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Huỳnh Mùi

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 ΩnSnX 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 so-called 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 well-known paper from 1982 "The mod p cohomology algebra of the extra-special group E(p3)", 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 well-known 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 20-year 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 10-year 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

List of participants
Group photograph
Titles of talks