Georgia Institute of Technology College of Sciences

I will discuss how various geometric categories (e.g. smooth manifolds, complex manifolds) can be be described in terms of locally ringed spaces. (A locally ringed space is a topological spaces endowed with a sheaf of rings whose stalks are local rings.) As an application of the notion of locally ringed space, I'll define what a scheme is.

We discuss comparison principle for viscosity solutions of fully nonlinear elliptic PDEs in $\R^n$ which may have superlinear growth in $Du$ with variable coefficients. As an example, we keep the following PDE in mind:$$-\tr (A(x)D^2u)+\langle B(x)Du,Du\rangle +\l u=f(x)\quad \mbox{in }\R^n,$$where $A:\R^n\to S^n$ is nonnegative, $B:\R^n\to S^n$ positive, and $\l >0$. Here $S^n$ is the set of $n\ti n$ symmetric matrices. The comparison principle for viscosity solutions has been one of main issues in viscosity solution theory. However, we notice that we do not know if the comparison principle holds unless $B$ is a constant matrix. Moreover, it is not clear which kind of assumptions for viscosity solutions at $\infty$ is suitable. There seem two choices: (1) one sided boundedness ($i.e.$ bounded from below), (2) growth condition.In this talk, assuming (2), we obtain the comparison principle for viscosity solutions. This is a work in progress jointly with O. Ley.

We discuss the convergence properties of first-order methods for two problems that arise in computational geometry and statistics: the minimum-volume enclosing ellipsoid problem and the minimum-area enclosing ellipsoidal cylinder problem for a set of m points in R^n. The algorithms are old but the analysis is new, and the methods are remarkably effective at solving large-scale problems to high accuracy.

In this talk we first present the exact asymptotics of the optimal error in the weighted L_p-norm, 1\leq p \leq \infty, of linear spline interpolation of an arbitrary bivariate function f \in C^2([0,1]^2). We further discuss the applications to numerical integration and adaptive mesh generation for finite element methods, and explore connections with the problem of approximating the convex bodies by polytopes. In addition, we provide the generalization to asymmetric norms. We give a brief review of known results and introduce a series of new ones. The proofs of these results lead to algorithms for the construction of asymptotically optimal sequences of triangulations for linear interpolation. Moreover, we derive similar results for other classes of splines and interpolation schemes, in particular for splines over rectangular partitions. Last but not least, we also discuss several multivariate generalizations.