Computational Group Theory

This was the seventh workshop on Computational Group Theory. It showed that Computational Group Theory has significantly expanded its range of activities. For example, symbolic computations with groups and their representations and computations with infinite groups play a major role nowadays. The talks also presented connections and applications to cryptography, number theory and the algorithmic theory of algebras

This Week's Finds in Mathematical Physics (1-50)

These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of $n$-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird).Comment: 242 page

Unsolved Problems in Group Theory. The Kourovka Notebook

This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update

ARF6 CONTROLS LYSOSOMAL TRANSPORT OF APP AND Ap42 PRODUCTION

Alzheimer’s disease (AD) is characterized by the deposition of Beta-Amyloid (AP) peptide plaques in the brain. Ap peptides are generated by the sequential cleavage of the Amyloid Precursor Protein (APP). The AP42 cleavage product is the most neurotoxic form. Previous studies in our lab have uncovered a novel rapid lysosomal APP trafficking pathway that bypasses the early and late endosomal compartments. We set out to characterize this transport pathway using APP constructs with an N-terminal HA-tag, allowing us to label APP at the cell surface with a fluorescently labeled antibody. SN56 cells and mouse cortical neurons were also co-transfected with fluorescently-tagged compartment marker proteins and a panel of endocytosis regulatory proteins bearing dominant negative and constitutively activating mutations. Rapid APP internalization to lysosomes is stimulated by antibody binding and is increased when Arfl activity was inhibited, but decreased when Arfó activity was inhibited. In addition, disruption of either Arfó or Arfl was able to significantly reduce Ap42 secretion into the media. Our findings suggest that rapid APP transport to lysosomes is regulated by Arfó and is an important, and potentially targetable, mechanism that regulates A(342 production, while Arfl regulates secretion of Ap42 into the media

Automated theory formation in pure mathematics

The automation of specific mathematical tasks such as theorem proving and algebraic manipulation have been much researched. However, there have only been a few isolated attempts to automate the whole theory formation process. Such a process involves forming new concepts, performing calculations, making conjectures, proving theorems and finding counterexamples. Previous programs which perform theory formation are limited in their functionality and their generality. We introduce the HR program which implements a new model for theory formation. This model involves a cycle of mathematical activity, whereby concepts are formed, conjectures about the concepts are made and attempts to settle the conjectures are undertaken.HR has seven general production rules for producing a new concept from old ones and employs a best first search by building new concepts from the most interesting old ones. To enable this, HR has various measures which estimate the interestingness of a concept. During concept formation, HR uses empirical evidence to suggest conjectures and employs the Otter theorem prover to attempt to prove a given conjecture. If this fails, HR will invoke the MACE model generator to attempt to disprove the conjecture by finding a counterexample. Information and new knowledge arising from the attempt to settle a conjecture is used to assess the concepts involved in the conjecture, which fuels the heuristic search and closes the cycle.The main aim of the project has been to develop our model of theory formation and to implement this in HR. To describe the project in the thesis, we first motivate the problem of automated theory formation and survey the literature in this area. We then discuss how HR invents concepts, makes and settles conjectures and how it assesses the concepts and conjectures to facilitate a heuristic search. We present results to evaluate HR in terms of the quality of the theories it produces and the effectiveness of its techniques. A secondary aim of the project has been to apply HR to mathematical discovery and we discuss how HR has successfully invented new concepts and conjectures in number theory