Talk of Gisbert Wüstholz

Black holes

Just recently, in connection with the new accelerator at CERN in Geneva, black holes ripped through Swiss and European newspapers. People got very worried and speculated about their potential danger. They saw miniature black holes flying around Switzerland, and harbingers of the end of the world in these scientific experiments. They even brought in the courts of law to prevent the scientists from starting their experiments.

But what have black holes to do with Robert D. MacPherson? Not much, it seems at first sight. Black holes are objects of the real world of physics and astrophysics, yet we know and celebrate MacPherson as a mathematician, even a pure mathematician ... maybe even a very pure mathematician.

Although he grew up in a family which was very much involved with physics and engineering, he decided to study mathematics - not particularly pleasing his father.

But at least his most successful paper, from the pragmatic point of view of his father, written as an undergraduate during an internship at Oak Ridge National Laboratory, dealt with random numbers and on a question which comes up in nuclear physics. The paper was so successful and fundamental that one of the leading computer scientists, Donald Knuth, devoted many pages in his famous book on the Art of Computer Programming to explain it.

But then Robert MacPherson turned into a real mathematician, becoming a leading world expert in singularities. To explain singularities and their exquisitely intricate nature to somebody who is not a mathematician is very difficult, yet they are essential objects not only in the real world but also in the mathematical world.

And these objects are exactly what relates MacPherson with black holes: Black holes are singularities!

Singularities

Indeed, it turns out that nature produces singularities everywhere. Just watch a small stream and you see how the current curls around and sinks, or simply watch in your bath the water disappearing down the drain, or perhaps observe at the beach how the waves crest at a certain critical moment.

If you try to describe these mostly harmless looking phenomena in nature in a mathematical language, in the effort to compute and to predict reality you run very quickly into serious problems. The mathematics just does not behave as you would expect. But you need not be a mathematician to experience what can happen. Just swim into such a turbulence in the stream or into a wave in the sea and see how you get yourself into problems.

Singularities can be studied in different ways using analysis, or you can regard them as geometric phenomena. For the latter, their study demands a deep geometric intuition and profound geometric insight; this is what MacPherson masters in a most striking and extremely artful way.

He combines geometric visions with algebraic rigidity. If you study his work you see that it is glowing with elegance and profound in its depth.

He consistently was ahead of his time, developing new ideas and new approaches - ones often not shaped by the main streams of mathematical thought of the day, but rather characterized by great vision. Repeatedly the mathematical community came to embrace, extend and apply his ideas and results as they caught up with that vision.

Cooperation

MacPherson very often collaborated with quite distinguished co-authors, amplifying the spectrum of his mathematical charisma. I mention only

• Fulton-MacPherson (1976-1995) (Intersection theory, Characteristic classes, enumerative geometry, singular spaces),
• Goresky-MacPherson (since 1977) (Intersection homology, stratified Morse theory)
• Borho-MacPherson(1981 - 1989)(幂零轨道,characteristic classes, resolution of singularities, representation theory of Weyl groups)

MacPherson's connections with the Swiss mathematical community date back to 1983 when he participated in and significantly contributed to the famous Borel seminar, a joint seminar organized by several Swiss Universities including ETH Zurich, Lausanne, Geneva, Bern and Basel. The seminar was initiated by Armand Borel, one of the most distinguished Swiss mathematicians of the last century. The topic of the seminar was the Goresky-MacPherson Intersection homology and its use for the cohomology of arithmetic groups, one of the main research areas of Armand Borel. In this connection I should also mention MacPherson's papers written jointly with Harder, which have a similar mathematical context.

This illustrates only a small part of the work of MacPherson, for it spans a wide spectrum of contributions in many very different areas.

• Algebraic geometry and topology
• Algebraic Groups, Group actions and Representation Theory
• Enumerative Geometry and Combinatorics
• Locally symmetric spaces, L^2-cohomology, arithmetic groups and the Langlands Program.

The list is by far not complete and we try only to give a representative selection of his contribution to mathematics. He influenced a whole generation of mathematicians by giving them new tools to attack difficult problems and teaching them novel geometric, topological and algebraic ways of thinking.

It is not a surprise that MacPherson was and still is an admirer of music and in particular of Johann Sebastian Bach. I sense a strong commonality in the work of Bach and the work of MacPherson: simplicity, elegance and beauty. I was told that he studied parts of Bach's work more intensively and carefully than mathematical papers. Another example of the very intimate relationship between mathematics and music!