notin p^{*}\textbf{B}_{+}(L)=\textbf{B}_{+}(p^{*}L)$, where $\textbf{B}_{+}(L)$ is the augmented base locus of $L$.

Abstract:
In this paper, we show that the canonical divisor of a smooth toroidal compactification of a complex hyperbolic manifold must be nef if the dimension is greater or equal to three. Moreover, if $n\geq 3$ we show that the numerical dimension of the canonical divisor of a smooth $n$-dimensional compactification is always bigger or equal to $n-1$. We also show that up to a finite \'etale cover all such compactifications have ample canonical class, therefore refining a classical theorem of Mumford and Tai. Finally, we improve in all dimensions $n\geq 3$ the cusp count for finite volume complex hyperbolic manifolds given in [DD15].

Abstract:
We exhibit the construction of a deterministic automaton that, given k > 0, recognizes the (regular) language of k-differentiable words. Our approach follows a scheme of Crochemore et al. based on minimal forbidden words. We extend this construction to the case of C\infinity-words, i.e., words differentiable arbitrary many times. We thus obtain an infinite automaton for representing the set of C\infinity-words. We derive a classification of C\infinity-words induced by the structure of the automaton. Then, we introduce a new framework for dealing with \infinity-words, based on a three letter alphabet. This allows us to define a compacted version of the automaton, that we use to prove that every C\infinity-word admits a repetition in C\infinity whose length is polynomially bounded.