We give a criterion for the hypoellipticity of a class of differentialoperators of Hormander type$$L = 1/2sum_{i = 0}^n X_i^2 + X_0.$$Our criterion is weaker than the usual finite-type Hormander condition and allows for decencies of $L$ of exponential order on hypersurfaces in the ambient space. The methods of proof are probabilistic, being based upon an analysis of the regularity of the diffusion process driven by $L$. These methods are also used to study the Dirichlet problem associated with exponentially degenerate operators $L$.

We consider the decomposition rules for the tensor powers of thesmallest representations of quantum groups of type E. For the caseswhere the rank is not 9, H. Wenzl has found uniform combinatorialbehavior for decomposing certain summands of the tensor powers usingLittelmann paths. From this he describes generators of part of thecentralizer algebras of these summands in terms of R-matrices acting onpath spaces and thus obtains a 2-parameter family of representations ofArtin's braid group that generalizes the BMW-algebras. In currentresearch, with an eye towards extending the results to the affine (rank9) case, we find a submodule of the tensor power whose simpledecomposition follows the same pattern as in the cases considered byWenzl. The summands we define have a finite decomposition, and we showthat the inclusion rules for these summands behave well with respect tothe Littelmann path formalism. We also note that the eigenvalues of theaction of the R-matrices are predicted by Wenzl's formulas and we can safely follow his program for describing the centralizer algebra.

A crystal is a shape which tessellates the plane (or space) in a periodicway, so that the pattern repeats at regular intervals in all directions.The group of symmetries (translational, rotational, reflectional) of sucha tessellation is called a crystallographic group. In two dimensions thereare exactly 17 different kinds of symmetry, the so-called "wallpapergroups", which I'll describe. I'll also talk about what happens in threedimensions, in hyperbolic space, and how you can make "quasiperiodic"tessellations (Penrose tilings) with five-fold symmetry.

Lattice polytopes are polytopes with lattice vertices (points with integral coordinates). One can associate with lattice polytopes algebraic and geometric structures such as affine semigroup rings and toric varieties. The aim of this talk is to explain this relationship between combinatorics, algebra and geometry by means of triangulations which involve only the lattice points of the polytop

Our good calculus students know that $sum_{n=1}^infty frac 1n$diverges and that $sum_{n=1}^infty frac{(-1)^n}{n}$ converges. Ourvery good students can even explain why $sum_{n=1}^infty frac{(-1)^{lfloor n /3
floor}}{n}$ converges. Our stellar calculusstudents may even be able to explain why $sum_{n=1}^infty frac{(-1)^{lfloor log n
floor}}{n}$ diverges. In joint work withBruce Reznick and Monika Serbinowska, we show that $$sum_{n=1}^infty frac{(-1)^{lfloor n sqrt{2}
floor}}{n}$$converges. Our proofs rely on Diophantine properties of $sqrt{2}$, and donot apply (for example) if $sqrt{2}$ is replaced by$frac{sqrt{5}+1}{2}$.

Bussemaker et al. (2000, PNAS) proposed the simple idea ofmodeling DNA non coding sequence as a concatenation of words and gavean algorithm to reconstruct deterministic words from an observedsequence. Moving from the same premises, we consider words that canbe spelled in a variety of forms (hence accounting for varying degreesof conservation of the same motif across genome locations).These ``words'' correspond to binding sites of regualtory proteins. Theoverall frequency of occurrence of each word in the sequence and theparameters describing the random spelling of words are estimated in amaximum-likelihood framework using an E-M gradient algorithm. Once these parameters are estimated, it is possible toevaluate the probability with which each motif occurs at a givenlocation in the sequence. These conditional probabilities can be used to predict whichgenes experience similar transcription regulations. Gene expression data can be used tovalidate/refine such predictions.