The German mathematician Friedrich (Fritz) Hirzebruch passed away a couple days ago, at the age of 84. Hirzebruch was perhaps the most important mathematician in the Germany of the postwar period, responsible for the founding of the Max Planck Institute in Bonn, as well as the yearly Bonn Arbeitstagung conference. The Mathematics Genealogy Project lists him as having 52 Ph.D. students and 368 descendants. There’s a wonderful interview and article about him at the Simons Foundation web-site.

Hirzebruch’s first great mathematical achievement was the proof in 1954 of the generalization of the classical Riemann-Roch theorem to higher dimensional complex manifolds, now known as the Hirzebruch-Riemann-Roch theorem. This used the new techniques of sheaf cohomology and was one of the centerpieces of the explosion of new results in geometry and topology during the 1950s. Further generalization of this led to the Grothendieck-Riemann-Roch theorem, and the Atiyah-Singer index theorem. Hirzebruch’s monograph on the subject *Topological Methods in Algebraic Geometry* was the essential textbook in this area for many years.

The last time I heard Hirzebruch talk was at the celebration of Atiyah’s 80th birthday in Edinburgh, where Hirzebruch gave a talk about his interactions with Atiyah. He displayed some of their correspondence from this period, which makes fascinating reading and is now available here.

With the loss of Raoul Bott a few years ago, and now Fritz Hirzebruch, the math and physics communities are deprived of two of the great figures who built parts of modern mathematics that appear crucially in the structure of the Standard Model. Much of this connection between math and physics remains a mystery, and it’s too bad they won’t be around to help make progress unraveling it.

**Update**: The New York Times has a very good obituary of Hirzebruch here.

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Could you elaborate a little on the contributions of Bott and Hirzebruch to the mathematics of the Standard Model?

Chris,

This is a long story, and I wrote a significant amount about it in my book. The basic objects in the SM are the Dirac operator on spinors, coupled to a connection. These are exactly the same basic objects that appear in the Atiyah-Singer index theorem. The Hirzebruch-Riemann-Roch theorem is in some sense the crucial case of the index theorem, using the Dolbeault operator instead of the closely related Dirac operator. Bott’s work made possible the general topological K-theory behind the index theorem. From the late 70s on, both Bott and Hirzebruch were very active in promoting the interaction of mathematics and physics, often giving beautiful expository lectures trying to explain this material to physicists, and relate the points of view of the two subjects.

Thanks, Peter

2 questions: (1) Is the problem mentioned at the end of Hirzebruch’s Atiyah80 video/lecture still open? (2) Of the mathematicians who are roughly of the generation of Hirzebruch & Atiyah, which of them made the most significant contributions to the mathematics of the Standard Model of particle physics?

David,

I have no idea if anyone took Hirzebruch up on that challenge (he was asking for an embedding of E6/(Spin(10)xU(1)) in dimension 61, or, failing that, 62 or 63, hopefully showing that the non-embedding result in dimension 60 is sharp).

Atiyah, Singer, Bott and Hirzebruch all have interacted strongly with physicists around issues raised by Yang-Mills theory, the Standard Model and attempts to extend it. Since the Standard Model was in place back in 1973, before they got involved, this isn’t activity that has actually changed the Standard Model, rather led to a better understanding of it, especially issues related to anomalies and instantons. Atiyah and Singer have both been extremely active in working with physicists, with Bott only a bit less so, Hirzebruch significantly less.

Hi Peter,

no disrespect to a great mathematician, no doubt he made a beautiful contribution to pure mathematics, but isn’t it a bit much to suggest the intricacies of the mathematical arguments Hirzebruch (et al) developed are relevant to Nature/Physics?

I mean isn’t this site devoted to the idea that over-elaborate mathematical arguments are not the way to understand Nature?

jg,

My problem is not with the use of mathematics to describe nature, quite the opposite. I believe that experience shows it is the deepest ideas in mathematics that are most likely to give us insight into fundamental physics at a deep level. The Hirzebruch-Riemann-Roch theorem is a very deep idea about mathematics, with close connections to things we have impressive physical evidence for (the Dirac operator, gauge fields), so this seems to me to be worth a lot of attention.

Yes, you can also use the HRR theorem to do some computations in horrifically complicated and ugly constructions used in failed attempts to get string theory unification, but there the problem is the construction, not the theorem. The HRR theorem ends up being useful in that case because it is so powerful and so general that it’s one of the few tools that can say something about such a complicated mess.

ah ok,

I personally don’t believe the breakthrough requires a deep result in pure mathematics, but I admire your position on this ðŸ™‚

fxqi’s 2012 essay question:

Which of Our Basic Physical Assumptions Are Wrong?

http://fqxi.org/community/essay

Peter,

HRR is very nice indeed, but perhaps your dislike of string compactification is causing you to leave out his surfaces, which are some of the simplest surfaces in complex algebraic geometry. ðŸ™‚

http://en.wikipedia.org/wiki/Hirzebruch_surface

Cheers,

P

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