arXiv subjects
Publications and source records attributed to Jeremy Taylor.
We characterize the kernel of Iwahori-Whittaker averaging in microlocal terms. Applying this to automorphic sheaves, we generalize Faergeman and Raskin's theorem that anti-temperedness is equivalent to having irregular singular support. Moreover, using a Radon transform argument of Bezrukavnikov and Morton-Ferguson, we extend the tilting property of Whittaker averaged central sheaves to integer coefficients.
We identify equivariant quasicoherent sheaves on the Grothendieck alteration of a reductive group $\mathsf{G}$ with universal monodromic Iwahori--Whittaker sheaves on the enhanced affine flag variety of the Langlands dual group $G$. This extends a similar result for equivariant quasicoherent sheaves on the Springer resolution due to Arkhipov--Bezrukavnikov. We further give a monoidal identification between adjoint equivariant coherent sheaves on the group $\mathsf{G}$ itself and bi-Iwahori--Whittaker sheaves on the loop group of $G$. These results are used in the sequel to this paper to prove the tame local Betti geometric Langlands conjecture of Ben-Zvi--Nadler. Our proof of fully faithfulness provides an alternative to the argument of Arkhipov--Bezrukavnikov. Namely, while they localize in unipotent directions, we localize in semi-simple directions, thereby reducing fully faithfulness to an order of vanishing calculation in semi-simple rank one.
We prove a monoidal equivalence between spectral and automorphic realizations of the universal affine Hecke category, thereby proving the tamely ramified local Betti geometric Langlands correspondence, as conjectured by Ben-Zvi--Nadler [BZN07, BZN18]. Specializing to the case of unipotent monodromy, this provides another argument for a fundamental theorem of Bezrukavnikov [B16].
We prove that the category of equivariant perverse sheaves on the affine Grassmannian of PGL(2, R) is highest weight and we construct the projective objects. Moreover we prove that the category of perverse sheaves on the odd component is equivalent to the principal block of Lusztig's quantum SL(2) at a primitive fourth root of unity.
We construct the universal monodromic big tilting sheaf on base affine space and calculate its endomorphisms. By formal completion, we recover Soergel's pro-unipotent Endomorphismensatz with arbitrary field coefficients. We give a Soergel bimodules description of the universal monodromic Hecke category and deduce a conjecture of Eberhardt that uncompletes Koszul duality.
Geometric Langlands predicts an isomorphism between Whittaker coefficients of Eisenstein series and functions on the moduli space of $\check{N}$-local systems. We prove this formula by interpreting Whittaker coefficients of Eisenstein series as factorization homology and then invoking Beilinson and Drinfeld's formula for chiral homology of a chiral enveloping algebra.
For a smooth projective curve $X$ and reductive group $G$, the Whittaker functional on nilpotent sheaves on $\text{Bun}_G(X)$ is expected to correspond to global sections of coherent sheaves on the spectral side of Betti geometric Langlands. We prove that the Whittaker functional calculates the (shifted) microstalk of nilpotent sheaves at the point in the Hitchin moduli where the Kostant section intersects the global nilpotent cone. In particular, the (shifted) Whittaker functional is exact for the perverse $t$-structure and commutes with Verdier duality. Our proof is topological and depends on the intrinsic local hyperbolic symmetry of $\text{Bun}_G(X)$. It is an application of a general result relating vanishing cycles to the composition of restriction to an attracting locus followed by vanishing cycles.
We consider the problem of inference for parameters selected to report only after some algorithm, the canonical example being inference for model parameters after a model selection procedure. The conditional correction for selection requires knowledge of how the selection is affected by changes in the underlying data, and current research explicitly describes this selection. In this work, we assume 1) we have in silico access to the selection algorithm and 2) for parameters of interest, the data input into the algorithm satisfies (pre-selection) a central limit theorem jointly with an estimator of our parameter of interest. Under these assumptions, we recast the problem into a statistical learning problem which can be fit with off-the-shelf models for binary regression. The feature points in this problem are set by the user, opening up the possibility of active learning methods for computationally expensive selection algorithms. We consider two examples previously out of reach of this conditional approach: stability selection and multiple cross-validation.
We address the microscopic origin of the current-induced forces by analyzing results of first principles density functional calculations of atomic gold wires connected to two gold electrodes with different electrochemical potentials. We find that current induced forces are closely related to the chemical bonding, and arise from the rearrangement of bond charge due to the current flow. We explain the current induced bond weakening/strengthening by introducing bond charges decomposed into electrode components.
We report first-principles studies of electronic transport in monolayers of Tour wires functionalized with different side groups. An analysis of the scattering states and transmission eigenchannels suggests that the functionalization does not strongly affect the resonances responsible for current flow through the monolayer. However, functionalization has a significant effect on the interactions within the monolayer, so that monolayers with NO$_2$ side groups exhibit local minima associated with twisted conformations of the molecules. We use our results to interpret observations of negative differential resistance and molecular memory in monolayers of NO$_2$ functionalized molecules in terms of a twisting of the central ring induced by an applied bias potential.
We describe an ab initio method for calculating the electronic structure, electronic transport, and forces acting on the atoms, for atomic scale systems connected to semi-infinite electrodes and with an applied voltage bias. Our method is based on the density functional theory (DFT) as implemented in the well tested Siesta approach (which uses non-local norm-conserving pseudopotentials to describe the effect of the core electrons, and linear combination of finite-range numerical atomic orbitals to describe the valence states). We fully deal with the atomistic structure of the whole system, treating both the contact and the electrodes on the same footing. The effect of the finite bias (including selfconsistency and the solution of the electrostatic problem) is taken into account using nonequilibrium Green's functions. We relate the nonequilibrium Green's function expressions to the more transparent scheme involving the scattering states. As an illustration, the method is applied to three systems where we are able to compare our results to earlier ab initio DFT calculations or experiments, and we point out differences between this method and existing schemes. The systems considered are: (1) single atom carbon wires connected to aluminum electrodes with extended or finite cross section, (2) single atom gold wires, and finally (3) large carbon nanotube systems with point defects.
Electronic transport properties of Au nano-structure are investigated using both experimental and theoretical analysis. Experimentally, stable Au nanowires were created using mechanically controllable break junction in air, and simultaneous current-voltage (I-V) and differential conductance $δI/δV$ data were measured. The atomic device scale structures are mechanically very stable up to bias voltage $V_b\sim0.6V$ and have a life time of a few $minutes$. Facilitated by a shape function data analysis technique which finger-prints electronic properties of the atomic device, our data show clearly differential conductance fluctuations with an amplitude $>1%$ at room temperature, and a nonlinear I-V characteristics. To understand the transport features of these atomic scale conductors, we carried out {\it ab initio} calculations on various Au atomic wires. The theoretical results demonstrate that transport properties of these systems crucially depend on the electronic properties of the scattering region, the leads, and most importantly the interaction of the scattering region with the leads. For ideal, clean Au contacts, the theoretical results indicate a linear I-V behavior for bias voltage $V_b<0.5V$. When sulfur impurities exist at the contact junction, nonlinear I-V curves emerge due to a tunnelling barrier established in the presence of the S atom. The most striking observation is that even a single S atom can cause a qualitative change of the I-V curve from linear to nonlinear. A quantitatively favorable comparison between experimental data and theoretical results is obtained. We also report other results concerning quantum transport through Au atomic contacts.
We present an {\it ab initio} analysis of electron conduction through a $C_{60}$ molecular device. Charge transfer from the device electrodes to the molecular region is found to play a crucial role in aligning the lowest unoccupied molecular orbital (LUMO) of the $C_{60}$ to the Fermi level of the electrodes. This alignment induces a substantial device conductance of $\sim 2.2 \times (2e^2/h)$. A gate potential can inhibit charge transfer and introduce a conductance gap near $E_F$, changing the current-voltage characteristics from metallic to semi-conducting, thereby producing a field effect molecular current switch.
Spin-coherent quantum transport in carbon nanotube magnetic tunnel junctions was investigated theoretically. A spin-valve effect is found for metallic, armchair tubes, with a magneto-conductance ratio ranging up to 20%. Because of the finite length of the nanotube junctions, transport is dominated by resonant transmission. The magnetic tunnel junctions are found to have distinctly different transport behavior depending on whether or not the length of the tubes is commensurate with a 3N+1 rule, with N the number of basic carbon repeat units along the nanotube length.