Abstract: In addition to formal definitions and theorems,
mathematical theories also contain clever, context-sensitive
notations, usage conventions, and proof methods. To mechanize
advanced mathematical results it is essential to capture
these...
The restricted 3-body problem has an intriguing dynamics. A deep
observation of Jacobi is that in rotating coordinates the problem
admits an integral. In joint work with P. Albers, G. Paternain and
O. van Koert, we...
The problem of capture in the planar restricted three-body
problem is addressed. In particular, weak capture is described,
which occurs at a complicated region called the weak stability
boundary, where the motion is...
Given data drawn from a mixture of multivariate Gaussians, a
basic problem is to accurately estimate the mixture parameters. We
provide a polynomial-time algorithm for this problem for any fixed
number ($k$) of Gaussians in $n$ dimensions (even...
A popular practical method of obtaining a good estimate of the
error rate of a learning algorithm is k-fold cross-validation.
Here, the set of examples is first partitioned into k equal-sized
folds. Each fold acts as a test set for evaluating...
Topological spaces given by either (1) complements of coordinate
planes in Euclidean space or (2) spaces of non-overlapping
hard-disks in a fixed disk have several features in common. The
main results, in joint work with many people, give...
We develop the droplet scaling theory for the low temperature
critical behavior of two-dimensional Ising spin glasses. The models
with integer bond energies vs. continuously-distributed bond
energies are in the...
To a regular algebraic cuspidal representation of GL(2) over a
quadratic imaginary field, whose central character is conjugation
invariant, Taylor et al. associated a two dimensional Galois
representation which is unramified at l different from p...
Iwasawa developed his theory for class groups in towers of
cyclotomic fields partly in analogy with Weil's theory of curves
over finite fields. In this talk, we present another such
conjectural analogy. It seems intertwined with Leopoldt's...
A knot is simply a tangled loop in ordinary three-dimensional
space, such as often causes us frustration in everyday life. Knots
are also the subject of a rather rich mathematical theory. In the
last three decades, it has unexpectedly turned out...
