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.