Subtle characteristic classes for Spin-torsors

Extending (Smirnov and Vishik, Subtle Characteristic Classes, arXiv:1401.6661), we obtain a complete description of the motivic cohomology with $${{\,\mathrm{\mathbb {Z}}\,}}/2$$ Z / 2 -coefficients of the Nisnevich classifying space of the spin group $$Spin_n$$ S p i n n associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel–Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from $$I^3$$ I 3 , which are closely related to $$Spin_n$$ S p i n n -torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel–Whitney class. Moreover, exploiting the relation between $$Spin_7$$ S p i n 7 and $$G_2$$ G 2 , we describe completely the motivic cohomology ring of the Nisnevich classifying space of $$G_2$$ G 2 . The result in topology was obtained by Quillen (Math Ann 194:197–212, 1971).

Relativistic fermions: Introduction

Some basic concepts needed for the discussion of Fermi fields have been introduced earlier, as in quantum mechanics (QM) with Grassmann variables, a representation by field integrals of the statistical operator e<συπ>−βH</συπ> for the non-relativistic Fermi gas in the formalism of second quantization, and an expression for the evolution operator. Here, it is first recalled how relativistic fermions transform under the spin group. The free action for Dirac fermions is analysed, the relation between fields and particles explained, an expression for the scattering matrix obtained, and the non-relativistic limit of a model of self-coupled massive Dirac fermions derived. A formalism of Euclidean relativistic fermions is then introduced. In the Euclidean formalism: fermions transform under the fundamental representation of the spin group Spin(d) associated with the SO(d) rotation group (spin 1/2 fermions for d = 4). As for the scalar field theory, the Gaussian integral, which corresponds to a free field theory is calculated. Then the generating functional of correlation functions is obtained by adding a source term to the action. The field integral corresponding to a general action with an interaction expandable in powers of the field, can be expressed in terms of a series of Gaussian integrals, which can be calculated, for example, with the help of Wick's theorem. The connection between spin and statistics is verified by a simple perturbative calculation. The appendix describes a few additional properties of the spin group, the algebra of γ matrices, and the corresponding spinors for Euclidean fermions.

Iterated Darboux Transformation for Isothermic Surfaces in Terms of Clifford Numbers

Isothermic surfaces are defined as immersions with the curvture lines admitting conformal parameterization. We present and discuss the reconstruction of the iterated Darboux transformation using Clifford numbers instead of matrices. In particulalr, we derive a symmetric formula for the two-fold Darboux transformation, explicitly showing Bianchi’s permutability theorem. In algebraic calculations an important role is played by the main anti-automorphism (reversion) of the Clifford algebra C(4,1) and the spinorial norm in the corresponding Spin group.

Arithmetic purity of strong approximation for semi-simple simply connected groups

In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with ${\mathrm {codim}}(G\setminus U, G)\geqslant 2$, we prove that (i) if $G$ is $k$-simple and $k$-isotropic, then $U$ satisfies strong approximation off any finite number of places; and (ii) if $G$ is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.

On the global Gan-Gross-Prasad conjecture for general spin groups

In the 1990s, Benedict Gross and Dipendra Prasad formulated an intriguing conjecture connected with restriction laws for automorphic representations of a particular group. More recently, Gan, Gross, and Prasad extended this conjecture, now known as the Gan-Gross-Prasad Conjecture, to the remaining classical groups. Roughly speaking, they conjectured the non-vanishing of a certain period integral is equivalent to the non-vanishing of the central value of a certain L- function. Ichino and Ikeda refined the conjecture to give an explicit relationship between this central value of a L-function and the period integral. We propose a similar conjecture for a nonclassical group, the general spin group, and prove one case.

Changes in Cortical Activation Patterns in Language Areas following an Aerobic Exercise Intervention in Older Adults

Previous work has shown that older adults who evidence increased right inferior frontal gyrus (IFG) activity during language tasks show decreased sematic verbal fluency performance. The current study sought to evaluate if an aerobic exercise intervention can alter patterns of brain activity during a semantic verbal fluency task assessed by functional magnetic resonance imaging (fMRI). Thirty-two community-dwelling, sedentary older adults were enrolled to a 12-week aerobic “Spin” exercise group or a 12-week nonaerobic exercise control condition (Balance). Thirty participants completed their assigned intervention (16 Spin; 14 Balance) with pre- and postintervention assessments of a semantic verbal fluency task during fMRI and estimated VO2max testing. There was a significant increase in the change scores for estimated VO2max of the Spin group when compared to the Balance group. Semantic verbal fluency output within the scanner was also improved in the Spin group as compared to controls at postassessment. Group fMRI comparisons of IFG activity showed lower activity in the right IFG following the intervention in the aerobic Spin group when compared to the Balance group. Regression analysis of imaging data with change in both estimated VO2max and semantic verbal fluency was negatively correlated with activity in right IFG. The current work is registered as clinical trial with NCT01787292 and NCT02787655.