Professor Jenny Harrison

I am a Professor of Mathematics at the University of California, Berkeley. You may want to check the Research section. Some other information is available from the About section.

You will need to browse the More section. There's some material there now, and I will be adding sections as I think of them. Right now you can find a few pictures and a little about my unusual math background.

The experimental undergraduate math class "Chainlet Geometry -- from discrete calculus to soap films, fractals, manifolds and beyond" has just come to a successful completion. This course was based on a graduate course I taught in the spring of 2005, but with recent advances, the material was more streamlined and geared towards undergraduates. It relied only on the basics of abstract linear algebra with no assumptions of classical coordinate calculus - not even the classical one variable derivative or Riemann integral! The theory is a blend of algebra, analysis and geometry, and unifies much of mathematics, both simplifying proofs and extending applications. Here is the syllabus. Our class completed essentially all of this, and were able to add some applications not on the syllabus, including solutions to Dirichlet's problem, new particle/field models of charged particles, and basic results in differential topology, including the Hairy Ball Theorem and the Brouwer Fixed Point Theorem. A text covering the material of this course is under preparation. This will include applications to real analysis, such as the intermediate value theorem, Rolle's theorem and the Mean Value Theorem. It should be suitable for a one semester honor's course in mathematics. This course was based on six lectures on Chainlet Theory given in the Summer School for Mathematical Physics in Ravello, Italy, September 2004. Notes from these six lectures are available, but they are sometimes a bit rough. here. A more recent preprint was posted on the arxiv in January 2006, and revised May 2006 Most recent chainlets preprint To date, the theory has spawned three new extensions of calculus:

• Calculus on nonsmooth domains (e.g., fractals)
• Bilayer calculus (e.g., soap films)
• Discrete exterior calculus

Interest in this theory seems to be growing. It comes from the heart of pure mathematics, but the applied world seemed to have appreciated it first. Besides the Ravello lectures for mathematical physicists, there has been a lecture on chainlets at Caltech in a meeting on discrete numerical methods (September, 2003), five lectures on chainlets given to engineers in Tampere Finland, and lectures given at Sandia labs (August, 2005), continuum mechanics symposium (Rome, October 2005), and in numerical methods (Parlett seminar Feb 2005). Recently, pure mathematicians have begun to enquire about chainlets with invitations to speak in Taubes' Harvard seminar, October 2005), the Northwest meeting on geometric analysis (Corvallis, November 2005), Algebra seminar, U.C. Berkeley (November, 2005), PDE seminar, U.C. Davis (April, 2006), U.C. Berkeley analysis seminar (April, 2006).

Chainlets form a predual to differential forms, whereas currents are a dual to differential forms. Chainlets form a proper subspace of currents. They are algebraically closed under basic operations and products of algebra and analysis. It is of great significance that chainlets form a normed space, whereas currents are only a topological vector space. For more details, see Chainlet geometry