Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

Neural correlates of wishful thinking.

Wishful thinking (WT) implies the overestimation of the likelihood of desirable events. It occurs for outcomes of personal interest, but also for events of interest to others we like. We investigated whether WT is grounded on low-level selective attention or on higher level cognitive processes including differential weighting of evidence or response formation. Participants in our MRI study predicted the likelihood that their favorite or least favorite team would win a football game. Consistent with expectations, favorite team trials were characterized by higher winning odds. Our data demonstrated activity in a cluster comprising parts of the left inferior occipital and fusiform gyri to distinguish between favorite and least favorite team trials. More importantly, functional connectivities of this cluster with the human reward system were specifically involved in the type of WT investigated in our study, thus supporting the idea of an attention bias generating WT. Prefrontal cortex activity also distinguished between the two teams. However, activity in this region and its functional connectivities with the human reward system were altogether unrelated to the degree of WT reflected in the participants' behavior and may rather be related to social identification, ensuring the affective context necessary for WT to arise.

The functional significance of EEG microstates-Associations with modalities of thinking

The momentary, global functional state of the brain is reflected by its electric field configuration. Cluster analytical approaches consistently extracted four head-surface brain electric field configurations that optimally explain the variance of their changes across time in spontaneous EEG recordings. These four configurations are referred to as EEG microstate classes A, B, C, and D and have been associated with verbal/phonological, visual, attention reorientation, and subjective interoceptive-autonomic processing, respectively. The present study tested these associations via an intra-individual and inter-individual analysis approach. The intra-individual approach tested the effect of task-induced increased modality-specific processing on EEG microstate parameters. The inter-individual approach tested the effect of personal modality-specific parameters on EEG microstate parameters. We obtained multichannel EEG from 61 healthy, right-handed, male students during four eyes-closed conditions: object-visualization, spatial-visualization, verbalization (6 runs each), and resting (7 runs). After each run, we assessed participants' degrees of object-visual, spatial-visual, and verbal thinking using subjective reports. Before and after the recording, we assessed modality-specific cognitive abilities and styles using nine cognitive tests and two questionnaires. The EEG of all participants, conditions, and runs was clustered into four classes of EEG microstates (A, B, C, and D). RMANOVAs, ANOVAs and post-hoc paired t-tests compared microstate parameters between conditions. TANOVAs compared microstate class topographies between conditions. Differences were localized using eLORETA. Pearson correlations assessed interrelationships between personal modality-specific parameters and EEG microstate parameters during no-task resting. As hypothesized, verbal as opposed to visual conditions consistently affected the duration, occurrence, and coverage of microstate classes A and B. Contrary to associations suggested by previous reports, parameters were increased for class A during visualization, and class B during verbalization. In line with previous reports, microstate D parameters were increased during no-task resting compared to the three internal, goal-directed tasks. Topographic differences between conditions concerned particular sub-regions of components of the metabolic default mode network. Modality-specific personal parameters did not consistently correlate with microstate parameters except verbal cognitive style which correlated negatively with microstate class A duration and positively with class C occurrence. This is the first study that aimed to induce EEG microstate class parameter changes based on their hypothesized functional significance. Beyond, the associations of microstate classes A and B with visual and verbal processing, respectively and microstate class D with interoceptive-autonomic processing, our results suggest that a finely-tuned interplay between all four EEG microstate classes is necessary for the continuous formation of visual and verbal thoughts, as well as interoceptive-autonomic processing. Our results point to the possibility that the EEG microstate classes may represent the head-surface measured activity of intra-cortical sources primarily exhibiting inhibitory functions. However, additional studies are needed to verify and elaborate on this hypothesis.

The algebraic density property for affine toric varieties

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén–Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X∖Y≠TX∖Y≠T. For toric surfaces we are able to classify those which possess a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.

The WTO beyond Doha: Thinking Ahead

This presentation by Pierre Sauvé was presented to the ICC Commission on Trade and Investment Policy in Paris.