Huỳnh Mùi was born on
January 21, 1944 and grew up in Huê, the former
capital of Vietnam in the 19th and early 20th
centuries. He became a student of the University of
Saigon in 1961. One year later, in 1962, he was
awarded a scholarship to study in the University of
Tokyo, Japan, where he received Bachelor degree in
1967, Master degree in 1969, and PhD degree in 1974
under the guidance of Professor Tokushi Nakamura.
After that, he did research in the University of
Tokyo for the period 1974–1976 before returning
to Vietnam in 1977.
In his work for the master
degree, 1967–1969, he defined an equivariant
cell decomposition of the direct product of a real
vector space with itself n times, on which the wreath
product of an order 2 cyclic group by the symmetric
group on n letters acts regularly. This is an
analogue of Nakamura's cell decomposition for the
same space under the regular action of the symmetric
group. The decomposition allowed Mùi to
determine the mod 2 homology of this wreath product
by using the homology of the Eilenberg–MacLane
spaces of Z/2.
Nakamura, by means of his cell
decomposition, successfully explained a certain type
of iterated bar construction not only as a
theoretical concept, but also as a geometrical
phenomenon. Therefore he gave a deeply geometrical
interpretation for the Cartan construction, which is
the main tool used to determine homology of the
Eilenberg–MacLane spaces. Using this, Nakamura
found a geometrical way to determine the homology of
the symmetric groups.
Mùi employed Nakamura's
ideas to compute the equivariant homology of the
configuration spaces and used this to determine the
homology of the iterated loop spaces
Ω^{n}S^{n}X by means of the
Nakamura decomposition and the Cartan operations.
Although Mùi's work in this field is published
late, in Acta Mathematica Vietnamica in 1980, it was
produced independently of F Cohen's work in the area.
Mùi's slogan was "from configuration spaces to
loop spaces", whereas that of Cohen was "from loop
spaces to configuration spaces".
The product in the cohomology of
the symmetric groups was continuously Mùi's main
interest. Then, his effort was to construct a
diagonal chain map for the free resolution of the
symmetric group given either by Nakamura's cell
decomposition or by Mùi's. Step by step,
Mùi recognized that, in the study of the
cohomology algebra of finite groups, the geometrical
method would lead to some very complicated problems.
So, he wanted to find another way to attack group
cohomology.
In the period of time
1969–1974 when he prepared for his doctoral
thesis, and was deeply influenced by Steenrod's works
on cohomology operations and cohomology of the wreath
products, Mùi focused on the restriction from
the cohomology of the symmetric group to the
cohomology of its maximal elementary abelian
p–subgroups. It turned out later that his idea
anticipated Quillen's in this field.
Cardenas, a student of Steenrod,
computed in 1965 the cohomology algebra of the
symmetric group of degree p². In this work,
Cardenas computed "directly by hand" the invariants
of the general linear group of degree 2. In order to
generalize Cardenas' result for arbitrary symmetric
group, Mùi recognized the role of modular
invariant theory. He extended Dickson's work from the
determination of the invariants under the general
linear group action in a polynomial algebra to that
of the invariants in the tensor product of an
exterior algebra and a polynomial algebra under the
action either of the general linear group or of its
Sylow subgroup. Further, he combined elegantly the
Steenrod construction with these invariants to study
the cohomology of the symmetric group and its Sylow
subgroup. Since this work, which is published in the
Journal of the University of Tokyo 1975, Dickson
invariants and Mùi invariants have became well
known tools in studying cohomology of finite groups
and Algebraic Topology in general.
Mùi also constructed
successfully homology operations and cohomology
operations derived from the socalled
Dickson–Mùi's invariants. In particular,
he proved that, for any prime p, the cohomology
operations derived from the Dickson–Mùi
invariants are exactly the elements in the Steenrod
algebra, which are dual to the elements in the Milnor
basis. For p=2, this result was also established by
Madsen–Milgram.
Mùi made a continuous effort
during the 1980s to examine various techniques for
studying p–group cohomology. In an unpublished
but nevertheless wellknown paper from 1982 "The mod
p cohomology algebra of the extraspecial group
E(p^{3})", he studied extensively the
transfer map from p–maximal subgroups to the
group in question by use of invariants to solve
completely the Hoschild–Serre spectral
sequence. (Note that Yagita first computed the
integral cohomology of this group up to the
Hoschild–Serre filtration.) In that paper,
Mùi focused on the cohomology classes, being
called now "essential classes", that is, the ones
whose restrictions are zero on the cohomology of any
proper subgroup. Also, he stated there the wellknown
conjecture that the square of the ideal of essential
classes is zero. This was attractive to many group
cohomology theorists, particularly to some in his
school. Recently, David Green gave a counter example
for Mùi's 20year old conjecture. Taking this
into account, Mùi recommenced that the
multiplicative triviality degree of the Essential
Ideal would be one of the next targets in the study
of group cohomology.
In 1977, returning to Vietnam
from Japan, Huỳnh Mùi started to build up
a working group in Algebraic Topology. It should be
noted that, in those days, Vietnam had just emerged
from a terrible long war and living conditions there
were extremely limited. One can imagine the
difficulties Mùi and his students had to face:
mathematical isolation and the lack of information,
particularly the lack of journals and books. Under
such circumstances, Mùi's working group started
to do research. With the strategy "Build up a
mathematical laboratory in every university of
Vietnam", Mùi often visited various universities
in Vietnam to give basic courses in Algebra and
Algebraic Topology, and then encouraged many young
faculties there to come to Hanoi to stay temporarily
for intensive study. Mùi's group is mainly
interested in using modular invariants to study
Algebraic Topology and group cohomology. Under
Mùi's guidance, there have been 10 persons
successfully defending PhD theses.
Huỳnh Mùi is not only
a mathematician but also a social activist. In the
period 1965–1972, he actively took part in the
movement against the war in Vietnam. This activity
occupied a lot of his time during this period.
In recent years, under the market
economy, the living conditions in Vietnam have
remarkably improved. However, because of the market
economy, there were very few young people who took
mathematics as major subject in the 10year period,
around 1990–2000. Therefore, in the early
1990's, Mùi moved from the Vietnam National
University to Thang Long University, a private one,
of which he was the Rector for a time. Since then, he
has mainly been being interested in Computer
Science.
Today, Huỳnh Mùi's
school in Algebraic Topology has became a very active
one in Vietnam and known through the mathematical
world.
John Hubbuck, Nguyễn H V
Hưng, and Lionel Schwartz
