Tancredi Schettini Gherardini

Talks

Invited Seminars

• Numerical Geometry as a Lens on AI for Pure Math
  IFS — Sydney Mathematical Research Institute, University of Sydney, 20 February 2026 — see here
  Abstract

  Machine learning techniques have recently emerged as powerful tools for progress in various branches of pure mathematics. Through the example of numerical differential geometry, we illustrate how AI and mathematics can form a meaningful synthesis and comment on more general features of this integration. Some of these are typically found appealing by mathematicians, while others are often viewed as less pleasant. We will reflect on several of these aspects, with the hope of offering new perspectives and stimulating discussion with the audience.
• AInstein: Machine Learning “Special” (Pseudo)-Riemannian Metrics
  High Energy Physics Seminar — Tor Vergata University of Rome, 15 December 2025 — see here
  Abstract

  A numerical scheme based on semi-supervised machine learning, “AInstein”, was recently introduced to approximate generic Riemannian Einstein metrics on a given manifold. Its versatility stems from encoding the differentiable structure directly in the loss function, making the method applicable to manifolds constructed in a “bottom-up” fashion that admit no natural embedding in Rⁿ. A limitation, however, is that the resulting numerical metric is not inherently global. To address this, we introduce a new approach for all real (n−1)-dimensional manifolds that can be embedded in Rⁿ, in which the neural-network ansatz is automatically globally defined. After a brief review of the original AInstein model, the talk presents novel results obtained with the new architecture, including applications to two open problems: the Kazdan–Warner (prescribed curvature) problem on S² and the search for negative-curvature metrics on S⁴ and S⁵. Finally, we focus on a further extension of the method to Lorentzian metrics, presenting some preliminary results concerning black holes.
• AInstein: Numerical Einstein Metrics through Machine Learning
  AI for QFT Seminar Series — Queen Mary University of London, 11 April 2025
  Abstract

  In this talk, we will discuss a very recent numerical scheme based on semi-supervised machine learning, “AInstein”, which approximates generic Riemannian Einstein metrics on a specified manifold (arXiv:2502.13043). We will begin by reviewing the first applications of machine learning to find numerical Calabi-Yau metrics, and then present our generalisation of those approaches. We will summarise the results obtained with AInstein so far, which concern Einstein metrics on spheres of various dimensions. A long-standing open question in this context is the existence of Ricci-flat metrics on S⁴ and S⁵, for which our results provide numerical evidence against. Finally, we will comment on the numerous possible extensions and further applications of AInstein.
• AInstein: Numerical Einstein Metrics through Machine Learning
  Geometry Seminar — Université Libre de Bruxelles (ULB), 7 April 2025 — see here
  Abstract

  In this talk, we will discuss a very recent numerical scheme based on semi-supervised machine learning, “AInstein”, which approximates generic Riemannian Einstein metrics on a specified manifold (arXiv:2502.13043). We will begin by reviewing the first applications of machine learning to find numerical Calabi-Yau metrics, and then present our generalisation of those approaches. We will summarise the results obtained with AInstein so far, which concern Einstein metrics on spheres of various dimensions. A long-standing open question in this context is the existence of Ricci-flat metrics on S⁴ and S⁵, for which our results provide heuristic numerical evidence against. Finally, we will comment on the numerous possible extensions and further applications of AInstein.
• Exotic spheres from different angles
  Special Seminar — Department of Applied Science and Technology, Politecnico di Torino, 4 February 2025
  Abstract

  This talk explores the mathematical properties of exotic (7-)spheres and their applications in theoretical physics. In particular, I will present a mix of foundational concepts and new research, starting with the geometry of Milnor’s bundles. This includes specific analytical results from arXiv:2309.01703 (J. High Energ. Phys. 2023, 100) and arXiv:2410.01909, as well as their significance in the realm of supergravity. The presentation will also cover the differentiable structures of exotic spheres and their conjectured connection to shock waves and general relativity (currently a work in progress). To conclude, I will introduce a machine learning-driven numerical strategy for computing Einstein metrics — a technique we plan to apply to exotic spheres (ongoing work with Edward Hirst and Alex Stapleton). I will also touch upon how this numerical scheme could be utilised in various (super)gravity models.
• Exotic spheres from different angles
  Seminar of the Quantum Gravity and Unified Theories Group — Max Planck Institute for Gravitational Physics, 27 January 2025
  Abstract

  During this presentation, I will explore various mathematical features of exotic (7-)spheres and their physical implications. Specifically, I will review established knowledge alongside recent findings. First, I will cover the geometric properties of Milnor’s bundles and several analytic outcomes presented in arXiv:2309.01703 (J. High Energ. Phys. 2023, 100) and arXiv:2410.01909, and explain how these findings apply to supergravity. I will also discuss the differentiable structure of exotic spheres, along with some speculative consequences for general relativity and shock waves (ongoing research). Finally, I will present a machine learning-based numerical method for identifying and approximating Einstein metrics, ultimately intended for application to exotic spheres (ongoing collaboration with Edward Hirst and Alex Stapleton). Furthermore, I will outline the potential uses of this computational approach within (super)gravity theories.
• Exotic spheres from different angles
  Mathematics and Theoretical Physics Seminars — University of Hertfordshire, 22 January 2025 — see here
  Abstract

  In this talk, I will discuss different mathematical properties of exotic (7-)spheres and their implications in physics. Namely, I will cover some known facts as well as recent results regarding:

  • The geometry of Milnor’s bundles and a number of analytic results obtained in arXiv:2309.01703 (J. High Energ. Phys. 2023, 100) and arXiv:2410.01909; I will discuss the relevance of these results in the context of supergravity.
  • The differentiable structure of exotic spheres together with its conjectured implications in general relativity and shock waves (work in progress).
  • A numerical technique for finding and approximating Einstein metrics, to be applied eventually to the case of exotic spheres, which is based on state-of-the-art machine learning tools (work in progress with Edward Hirst and Alex Stapleton). The possible uses of this numerical scheme in (super)gravity theories are also discussed.
• Exotic Spheres and Multi-dimensional Physics
  Research Seminar “Differential Geometry and Analysis” — University of Marburg, 4 June 2024 — see here
  Abstract

  Exotic spheres are seven-dimensional compact manifolds that, by a non-constructive existence theorem, are known to admit many Sasaki–Einstein metrics. For this reason, they are promising candidates for a range of applications in string theory, in particular in M-theory compactifications, but they have not previously been studied in this context because an appropriate description has been lacking. In this talk, I will discuss metrics on exotic spheres regarded as non-principal S³ bundles over S⁴, namely Milnor bundles, and summarise the results presented in arXiv:2309.01703 (published as J. High Energ. Phys. 2023, 100). I will review the properties of these manifolds, mention the ways in which exotic differentiable structures have appeared in physics so far, and describe their potential application in string theory. All necessary concepts will be introduced, and no prior background in string theory will be assumed. Finally, I will present in detail an explicit metric on one of the exotic spheres and comment on how this result could be used in the compactification mechanism. I will conclude by discussing some open questions and possible extensions of this work.
• Exotic Spheres and Multi-dimensional Physics
  Geometry Seminar, DIMAI — Università degli Studi di Firenze, 27 March 2024 — see here and here
  Abstract

  Exotic spheres are seven-dimensional compact manifolds that have been shown (through a non-constructive existence theorem) to admit numerous Sasaki–Einstein metrics. For this reason, they are interesting candidates for various applications in string theory (specifically, within compactifications of M-theory), but have never been considered in this context due to the lack of a suitable description. In this talk, I will discuss metrics on exotic spheres viewed as non-principal S³ bundles over S⁴, i.e. Milnor’s bundles, and summarise the findings presented in arXiv:2309.01703 (J. High Energ. Phys. 2023, 100). I will outline the properties of these manifolds, mention the appearance of exotic differentiable structures in physics so far, and describe the potential application within the realm of string theory. All the necessary concepts will be introduced, and no prior knowledge on string theory will be required. Finally, I will present in detail an explicit metric for one of the exotic spheres, and comment on how this result might be applied in the compactification mechanism. I will end by discussing some open questions and interesting extensions of this work.

Invited Conference Talks

• DANGER: Data, Numbers, and Geometry
  Banff International Research Station — upcoming
• A Physicist’s Visit to Exotic Spheres
  Geometry at Large — International Centre for Mathematical Sciences (ICMS-Sofia), 11 October 2025 — see here
  Abstract

  In this talk, I will present some differential-geometric properties of exotic 7-spheres through the lens of Kaluza-Klein formalism. I will review both established facts and recent developments, beginning with Milnor’s bundles and their geometry, and proceeding with the results from arXiv:2309.01703 (J. High Energ. Phys. 2023, 100) and arXiv:2410.01909 (J. Geom. Phys. 216 2025, 105590); comments about the role of Kaluza-Klein theory and (possible) appearance of exotic spheres in supergravity will also be made. Additionally, I will discuss some aspects of the differential topology of these exotic spheres, including their proposed implications for shock waves and general relativity (work in progress).
• AInstein: Numerical Einstein Metrics through Machine Learning
  23rd South East Mathematical Physics Seminar, 11 July 2025 — see here
  Abstract

  In this talk, we will discuss a very recent numerical scheme based on semi-supervised machine learning, “AInstein”, which approximates generic Riemannian Einstein metrics on a specified manifold (arXiv:2502.13043). We will begin by reviewing the first applications of machine learning to find numerical Calabi-Yau metrics, and then present our generalisation of those approaches. We will summarise the results obtained with AInstein so far, which concern Einstein metrics on spheres of various dimensions. A long-standing open question in this context is the existence of non-round metrics on S⁴ and S⁵, for which our results provide heuristic numerical evidence against. Finally, we will comment on the numerous possible extensions and further applications of AInstein.

Contributed Conference Talks

• AInstein: atlas architecture vs embedding architecture
  String Data 2025 — Infosys Canary Wharf & London Institute for Mathematical Sciences, 8 December 2025 — see here
  Abstract

  A numerical scheme based on semi-supervised machine learning, “AInstein”, was recently introduced to approximate generic Riemannian Einstein metrics on a given manifold. Its versatility stems from encoding the differentiable structure directly in the loss function, making the method applicable to manifolds constructed in a “bottom-up” fashion that admit no natural embedding in Rⁿ. After a brief review of the original AInstein model, we focus on a new architecture, adapted to the special case of real (n−1)-dimensional manifolds that can be embedded in Rⁿ; this has the advantage that the neural-network ansatz is automatically globally defined. We present novel preliminary results obtained with the new architecture, concerning two open problems: the Kazdan–Warner (prescribed curvature) problem on S² and the existence of negative-curvature metrics on S⁴, S⁵. Finally, if time permits, we will briefly comment on a further ongoing extension of AInstein to Lorentzian metrics and black holes.
• Exotic Spheres, Kaluza-Klein Formalism and Supergravity
  Eurostrings meets fpuk 2024 — University of Southampton, 3 September 2024 — see here and here
  Abstract

  Exotic spheres are seven-dimensional compact manifolds that have been shown (through a non-constructive existence theorem) to admit numerous Sasaki–Einstein metrics. Hence, they are suitable candidates for compactifications of M-theory, but have never been considered in this context due to the lack of a suitable description. In this talk, I will discuss metrics on exotic spheres viewed as non-principal S³ bundles over S⁴, i.e. Milnor’s bundles, summarising what was found in arXiv:2309.01703 (J. High Energ. Phys. 2023, 100) and some ongoing work with David Berman and Martin Cederwall. I will outline the importance of these manifolds in differential geometry, and then present in detail an explicit Kaluza-Klein metric for one of the exotic spheres. I will discuss its properties, in terms of geometric quantities and physical energy conditions, then comment on its relation to 7-dimensional Euclidean gravity and its role in supergravity theories. Finally, I will discuss some interesting extensions of this work, which lies within a little-explored, but potentially fruitful, territory.
• Exotic Spheres and Kaluza-Klein Formalism
  Mathematical Supergravity — UNED Madrid, 23 February 2024 — see here
  Abstract

  Exotic spheres are seven-dimensional compact manifolds that have been shown (through a non-constructive existence theorem) to admit numerous Sasaki–Einstein metrics. Hence, they are suitable candidates for compactifications of M-theory, but have never been considered in this context due to the lack of a suitable description. In this talk, I will discuss metrics on exotic spheres viewed as non-principal S³ bundles over S⁴, i.e. Milnor’s bundles, and summarise the findings presented in arXiv:2309.01703 (J. High Energ. Phys. 2023, 100). I will outline the importance of these manifolds in differential geometry, mention the appearance of exotic differentiable structures in physics so far, and then present in detail an explicit Kaluza-Klein metric for one of the exotic spheres. I will comment on its relation to 7-dimensional Einstein gravity with a cosmological constant and its possible role in supergravity theories. Finally, I will discuss some interesting extensions of this work, which lies within a little-explored, but potentially fruitful, territory.
• Exotic Spheres and Kaluza-Klein Formalism
  Fundamental Physics UK (fpuk) — King’s College London, 10 January 2024 — see here
  Abstract

  I will discuss metrics on exotic spheres viewed as non-principal S³ bundles over S⁴, i.e. Milnor’s bundles, summarising the findings presented in arXiv:2309.01703 (J. High Energ. Phys. 2023, 100). I will outline the importance of these manifolds in differential geometry, mention the appearance of exotic differentiable structures in physics so far, and then present in detail an explicit Kaluza-Klein metric for one of the exotic spheres. I will comment on its relation to 7-dimensional Einstein gravity with a cosmological constant and its possible role in supergravity theories. Finally, I will discuss some interesting extensions of this work, which lies within a little-explored, but potentially fruitful, territory.