Georgia Institute of Technology College of Sciences

This is the first in the series of two talks aimed to discuss a recent work of Ontaneda which gives a poweful method of producing negatively curved manifolds. Ontaneda's work adds a lot of weight to the often quoted Gromov's prediction that in a sense most manifolds (of any dimension) are negatively curved.

One of the basic problems of Harmonic analysis is to determine ifa given collection of functions is complete in a given Hilbert space. Aclassical theorem by Beurling and Malliavin solved such a problem in thecase when the space is $L^2$ on an interval and the collection consists ofcomplex exponentials. Two closely related problems, the so-called Gap andType Problems, studied by Beurling, Krein, Kolmogorov, Levinson, Wiener andmany others, remained open until recently.In my talk I will  present solutions to the Gap and Type problems anddiscuss their connectionswith adjacent fields.

Knowledge of the distribution of class groups is elusive -- it is not even known if there are infinitely many number fields with trivial class group. Cohen and Lenstra noticed a strange pattern -- experimentally, the group \mathbb{Z}/(9) appears more often than \mathbb{Z{/(3) x \mathbb{Z}/(3) as the 3-part of the class group of a real quadratic field \Q(\sqrt{d}) - and refined this observation into concise conjectures on the manner in which class groups behave randomly. Their heuristic says roughly that p-parts of class groups behave like random finite abelian p-groups, rather than like random numbers; in particular, when counting one should weight by the size of the automorphism group, which explains why \mathbb{Z}/(3) x \mathbb{Z}/(3) appears much less often than \mathbb{Z}/(9) (in addition to many other experimental observations). While proof of the Cohen-Lenstra conjectures remains inaccessible, the function field analogue -- e.g., distribution of class groups of quadratic extensions of \mathbb{F}_p(t) -- is more tractable. Friedman and Washington modeled the \el$-power part (with \ell \neq p) of such class groups as random matrices and derived heuristics which agree with experiment. Later, Achter refined these heuristics, and many cases have been proved (Achter, Ellenberg and Venkatesh). When $\ell = p$, the $\ell$-power torsion of abelian varieties, and thus the random matrix model, goes haywire. I will explain the correct linear algebraic model -- Dieudone\'e modules. Our main result is an analogue of the Cohen-Lenstra/Friedman-Washington heuristics -- a theorem about the distributions of class numbers of Dieudone\'e modules (and other invariants particular to \ell = p). Finally, I'll present experimental evidence which mostly agrees with our heuristics and explain the connection with rational points on varieties.

Knowledge of rays and critical points of infinity in von Mangoldt planes can be applied to understanding the structure of open complete manifolds with lower radial curvature bounds. We will show how the set of souls is computed for every von Mangoldt plane of nonnegative curvature. We will also make some observations on the structure of the set of critical points of infinity for von Mangoldt planes with negative curvature.