Focal neural perturbations reshape low-dimensional trajectories of brain activity supporting cognitive performance

The emergence of distributed patterns of neural activity supporting brain functions and behavior can be understood by study of the brain’s low-dimensional topology. Functional neuroimaging demonstrates that brain activity linked to adaptive behavior is constrained to low-dimensional manifolds. In human participants, we tested whether these low-dimensional constraints preserve working memory performance following local neuronal perturbations. We combined multi-session functional magnetic resonance imaging, non-invasive transcranial magnetic stimulation (TMS), and methods translated from the fields of complex systems and computational biology to assess the functional link between changes in local neural activity and the reshaping of task-related low dimensional trajectories of brain activity. We show that specific reconfigurations of low-dimensional trajectories of brain activity sustain effective working memory performance following TMS manipulation of local activity on, but not off, the space traversed by these trajectories. We highlight an association between the multi-scale changes in brain activity underpinning cognitive function.

Chebyshev polynomials and inequalities for Kleinian groups

The principal character of a representation of the free group of rank two into [Formula: see text] is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is, a Kleinian group. A classical necessary condition is Jørgensen’s inequality. Here, we use certain shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.

The Cohomological Genus and Symplectic Genus for 4-Manifolds of Rational or Ruled Types

An important problem in low dimensional topology is to understand the properties of embedded or immersed surfaces in 4-dimensional manifolds. In this article, we estimate the lower genus bound of closed, connected, smoothly embedded, oriented surfaces in a smooth, closed, connected, oriented 4-manifold with the cohomology algebra of a rational or ruled surface. Our genus bound depends only on the cohomology algebra rather than on the geometric structure of the 4-manifold. It provides evidence for the genus minimizing property of rational and ruled surfaces.

Orbital chemistry of high valence band convergence and low-dimensional topology in PbTe

The tight-binding method provides insight into the orbital interactions that lead to the exceptional thermoelectric performance of PbTe. Using this framework, we can predict strategies to achieve enhanced thermoelectric performance in new alloys.