Recent breakthroughs with applied origins

Question

Historically, the boundary between pure mathematics and its applications was much less defined. However, with the increasing complexity of modern mathematics and the resulting need for specialization, mathematical research often evolves away from its original motivations. While there are numerous cases where theoretical results have found unexpected applications, I am interested in the converse: instances where applied fields have directly inspired breakthroughs in pure mathematics.

One well-known example is the work of Candelas et al. [1], who calculated the number of degree-three curves on a quintic Calabi–Yau manifold (317,206,375). String Theory may be controversial among physicists, but I think this serves as an example of how insights from outside mathematics can drive advancements in enumerative geometry.

For the purposes of this question, let’s define "recent" as within the past 50 years. Are there other examples where insights or techniques from an applied field have led to major progress in pure mathematics?

Candelas, Philip; de la Ossa, Xenia C.; Green, Paul S.; Parkes, Linda, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory., Nucl. Phys., B 359, No. 1, 21-74 (1991). ZBL1098.32506.

Answer 1

If you include mathematical physics (string theory, QFT) to "applied mathematics", the breakthroughs are too numerous to list them here. Statistical mechanics is especially productive in its influence on pure mathematics.

Here is a very recent example not related to QFT or string theory. An important Connes' Embedding Conjecture from the theory of von Neumann algebras, has been apparently disproved by the arguments from Computer Science (quantum computing):

MIP*=RE

(The complete paper is not published yet, given its size, it will take some time to verity, but it has already 348 citations on Google Scholar at the time of this writing).

More examples (just a selection of those which I encountered during my career). All of these are examples of breakthroughs in pure mathematics inspired by physics.

In 1987, Zograf and Takhtadzhyan made a breakthrough in the classical accessory parameter problem of uniformization theory by proving a conjecture of physicists Belavin, Polyakov and Zamolodchikov inspired by string theory.

In 1993, Kari Astala proved the old conjecture of Gehring and Reich by using ``thermodynamic formalism''. The proof has been substantially simplified since, but still uses the main idea which comes from statistical physics. More generally "thermodynamic formalism" is a part of pure mathematics directly inspired by physics, see, for example this survey paper by David Ruelle on the influence of statistical mechanics on pure mathematics.

In the early 2000s Oded Schramm created the SLE theory (Stochastic Loewner Equation, aka Schramm-Loewner evolution) on the boundary of probability and complex analysis. It was directly inspired by questions from physics (statistical mechanics, percolation theory, Cardy formula).

In 2005, Mukhin, Tarasov and Varchenko proved the B. and M. Shapiro Conjecture from real enumerative geometry by unexpected argument from mathematical physics (Bethe-ansatz from statistical mechanics). In 2020 a surprising connection of this former Shapiro conjecture with SLE has been found (Peltola and Wang, https://arxiv.org/pdf/2006.08574 )

In cases 1,2,4, the initial problem had nothing to do with physics, and solving them by methods coming from physics was quite unexpected.

There are various opinions on this "unreasonable effectiveness of physics in pure mathematics", for example, Ruelle (loc. cit.) ascribes a deep philosophical meaning to it. Another opinion that I've heard is simply that "some physicists are very clever people, and when they start thinking of mathematics, they come with clever ideas" (this was proposed by V. G. Drinfeld when he was listening my talk on Astala's theorem).

Answer 2

I would draw attention to anything connected to Einstein manifolds. While proposed more than fifty years ago, it is a very active area of research in physics and mathematics, so I think one could argue it is, at least partially, modern.

The concept of Einstein manifolds came directly from Einstein's theory of general relativity. The Einstein condition on a smooth Riemannian manifold is given by:

$\text{Ric} = kg$

In physics, this condition represents a solution to the Einstein field equations in a vacuum, possibly with a cosmological constant. Thus, Einstein manifolds had its origins in physics.

This physics-driven concept intrigued mathematicians, who were, and continue to be, fascinated by:

• Sign variations of $ k $: Different signs (i.e. positive, negative, or zero) leads to varying curvature properties, which affect the global structure of the manifold.

• Dimensional challenges: The complexity of constructing Einstein manifolds, and the observation of exotic structures, can increase or decrease depending on the dimension. For example, understanding compact Einstein manifolds in four dimensions presents immense difficulty.

• Compact construction: Constructing compact Einstein manifolds is especially tough, and thus represents a rich area of mathematical research.

Moving forward, the impact of mathematical breakthrough provided by physics increased, as gauge theory began to take centre stage. Edward Witten's work on Seiberg-Witten theory, a supersymmetric gauge theory in physics, opened new doors to understanding 4-manifolds, including Einstein manifolds in the fourth dimension. Witten discovered that Seiberg-Witten invariants could be sued to study 4-manifolds, providing a more accessible alternative to Donaldson invariants for proofs and analysis of 4-manifolds. The transition marked a major advance, as these invariants allowed mathematicians to prove results on 4-dimensional Einstein manifolds, which were previously out of reach.

Another example of where physics has continued to increase understanding in this field is provided in the way that gauge theory, in particular now that it has been converted into a more pure mathematical language in this context, has led to research in using numerical analysis for the construction of Einstein metrics. Physicist' methods of rewriting equations in gauge notation has led to computational techniques to simulate and study Einstein metrics, helping to generate potential examples.