We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

Decomposing p-groups via Jordan algebras

A fast isomorphism test for groups whose Lie algebra has genus 2

A theorem is proved on the structure of the group of isometries of a Hermitian map b : V × V → W , where V and W are vector spaces over a finite field of odd order. Also a Las Vegas polynomial-time algorithm is presented which, given a Hermitian map, finds generators for, and determines the structure of its isometry group. The algorithm can be adapted to construct the intersection over a set of classical subgroups of GL(V ), giving rise to the first polynomial-time solution of this old problem. The approach yields new algorithmic tools for algebras with involution, which in turn have applications to other computational problems of interest. Implementations of the various algorithms in the Magma system demonstrate their practicability.

https://www.ams.org/tran/2012-364-04/S0002-9947-2011-05388-2/S0002-9947-2011-05388-2.pdf

Computing isometry groups of Hermitian maps

For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ pairwise nonisomorphic quotients of order $p^{n}$. Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most $p$. They are also directly and centrally indecomposable and of the same indecomposability type. The recognized portions of their automorphism groups are isomorphic, represented isomorphically on their abelianizations, and of small index in their full automorphism groups. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.

Isomorphism in expanding families of indistinguishable groups

The system and method are provided for assessing a supplier's compliance with a customer's contract terms, conditions, and applicable regulations. The method includes the operation of identifying a complete set of compliance provisions from contract terms, conditions, and applicable regulations for a customer's contract. Supplier compliance information can be gathered to quantify compliance by the supplier with the compliance provisions. The supplier compliance information can be stored in a database that is accessible over a network. A further operation is verifying the supplier compliance information using a verification provider in order to produce qualitative verification of supplier compliance information. Another operation is facilitating customer access over the network to the supplier compliance information for the compliance provisions.

James B. Wilson

Papers

Decomposing p-groups via Jordan algebras

A fast isomorphism test for groups whose Lie algebra has genus 2

Computing isometry groups of Hermitian maps

Isomorphism in expanding families of indistinguishable groups

System and method for assessing a supplier's compliance with a customer's contract terms, conditions, and applicable regulations