I will first recall some examples of L-functions and indicate some of the ways they have been important in algebraic number theory. I will then describe what appears to be their intimate connection with Galois theory (eg the Fontaine-Mazur conjectures), as well as touching on their relationship with algebraic geometry and automorphic forms. Finally, I will discuss what can be proved in this direction.

The large-scale flow of the mid-latitude atmosphere and oceans is governed by systems of PDEs that approximate the Euler and Navier-Stokes equations in the presence of rotation and stratification. These PDEs include the quasi-geostrophic and shallow-water equations in two and three dimensions (2-D and 3-D), as well as the so-called primitive equations in 3-D; the latter are used, in discretized form, to formulate the general circulation models used in numerical weather prediction and climate studies.
We illustrate the mathematical issues that arise in solving these PDEs by the specific problem of the oceans' wind-driven circulation. This circulation is dominated by a large, anticyclonic and a smaller, cyclonic gyre in each mid-latitude basin on Earth. These two gyres are induced by the shear in the winds that cross the respective ocean basins. They share the eastward extension of western boundary currents, such as the Gulf Stream or Kuroshio. The boundary currents and eastward jets carry substantial amounts of heat and momentum. The jets also contribute to mixing in the oceans by their "whiplashing" oscillations and the detachment of eddies from them.
We study the low-frequency variability of this double-gyre circulation, for time-constant and purely periodic wind stress. Both analytical and numerical methods of nonlinear dynamics are applied in our study. Symmetry-breaking bifurcations occur, from steady to periodic and aperiodic flows, as wind stress increases or dissipation decreases. The first bifurcation is of pitchfork or perturbed-pitchfork type, depending on the model's degree of realism. Two types of oscillatory instabilities arise by supercritical Hopf bifurcation, with periods of a few months and a few years, respectively. Numerical evidence points to homoclinic orbits that connect high- and low-energy branches of steady-state solutions. The results are compared with decade-long in situ and more recent, satellite observations of three ocean basins, the North and South Atlantic, and the North Pacific, and their significance for climate variability is discussed.
This talk reflects collaborative work with K.-I. Chang (KORDI), H. Dijkstra (CSU and Utrecht U.), Y. Feliks (IIBR, Israel), K. Ide (UCLA), S. Jiang (Wall Street), F.-f. Jin (FSU), C. A. Lai (LANL), G. Loeper (ENS, Paris), E. Simonnet (INLN, Nice), S. Speich (UBO/Ifremer, Brest), L. U. Sushama (UQAM), R. Temam (Indiana U. & Paris Sud/Orsay), and S. Wang (Indiana U.).

It is well-known (Liouville's Theorem) that a complex projective manifold X does not admit any non-constant algebraic (or holomorphic) functions. We explain how the collection of all algebraic vector bundles on X and morphisms between them gives rise to a structure (the derived category of X) which serves as a replacement -in many interesting ways - of the L^2 space of functions in analysis. Several results describing the properties of this structure will be explained.

For the Schrödinger operator $-\Delta + V$ on $L^2 (\mathbb{R}^n)$, let $N(V)$ be the number of bound states. We will review a few classical bounds for $N(V)$: Birman-Schwinger, Cwikel-Lieb-Rosenbljum, Birman-Solomjak. We will then present new bounds for $N(V)$ in dimension two. This work was motivated by a conjecture of Khuri, Martin and Wu.