NEWSLETTER

History
[K 1 ] S. Kumar, Kac-Moody Groups, their Flag Varieties and
Representation Theory, Progress in Mathematics, vol.
204, Birkhäuser, 2002.
[K 2 ] S. Kumar, Tensor product decomposition, Proc. of the International
Congress of Mathematicians, Hyderabad (India),
(2010), 1226–1261.
[K 3 ] S. Kumar, A Survey of the Additive Eigenvalue Problem
(with Appendix by M. Kapovich), Transformation Groups
19 (2014), 1051–1148.
[KuLM] S. Kumar, B. Leeb and J. J. Millson, The generalized triangle
inequalities for rank 3 symmetric spaces of noncompact
type, Contemp. Math. 332 (2003), 171–195.
[Le] B. Lee, A Comparison of Eigencones Under Certain Diagram
Automorphisms, PhD Thesis (under the supervision
of S. Kumar), University of North Carolina, 2012.
[Li] B.V. Lidskii, The proper values of the sum and product of
symmetric matrices, Dokl. Acad. Nauk SSSR 74 (1950),
769–772.
[N] L. Ness, A stratification of the null cone via the moment
map (with an appendix by D. Mumford), Amer. J. Math.
106 (1984), 1281–1329.
[RR] S. Ramanan and A. Ramanathan, Some remarks on the
instability flag, Tôhoku Math. J. 36 (1984), 269–291.
[R] N. Ressayre, Geometric invariant theory and the generalized
eigenvalue problem, Inventiones Math. 180 (2010),
389–441.
[Sj]
[S]
[TF]
[W]
[Wi]
[Z]
R. Sjamaar, Convexity properties of the moment mapping
re-examined, Adv. Math. 138 (1998), 46–91.
S. Sam, Symmetric quivers, invariant theory, and saturation
theorems for the classical groups, Adv. Math. 229
(2012), 1104–1135.
R. C. Thompson and L. Freede, On the eigenvalues of
sums of Hermitian matrices, Linear Algebra Appl. 4
(1971), 369–376.
H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte
linearer partieller Differentialgleichungen, Math.
Annalen 71 (1912), 441–479.
H. Wielandt, An extremum property of sums of eigenvalues,
Proc. Amer. Math. Soc. 6 (1955), 106–110.
A. Zelevinsky, Littlewood-Richardson semigroups, In:
“New Perspectives in Algebraic Combinatorics”, MSRI
Publ. 38 (1999), Cambridge Univ. Press, Cambridge, 337–
345.
Shrawan Kumar [shrawan@email.unc.edu]
is the John R. and Louise S. Parker Distinguished
Professor at the University of
North Carolina, Chapel Hill, USA. He was
an invited speaker at ICM 2010. He is a
Fellow of the American Mathematical Society.
George Boole and andBoolean BooleanAlgebra
Stanley
Stanley
Burris
Burris
(University
(University
of
of
Waterloo,
Waterloo,
Waterloo,
Ontario,
Ontario,
Canada)
Canada)
George Boole (1815–1864) was responsible, in the years
1847–1854, for initiating the revolution in the subject of logic
by creating an algebra of logic for classes. This is all the more
remarkable because Boole was largely self-educated in mathematics
(and several languages), having had to give up attending
school at the age of 16 to start his career as a schoolteacher
to provide financial support for his parents and siblings. He
started publishing mathematical papers, mainly on analysis,
in 1841. Three years later, in 1844, at the age of 29, he won
the first gold medal awarded in mathematics by the Royal Society.
Boole struck up a friendly correspondence with Augustus
De Morgan (1806–1871). Subsequently, De Morgan’s noisy
feud (over a rather trivial matter in logic) with the respected
philosopher Sir William Hamilton (1788–1856) of Edinburgh
inspired Boole to write a booklet [2] in 1847 applying algebra
to logic. In 1849, at the age of 34, Boole left school-teaching
in Lincolnshire, England, for a professorship at Queen’s College
in Cork, Ireland. For fascinating details on Boole’s life,
see the excellent biography [23] by Desmond MacHale –
including such remarkable details as the fact that Boole’s
youngest daughter Ethel Lilian (1864–1960) wrote a novel
called The Gadfly which essentially became the ‘bible of the
Russian revolution’.
Although Boole was primarily an algebraist and analyst,
today he is best known for his work in logic, in particular
for his 1854 book An Investigation of the Laws of Thought
on which are founded the Mathematical Theories of Logic
and Probabilities. We will refer to this book as LT. The first
two-thirds of LT are on Boole’s algebra of logic for classes
(which will henceforth be called “Boole’s algebra") and the
last third on applications of this algebra to probability theory.
This article is only concerned with the logic portion of LT.
Boole’s algebra had equational laws, rules of inference for
equational reasoning and a powerful Rule of 0 and 1, which
has only recently been deciphered. Indeed, it is remarkable
how long it has taken to properly understand Boole’s algebra
– the breakthrough came in 1976 with the publication of [15]
by Theodore Hailperin (1916–2014). 1
1 Using Ordinary Algebra
One of the distinguishing features of Boole’s algebra of logic
is the extent to which it looks like ordinary algebra. Indeed,
this fact is beautifully summarised in the following two
quotes:
That the symbolic processes of algebra, invented as tools of numerical
calculation, should be competent to express every act of
1 For a modern introduction to Boole’s algebra of logic, see the author’s
article [7] on Boole in the online Stanford Encyclopedia of Philosophy,
as well as the texts [8], [9] of two recent talks by the author on Boole’s
algebra of logic.
EMS Newsletter December 2015 27