Hans Aberg
Hans AbergMy first research concerned finiteness properties of metabelian groups: For example, among the set of arithmetic groups with invertible primes, this is the first non-trivial case. Among other things, I proved a difficult conjecture by Robert Bieri and Ralph Strebel, relating the finiteness properties to number theory. The proof made heavy use of algebra, such as group theory, ring theory, commutative algebra, number theory, homological algebra and category theory, but also relied on topology, algebraic topology, homotopy theory, and geometry. These latter topics became a part by work originated by Kenneth S. Brown.
I discovered a formula (equation (1.16) in the manuscript) that suggests that in order to perform the metric variations of the GR Lagrangian when including QM statistical waves, then this kind of intrinsic spin theory is necessary. I then developed the intrinsic spin theory, so that it can be used for translating QM formulas into Lorentz coordinate independent formulas. This is what is described in the manuscript.
Recently, I put the intrinsic spin theory to work, and made additional findings relating GR and QM, a construction of a wave based Fock space on a suitably modified Lorentz cotangent bundle manifold. The intrinsic spin makes it possible to construct a coordinate independent, Lorentz-covariant positive definite metric on this Fock space. (If one, for example, converts the formula (1-57) in the Streater-Wightman book on PCT, that scalar product will be Lorentz covariant and positive definite, but not coordinate independent.) Other parts of the construction correspond to QFT axioms. In principle, by supplying a suitable wave equation, one could extract QFT via Sobolev spaces by such a construction. (To be written up.)
For an overview of physics unification theories in general, and non-perturbative quantum gravity in particular, see the paper by Abhay Ashtekar, gr-qc/9302024.I made an election method, called Voting By Priorities (VBP), extending the Single Transferrable Votes (STV) person and the Greatest Remainder party election methods. It sports a new quota, and is very suitable for computer implementation. One variation of the election method uses the Meek vote redistribution method. Several possible generalizations are at hand. The cross-sum order I introduced, leads to the same kind of majority principle as in Bram's Approval Voting.
The earlier general public easy access version is still available, but I have decided to not post it.
The C++ source codes of my election system program are also available.
Lately, I have been interested in relations between pure mathematics and computer science, the semantic theory used in constructing computer languages:
I have made a computer semantics, successfully implemented, that can be used to produce both functional (Church lambda-theory based) and imperative programming styles, and I am currently writing this up as a presentable theory. Church's lambda-theory describes the variable substitutions used in mathematics, and could be described as a combined syntactic/semantic description of a common (meta-)mathematical feature, similar to the one used in the construction of computer languages.
A computer language consists of a grammar and a set of rules, translating syntactic components into a semantics (for a runnable program, usually binary code, but it could also be another computer language). My computer semantics is syntax-independent, so it is possible to use different syntaxes translating into the same semantics, something which is difficult to achieve with computer languages.
My ideas are not based on lambda theory, but could be described as being based on an algebra of function pointers; if this algebra is sufficiently rich, both lambda theory and imperative programming styles can be produced by it. I used C++ to make an implementation of this semantics; the working name of this program is Synerge. The effect of lazy evaluation is produced by letting strict evaluations return semantic expressions representing future evaluations. The idea of multimethods is a crucial part of getting these ideas to work.
I have also found an object oriented description which I feel satisfied with: It is based on splitting the traditional object orientation concept into several parts, type restriction casts, implementation, and request inheritance, which gives some logical leeway for producing a more flexible object oriented model. Some of these ideas come from pure mathematics, like the restriction maps of sheaves. This object oriented model is not implemented as of yet.
In another direction, I have made some new investigations of the use of monads in implementing computer languages. A monad (also called a triple) in a category C is just a monoid in the monoidal endofunctor category End(C), and is typically used for describing an adjoint category to C.
My line of investigation follows the use of monads as describing a certain code extension property: The ability to ensure that old and new language features can work together is of course very important in the design of computer languages, and monads look to be the right tool for describing this. Another example: the different exception handling implementation styles of C++ and Haskell will have the same monad, as they have the same code extension property.
The monads used in computer science so far deals only with the Kleisli category, the full subcategory of free monad algebras in the category of monad algebras. So, in another line of investigation, I have investigated the category of all monad algebras, with the idea that this might be developed as a tool for implementing categories in computer languages.
Some background:The use of monads in computer science started with Moggi; he had the idea that monads should supply the extra semantic information needed to implement lambda-theory. One of Moggi's papers labeled abs-view is an introduction to the subject.