I'll be on campus giving a technical seminar on utilizing 
Mathematica, Wolfram|Alpha, and other Wolfram technologies for 
teaching and research, and I thought you might be interested in 
attending.

I like to begin with a technical overview of Mathematica, as well 
as briefly touching on the creation of Wolfram|Alpha. Next, we 
can discuss emerging trends in technology and what is currently 
available (or being developed) to support those trends. Then, to 
give you a sense of what's possible, I'll discuss how other 
organizations use these tools for teaching and research.

I will first discuss why Hamilton-Jacobi equations for nonconvex Hamiltonians are so interesting, and then explain some recent progress in characterizing the geometric structure and other properties of viscosity solutions.

Nucleation and molecular aggregation are important processes in numerous physical and biological systems. In many applications, these processes often take place in confined spaces, involving a finite number of particles.  We examine the classic problem of homogeneous nucleation and self-assembly by deriving and analyzing a fully discrete stochastic master equation.  By enumerating the highest probability steady-states, we derive exact analytical formulae for quenched and equilibrium mean cluster size distributions. Comparing results with those from mass-action models reveals striking differences between the two corresponding equilibrium mean cluster concentrations. These differences depend primarily on the divisibility of the total available mass by the maximum allowed cluster size, and the remainder. When such mass "incommensurability'' arises, a single remainder particle can "emulsify'' the system by significantly broadening the equilibrium mean cluster size distribution. This discreteness-induced broadening effect is periodic in the total mass of the system but arises even when the system size is asymptotically large, provided the ratio of the total mass to the maximum cluster size is finite.  Our findings define a new scaling regime in which results from classic mass-action theories are qualitatively inaccurate, even in the limit of large total system size.  First passage times to the formation of the largest cluster will also be discussed.
 
 

The classical Huaxin Lin's theorem shows that the distance from a matrix A to the set of normal matrices can be estimated in terms of its self-commutator [A,A*]. We obtain a quantitative version of this theorem, "optimal" with respect to the power of self-commutator. Under certain assumptions on A, our approach can be extended to the case of general bounded operators in Hilbert spaces and to elements of C*-algebras of real rank zero. The results are joint with Professor Yuri Safarov from King's College London.