A true connoisseur of geometric shapes

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Alessandro Carlotto uses 3D models to show what shapes the more complex minimal

Alessandro Carlotto uses 3D models to show what shapes the more complex minimal surfaces between two liquids can have.

He investigates the mysteries of shapes and curvature in higher dimensions: mathematician Alessandro Carlotto will receive the 2022 ETH Zurich Latsis Prize for his original research at the frontier of mathematics and physics.

In his research, Alessandro Carlotto often operates on the border of mathematics and physics: his perspective is that of geometric analysis, which - put very simply - employs the tools of mathematical analysis to explore the shape of objects in space and how they deform over time and under the influence of curvature. Within that field, Carlotto is considered a rising star and ’the best young researcher in continental Europe’. He will receive the 2022 ETH Zurich Latsis Prize on ETH Day.

The award is acknowledgement of the Italian mathematician’s pursuit of a very independent and rich research agenda, his reputation as an excellent lecturer and his award of an ERC Starting Grant 2020, which is considered a seal of quality for excellent research. In its laudation, the ETH Zurich Research Commission justifies Carlotto’s award as follows: ’His deep and highly original results cover a broad spectrum from differential geometry to general relativity. His scientific work has not only had a strong impact on mathematics, where he has tackled numerous longstanding problems, but also on theoretical physics.’

Thinking physics models through to the end

In fact, Albert Einstein’s General Theory of Relativity and the mathematical problems that arise in physics are an inspiration for Carlotto: ’I am extremely intrigued by the interplay between mathematics and physics, and I like the predictive power of mathematical reasoning.’ In his research, Carlotto usually starts from a given physical model that describes certain natural phenomena. Unlike a physicist, however, he is not really concerned with revising or refining the model in question based on empirical data, but rather - using the sole force of abstraction - investigates its mathematical implications and, hence, what the inferred results say about the corresponding phenomena.

Following this approach and working jointly with his doctoral supervisor, 2017 Heinz Hopf Prize winner Richard M. Schoen of Stanford University and the University of California, Irvine, he discovered novel, localised solutions of the Einstein field equations. These equations describe gravitational forces within the context of general relativity. This theory is now well established. However, 100 years after Albert Einstein published his General Theory of Relativity, in 2015 Carlotto and Schoen were able to provide rigorous proof of the existence of spacetimes that satisfy the Einstein field equations and have large unbounded regions where no gravity is perceivable while also containing black hole regions and thus exhibiting extremely strong gravitational forces. In research, this phenomenon is called gravitational shielding because objects in the former regions are totally shielded against the influence of the gravitational field of the latter. In classical Newtonian field theory, this is impossible.

Carlotto and Schoen’s result is a good illustration of the aforementioned predictive power of mathematics, as their work introduced this gravitational shielding before the design of whatever accepted, repeatable experiment that empirically displays it. ’Our solutions are perfectly consistent with the axioms of general relativity. However, they explicitly exhibit new phenomena in a way that is extremely surprising and counterintuitive,’ says Carlotto.

Enigmatic shapes and curvature

The concepts of shape and curvature have been essential throughout his research - despite the diversity of topics. In the simplest sense, mathematicians understand curvature as being the local deviation of a curve from a straight line. In three-dimensional space that people are familiar with in everyday life, the surface of a sphere (unlike that of a doughnut), for example, exhibits constant curvature, as does a flat plane (the difference being that the sphere is positively curved).

In mathematics, it is very natural and almost necessary to abstract from two-dimensional spaces to higher-dimensional spaces and to formulate generalised notions of curvature for objects with three or more dimensions. Albert Einstein again provides good motivation for this: the 1921 Nobel Prize winner in physics used the language of differential geometry to describe the curvature of spacetime, which is the four-dimensional ’theatre’ where events take place and thus the background for his celebrated theory of gravitation. String theory provides other excellent reasons to ascend to higher dimensions, as do other theories in contemporary physics. ’In these higher dimensions, things are still mysterious in many ways,’ says Carlotto.

Of all the notions of curvature or ways of measuring the shape of spaces (of any dimension), Carlotto favours scalar curvature. This is the object of two of his recent, most influential works. ’Lately I have been thinking a lot about ’spaces of solutions’ to certain geometric problems and how they look ’in the large’,’ Carlotto comments.

For instance, in 2021 he completed a four-year project with Chao Li, a researcher working at the Courant Institute of the New York University. Building on the pioneering works of many scientists (notably Hamilton, Perelman, Kleiner, Lott and Bamler), they proved that on any compact three-dimensional manifold the space of positive scalar curvature metrics with minimal boundary is either empty or contractible. ’Put simply, deformations that respect these constraints dictated by curvature are unobstructed in the strongest possible sense. That proof actually employs many of the tools I have learned throughout my career,’ Carlotto states.

The secret of interfaces

Carlotto’s research is not limited to mathematical physics: ’I am always excited about contemplating new problems,’ he says. ’Like Albert Einstein, I believe that scientific progress requires the adoption of an opportunistic attitude.’

In fact, in recent years his interest has embraced various classical themes and problems in differential geometry. For example, he has extensively studied minimal surfaces, whose exemplifying ’toy models’ are soap films. Classically, they are presented as surfaces that minimise the area (roughly comparable to ’elastic energy’) among all surfaces that have the same boundary.

Such minimal surfaces and similar interfaces also arise in other contexts. Consider the following idealised model: two immiscible fluids are in a spherical container, each accounting for an equal amount of volume (half of the total) and assume gravity to be negligible compared to the other forces in play. In such a case, there are energetically optimal interfaces separating the fluids. For instance, in a fishbowl filled with an equal amount of air and water the simplest such interface (in fact the one with least possible area) would be a flat disc passing through the centre of the bowl.

Mathematicians now ask: What equilibrium configurations are possible, and what shapes might these interfaces have? This question of how other possible, ’exotic’ interfaces might look has occupied mathematicians for almost 40 years. In 2020, Alessandro Carlotto, Giada Franz and Mario Schulz solved this problem and proved that there does indeed exist an infinite number of further, complex equilibrium states in a Euclidean sphere. Their boundary surfaces are formed by minimal surfaces with a connected boundary on the sphere’s surface and an arbitrary number of ’handles’. Mario Schulz, one of Carlotto’s closest collaborators and an alumnus of ETH Zurich, has designed accurate, numerical approximations of such surfaces that can - among other things - be printed by a 3D printer, thereby providing concrete, tangible models for such seemingly exotic surfaces (cf. images below).

Like 2018 Fields Medal winner Alessio Figalli, Alessandro Carlotto graduated from the Scuola Normale Superiore in Pisa: ’La Normale - as it is often called in Italian - is a very special place that has changed my life in many respects.’

With respect to Carlotto’s award, Professor Emeritus of Mathematics Michael Struwe comments as follows: ’While still at a fairly young age, Alessandro is already a fully accomplished mathematician. He has a vast overview not only of the field of geometric analysis but also of modern mathematics as a whole, which is one of his passions.’ He adds: ’Alessandro provides a valuable service to the department, and his highly engaged role as an academic teacher is perhaps best illustrated by the fact that in recognition of the help that he gave to students throughout the toughest phases of the pandemic, and in view of the overwhelmingly positive feedback that he received for his efforts, he was invited to lecture in the ’Refresh Teaching’ series of former rector Sarah Springman on the topic of ’Connecting to your students’.’

Carlotto joined ETH Zurich in 2015 as a Junior Fellow of the Institute for Theoretical Studies. In May 2016, the ETH Board appointed him Assistant Professor of Mathematics. By the end of August 2022, he had returned to Italy to take up a professorship at the UniversitÓ degli Studi di Trento.

References

Carlotto, A., Schoen, R. Localizing solutions of the Einstein constraint equations. Inventiones Mathematicae 205, 559’615 (2016). DOI: external page 10.1007/s00222’015 -0642-4

Ambrozio, L., Carlotto, A., Sharp, B. Comparing the Morse index and the first Betti number of minimal hypersurfaces. Journal of Differential Geometry 108 (3), 379’410 , March 2018. DOI: external page 10.4310/jdg/1519959621

Carlotto, A., Li, C. Constrained deformations of positive scalar curvature metrics, II. arXiv:2107.11161v2 [math.DG]. DOI: external page 10.48550/arXiv.2107.11161

Carlotto A., Franz, G., Schulz, M.B. Free boundary minimal surfaces with connected boundary and arbitrary genus. Cambridge Journal of Mathematics, 10 (4), 835’857 , 2022. DOI: external page 10.4310/CJM.2022.v10.n4.a3

Florian Meyer

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