Lectures on Duflo isomorphisms in Lie algebra and complex geometry

International audienceDuflo isomorphism first appeared in Lie theory and representation theory. It is an isomorphism between invariant polynomials of a Lie algebra and the center of its universal enveloping algebra, generalizing the pioneering work of Harish-Chandra on semi-simple Lie algebras. Later on, Duflo’s result was refound by Kontsevich in the framework of deformation quantization, who also observed that there is a similar isomorphism between Dolbeault cohomology of holomorphic polyvector fields on a complex manifold and its Hochschild cohomology. The present book, which arose from a series of lectures by the first author at ETH, derives these two isomorphisms from a Duflo-type result for Q-manifolds.All notions mentioned above are introduced and explained in the book, the only prerequisites being basic linear algebra and differential geometry. In addition to standard notions such as Lie (super)algebras, complex manifolds, Hochschild and Chevalley–Eilenberg cohomologies, spectral sequences, Atiyah and Todd classes, the graphical calculus introduced by Kontsevich in his seminal work on deformation quantization is addressed in details.The book is well-suited for graduate students in mathematics and mathematical physics as well as for researchers working in Lie theory, algebraic geometry and deformation theory

A concave-convex problem with a variable operator

We study the following elliptic problem $-A(u) = \lambda u^q$ with Dirichlet boundary conditions, where $A(u) (x) = \Delta u (x) \chi_{D_1} (x)+ \Delta_p u(x) \chi_{D_2}(x)$ is the Laplacian in one part of the domain, $D_1$, and the $p-$Laplacian (with $p>2$) in the rest of the domain, $D_2$. We show that this problem exhibits a concave-convex nature for $1<q<p-1$. In fact, we prove that there exists a positive value $\lambda^*$ such that the problem has no positive solution for $\lambda > \lambda^*$ and a minimal positive solution for $0<\lambda < \lambda^*$. If in addition we assume that $p$ is subcritical, that is, $p<2N/(N-2)$ then there are at least two positive solutions for almost every $0<\lambda < \lambda^*$, the first one (that exists for all $0<\lambda < \lambda^*$) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every $0<\lambda < \lambda^*$) comes from an appropriate (and delicate) mountain pass argument

Electron temperature in electrically isolated Si double quantum dots

Charge-based quantum computation can be attained through reliable control of single electrons in lead-less quantum systems. Single-charge transitions in electrically-isolated double quantum dots (DQD) realised in phosphorus-doped silicon can be detected via capacitively coupled single-electron tunnelling devices. By means of time-resolved measurements of the detector's conductance, we investigate the dots' occupancy statistics in temperature. We observe a significant reduction of the effective electron temperature in the DQD as compared to the temperature in the detector's leads. This sets promises to make isolated DQDs suitable platforms for long-coherence quantum computation.Comment: 4 pages, 3 figure

Discontinuous gradient constraints and the infinity Laplacian

Motivated by tug-of-war games and asymptotic analysis of certain variational problems, we consider a gradient constraint problem involving the infinity Laplace operator. We prove that this problem always has a solution that is unique if a certain regularity condition on the constraint is satisfied. If this regularity condition fails, then solutions obtained from game theory and $L^p$-approximation need not coincide

The first nontrivial eigenvalue for a system of p−p-Laplacians with Neumann and Dirichlet boundary conditions

We deal with the first eigenvalue for a system of two $p-$Laplacians with Dirichlet and Neumann boundary conditions. If \Delta_{p}w=\mbox{div}(|\nabla w|^{p-2}w) stands for the $p-$Laplacian and $\frac{\alpha}{p}+\frac{\beta}{q}=1,$ we consider $$\begin{cases} -\Delta_pu= \lambda \alpha |u|^{\alpha-2} u|v|^{\beta} &\text{ in }\Omega,\\ -\Delta_q v= \lambda \beta |u|^{\alpha}|v|^{\beta-2}v &\text{ in }\Omega,\\ \end{cases}$$ with mixed boundary conditions $$u=0, \qquad |\nabla v|^{q-2}\dfrac{\partial v}{\partial \nu }=0, \qquad \text{on }\partial \Omega.$$ We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem $$\lambda_{p,q}^{\alpha,\beta} = \min \left\{\dfrac{\displaystyle\int_{\Omega}\dfrac{|\nabla u|^p}{p}\, dx +\int_{\Omega}\dfrac{|\nabla v|^q}{q}\, dx} {\displaystyle\int_{\Omega} |u|^\alpha|v|^{\beta}\, dx} \colon (u,v)\in \mathcal{A}_{p,q}^{\alpha,\beta}\right\},$$ where $$\mathcal{A}_{p,q}^{\alpha,\beta}=\left\{(u,v)\in W^{1,p}_0(\Omega)\times W^{1,q}(\Omega)\colon uv\not\equiv0\text{ and }\int_{\Omega}|u|^{\alpha}|v|^{\beta-2}v \, dx=0\right\}.$$ We also study the limit of $\lambda_{p,q}^{\alpha,\beta}$ as $p,q\to \infty$ assuming that $\frac{\alpha}{p} \to \Gamma \in (0,1)$, and $\frac{q}{p} \to Q \in (0,\infty)$ as $p,q\to \infty.$ We find that this limit problem interpolates between the pure Dirichlet and Neumann cases for a single equation when we take $Q=1$ and the limits $\Gamma \to 1$ and $\Gamma \to 0$.Comment: 21 pages, 1 figur

Afterglow lightcurves, viewing angle and the jet structure of gamma-ray bursts

Gamma ray bursts are often modelled as jet-like outflows directed towards the observer; the cone angle of the jet is then commonly inferred from the time at which there is a steepening in the power-law decay of the afterglow. We consider an alternative model in which the jet has a beam pattern where the luminosity per unit solid angle (and perhaps also the initial Lorentz factor) decreases smoothly away from the axis, rather than having a well-defined cone angle within which the flow is uniform. We show that the break in the afterglow light curve then occurs at a time that depends on the viewing angle. Instead of implying a range of intrinsically different jets - some very narrow, and others with similar power spread over a wider cone - the data on afterglow breaks could be consistent with a standardized jet, viewed from different angles. We discuss the implication of this model for the luminosity function.Comment: Corrected typo in Eq. 1