A tribute to two premier analysts

 From Flanders Today src1 and Ryle Trust Lecture src2

Baron Jean Bourgain and Sir Michael Atiyah passed away within the past three weeks. They became mathematical nobility by winning the Fields Medal, Atiyah in 1966 and Bourgain in 1994. Bourgain was created Baron by King Philippe of Belgium in 2015. Atiyah’s knighthood did not confer nobility, but he held the dynastic Order of Merit, which is limited to 24 living members and has had fewer than 200 total since its inception in 1902. Atiyah had been #2 by length of tenure after Prince Philip and ahead of Prince Charles.

Today we discuss how they ennobled mathematics by their wide contributions.

Bourgain was affiliated to IAS by the IBM John von Neumann Professorship. He had been battling cancer for a long time. Here is the middle section of the coat of arms he created for his 2015 investiture:

 Detail from IAS source

The shield shows the beginning of an Apollonian circle packing, in which every radius is the reciprocal of an integer. This property continues as circles are recursively inscribed in the curvilinear regions—see this 2000 survey for a proof. To quote Bourgain’s words accompanying his design:

The theory of these [packings] is today a rich mathematical research area, at the interface of hyperbolic geometry, dynamics, and number theory.

Bourgain’s affinity to topics we hold dear in computing theory is shown by this 2009 talk titled, “The Search for Randomness.” It covers not only PRNGs and crypto but also expander graphs and succinctness in quantum computing. He has been hailed for diversity in other mathematical areas and editorships of many journals. We will talk about a problem in analysis which he helped solve not by analytical means but by connecting the problem to additive combinatorics.

## From Analysis to Combinatorics

Sōichi Kakeya posed the problem of the minimum size of a subset of ${\mathbb{R}^{2}}$ in which a unit-length needle can be rotated through 360 degrees. Abram Besicovitch showed in 1928 that such sets can have Lebesgue measure ${\epsilon}$ for any ${\epsilon > 0}$. He had already shown that one can achieve measure zero with a weaker property, which he had used to show a strong failure of Fubini’s theorem for Riemann integrals:

For all ${d}$ there is a measure-zero subset of ${\mathbb{R}^d}$ that contains a unit line segment in every direction.

The surprise to many of us is that such strange sets would have important further consequences in analysis. A 2008 survey in the AMS Bulletin by Izabella Łaba, titled “From Harmonic Analysis to Arithmetic Combinatorics,” brings out breakthrough contributions by Bourgain to conjectures and problems that involve further properties of these sets, which seem to retain Kakeya’s name:

Conjecture: A Kakeya set in ${R^{d}}$ must have Hausdorff dimension ${d}$.

This and the formally weaker conjecture that the set must have Minkowski dimension ${d}$ are proved in ${\mathbb{R}^2}$ but open for all ${d \geq 3}$. Bourgain first proved that the restriction conjecture of Elias Stein, which is about extensions of the Fourier transform from certain subspaces of functions from ${\mathbb{R}^d}$ to ${\mathbb{C}}$ to operators from ${L^q}$ to ${L^p}$ functions on ${\mathbb{R}^d}$, implies the Kakeya conjecture. It is likewise open for ${d,p \geq 3}$. As Łaba writes, associated estimates “with ${p > 2}$ require deeper geometrical information, and this is where we find Kakeya sets lurking under the surface.”

What Bourgain showed is that the restriction estimates place constraints on sets of lower Hausdorff dimension that force them to align “tubes” along discrete directions that can be approximated via integer lattices. This led to the following “key lemma”:

Lemma 1 Consider subsets ${S}$ of ${A \times B}$, where ${A}$ and ${B}$ are finite subsets of ${\mathbb{Z}^d}$, and define

$\displaystyle S^{+} = \{a+b: (a,b) \in S\}, \qquad S^{-} = \{a - b: (a,b) \in S\}.$

For every ${C > 0}$ there is ${C' > 0}$ such that whenever ${|S^{+}| \leq Cn}$, where ${n = \max\{|A|,|B|\}}$, we have ${|S^{-}| \leq C'n^{2 - \frac{1}{13}}}$.

To quote Łaba: “Bourgain’s approach, however, provided a way out. Effectively, it said that our hypothetical set would have structure, to the extent that many of its lines would have to be parallel instead of pointing in different directions. Not a Kakeya set, after all.” She further says:

Bourgain’s argument was, to this author’s knowledge, the first application of additive number theory to Euclidean harmonic analysis. It was significant, not only because it improved Kakeya bounds, but perhaps even more so because it introduced many harmonic analysts to additive number theory, including [Terence] Tao who contributed so much to the subject later on, and jump-started interaction and communication between the two communities. The Green-Tao theorem [on primes] and many other developments might have never happened, were it not for Bourgain’s brilliant leap of thought in 1998.

Among many sources, note this seminar sponsored by Fan Chung and links from Tao’s own memorial post.

## Michael Atiyah

Michael Atiyah was also much more than an analyst—indeed, he was first a topologist and algebraic geometer. He was also a theoretical physicist. Besides all these scientific hats, he engaged with society at large. After heading Britain’s Royal Society from 1990 to 1995, he became president of the Pugwash Conferences on Science and World Affairs. This organization was founded by Joseph Rotblat and Bertrand Russell in the 1950s to avert nuclear war and misuse of science, and won the 1995 Nobel Peace Prize.

The “misuse of science” aspect comes out separately in Atiyah’s 1999 article in the British Medical Journal titled, “Science for evil: the scientist’s dilemma.” It lays out a wider scope of ethical and procedural concerns than the original anti-war purpose. This is furthered in his 1999 book chapter, “The Social Responsibility of Scientists,” which laid out six points including:

• First there is the argument of moral responsibility. If you create something you should be concerned with its consequences. This should apply as much to making scientific discoveries as to having children.

• Scientists will understand the technical problems better than the average politician or citizen and knowledge brings responsibility.

• [T]here is need to prevent a public backlash against science. Self-interest requires that scientists must be fully involved in public debate and must not be seen as “enemies of the people.”

As he says in its abstract:

In my own case, after many years of quiet mathematical research, working out of the limelight, a major change occurred when unexpectedly I found myself president of the Royal Society, in a very public position, and expected to act as a general spokesman for the whole of science.

Within physics and mathematics, he also ventured into a debate that comes closer to the theory-as-social-process topic we have discussed on this blog. In 1994 he led a collection of community responses to a 1993 article by Arthur Jaffe and Frank Quinn that began with the question, “Is speculative mathematics dangerous?” Atiyah replied by saying he agreed with many of their points, especially the need to distinguish between results based on rigorous proofs and heuristic arguments,

…But if mathematics is to rejuvenate itself and break exciting new ground it will have to allow for the exploration of new ideas and techniques which, in their creative phase, are likely to be as dubious as in some of the great eras of the past. …[I]n the early stages of new developments, we must be prepared to act in more buccaneering style.

Now we cannot help recalling his claim last September of heuristic arguments that will build a proof of the Riemann Hypothesis, which we covered in several posts. As we stated in our New Year’s post, nothing more of substance has come to our attention. We do not know how much more work was done on the promised longer paper. We will move toward discussing briefly how his most famous work is starting to matter in algorithms and complexity.

## Indexes and Invariants

We will not try to go into even as much detail as we did for Kakeya sets about Atiyah’s signature contributions to topological K-theory, physical gauge theory, his celebrated index theorem with Isadore Singer, and much else. But we can evoke reasons for us to be interested in the last. We start with the simple statement from the essay by John Rognes of Oslo that accompanied the 2004 Abel Prize award to Atiyah and Singer:

Theorem 2 Let ${P(f) = 0}$ be a system of differential equations. Then

$\displaystyle \text{analytical index}(P) = \text{topological index}(P).$

Here the analytical index equals the dimension ${d_k}$ of the kernel of ${P}$ minus the dimension ${d_c}$ of the co-kernel of ${P}$, which (again quoting Rognes) “is equal to the number of parameters needed to describe all the solutions of the equation, minus the number of relations there are between the expressions ${P(f)}$.” The topological index has a longer laundry list of items in its definition, but the point is, those items are usually all easily calculable. It is further remarkable that in many cases we can get ${d_k - d_c}$ without knowing how to compute ${d_k}$ and ${d_c}$ individually. The New York Times obituary quotes Atiyah from 2015:

It’s a bit of black magic to figure things out about differential equations even though you can’t solve them.

One thing it helps figure out is satisfiability. Besides cases where knowing the number of solutions does help in finding them, there are many theorems that needed only information about the number and the parameterization.

We have an analogous situation in complexity theory with the lower bound theorem of Walter Baur and Volker Strassen, which we covered in this post: The number of multiplication gates needed to compute an arithmetical function ${f}$ is bounded below by a known constant times the log-base-2 of the maximum number of solutions to a system formed from the partial derivatives of ${f}$ and a certain number of linear equations, over cases where that number is finite. Furthermore, both theorems front on algebraic geometry and geometric invariant theory, whose rapid ascent in our field was witnessed by a workshop at IAS that we covered last June. That workshop mentioned not only Atiyah but also the further work in algebraic geometry by his student Frances Kirwan, who was contemporaneous with Ken while at Oxford. Thus we may see more of the kind of connections in which Atiyah delighted, as noted in current tributes and the “matchmaker” label which was promoted at last August’s ICM.

## Open Problems

Our condolences go out to their families and colleagues.