On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.

Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

Congruences between modular forms: Raising the level and dropping Euler factors

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida’s result accounts for congruences in terms of the value of an L-function, and Ribet’s result is related to the behavior of the period that appears there. Wiles’ theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at “nonminimal level” is obtained from one at “minimal level” by dropping Euler factors from the L-function.

Human γ-aminobutyric acid type B receptors are differentially expressed and regulate inwardly rectifying K+ channels

γ-Aminobutyric acid type B receptors (GABABRs) are involved in the fine tuning of inhibitory synaptic transmission. Presynaptic GABABRs inhibit neurotransmitter release by down-regulating high-voltage activated Ca2+ channels, whereas postsynaptic GABABRs decrease neuronal excitability by activating a prominent inwardly rectifying K+ (Kir) conductance that underlies the late inhibitory postsynaptic potentials. Here we report the cloning and functional characterization of two human GABABRs, hGABABR1a (hR1a) and hGABABR1b (hR1b). These receptors closely match the pharmacological properties and molecular weights of the most abundant native GABABRs. We show that in transfected mammalian cells hR1a and hR1b can modulate heteromeric Kir3.1/3.2 and Kir3.1/3.4 channels. Heterologous expression therefore supports the notion that Kir3 channels are the postsynaptic effectors of GABABRs. Our data further demonstrate that in principle either of the cloned receptors could mediate inhibitory postsynaptic potentials. We find that in the cerebellum hR1a and hR1b transcripts are largely confined to granule and Purkinje cells, respectively. This finding supports a selective association of hR1b, and not hR1a, with postsynaptic Kir3 channels. The mapping of the GABABR1 gene to human chromosome 6p21.3, in the vicinity of a susceptibility locus (EJM1) for idiopathic generalized epilepsies, identifies a candidate gene for inherited forms of epilepsy.

Behavior of several two-dimensional fluid equations in singular scenarios

We give conditions that rule out formation of sharp fronts for certain two-dimensional incompressible flows. We show that a necessary condition of having a sharp front is that the flow has to have uncontrolled velocity growth. In the case of the quasi-geostrophic equation and two-dimensional Euler equation, we obtain estimates on the formation of semi-uniform fronts.

Zeta functions and Eisenstein series on classical groups

We construct an Euler product from the Hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a CM field and the eigenform is holomorphic. We also prove analytic continuation of an Eisenstein series on another unitary group, containing the group just mentioned defined with such an eigenform. As an application of our methods, we prove an explicit class number formula for a totally definite hermitian form over a CM field.

Vascular endothelial growth factor B, a novel growth factor for endothelial cells.

We have isolated and characterized a novel growth factor for endothelial cells, vascular endothelial growth factor B (VEGF-B), with structural similarities to vascular endothelial growth factor (VEGF) and placenta growth factor. VEGF-B was particularly abundant in heart and skeletal muscle and was coexpressed with VEGF in these and other tissues. VEGF-B formed cell-surface-associated disulfide-linked homodimers and heterodimerized with VEGF when coexpressed. Conditioned medium from transfected 293EBNA cells expressing VEGF-B stimulated DNA synthesis in endothelial cells. Our results suggest that VEGF-B has a role in angiogenesis and endothelial cell growth, particularly in muscle.