Applied Math Colloquium

Solidifying metal alloys develop complex microstructures by the interplay between surface energy and heat and solute transport. The phase field model replaces the sharp solid/liquid interface by a diffuse one and trades a geometrically complicated free boundary problem for a stiff set of partial differential equations with a small parameter. Recent advances in the asymptotic analysis have greatly improved the accuracy and hence the potential of this model to provide quantitative information for real materials. We show how this new second-order asymptotics can be constructed for the case of asymmetric diffusivities, important for real alloy solidification, and discuss other interesting aspects of the model.

Dispersive approximations to nonlinear hyperbolic systems have solitions that are highly oscillatory in regions where the solution of the corresponding hyperbolic equations have a shock. As the dispersive perturbation tends to zero, the solutons of the dispersive approximation with fixed initial values tend weakly, but not strongly, to a limit. These weak limits satisfy not the given hyperbolic equations but some modifications of them. Some completely integrable cases can be analyzed exactly. The modified eqations bear a resemblance to the appearance of Reynolds stresses in turbulence theory.

The Busemann-Petty problem (posed in 1956) asks the following question. Suppose that K and L are two origin-symmetric convex bodies in n-dimensional Euclidean space such that the ((n-1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L. Does it follow that the (n-dimensional) volume of K is smaller than the volume of L?

The answer is negative in dimensions 5 and larger (it was proved during the period from 1975 to 1993 by individual counterexamples lowering the dimension from 12 to 5). The answer is affirmative for dim=3 (R.J.Gardner, 1994). Until very recently, the answer in the case dim=4 was considered to be negative, but several months ago G. Zhang revised the solution and showed that the opposite is true.

We present a new approach to the Busemann-Petty problem, based on the use of the Fourier transform, which initiated the revision of the case dim=4, provided a large class of bodies K for which the answer is positive for every choice of L with larger sections, and put the production of counterexamples in dim=5 and higher on the industrial level. This approach also leads to a new simple solution to the problem (in all dimensions), and explains why the transition occurs between dim=4 and dim=5.

Computer arithmetic can behave in apparently strange ways, even though each elementary floating point operation (+,-,/,*) is performed to high accuracy. We describe various examples of how rounding errors can have surprising or even calamitous consequences, including puzzling differences between results from similar Cray computers, and the Patriot missile software problem in the Gulf war. On the positive side, we describe IEEE standard arithmetic and its advantages for the programmer. As an example of how the effects of rounding errors can be understood we explain how to obtain a sharp bound for the error in summing floating point numbers.

We present geometric tools for algorithms. These tools illustrate that in algorithm discovery a geometric perspective can be an insightful one.

(i) Outlier Removal: A set of points in n-dimensional space (data with multiple attributes), may contain "outliers". They could be due to error in collecting the data (experimental error) or could correspond to an interesting pattern. In either case, it would be useful to find (and separate) outliers.

What precisely is an outlier? We present a robust notion of outliers and a polynomial-time algorithm to remove a small fraction of any set of points (or more generally, any distribution on rationals) so that the remaining set has no outliers. As an application, we give a polynomial-time guarantee for the classical perceptron algorithm (for learning a half-space in n-dimensional space).

(ii) Random Projection: Our second tool is a simple method for reducing the dimensionality of a set of points. Choose at random a low-dimensional subspace (a hyperplane through the origin), and project the set of points to this subspace.

Does this way of reducing dimensionality retain relevant properties of the point set? We show that in several different contexts, random projection is the key to finding efficient algorithms. The first is an old problem from learning theory. Given a set of points labelled "red" or "blue", find a set of k half-spaces such that the "red" points all lie in one intersection of the half-spaces, and the "blue" points all lie outside it. Two other problems that we will mention include text retrieval and minimizing graph bandwidth.

Various properties of the DNA molecule are comprehensively treated within the framework of highly simplified but very useful theoretical models. Some of these models, the results obtained and their comparison with experimental data will be considered in the talk.

Perhaps the most successful of DNA models treats the molecule as a homogeneous and isotropic elastic rod. This model quantitatively explains the macromolecular behavior of linear DNA. It also provides a major clue to understand closed circular DNA molecules (ccDNA). The theoretical treatment of ccDNA heavily relies on the knot theory. Monte Carlo simulations of ccDNA make it possible to predict the equilibrium probability of knot formation in DNA as a function of its length and width, which quantitatively agrees with experiment.

Other DNA theoretical models, the polyelectrolyte model and the helix-coil model, which make it possible to consider DNA behavior depending on the ambient conditions, will be discussed, if time permits.