Yang Li

Education and employment: 

• 2011-2014: University of Cambridge, Downing College, BA degree in mathematics

• 2014-2015: University of Cambridge, MMath degree (‘Cambridge Part III’), with Distinction 

• 2015-2019: Imperial College London (CDT London school of Geometry and Number Theory), Ph.D. supervised by Professor Simon Donaldson and co-supervised by Professor Mark Haskins. Thesis title: Metric collapsing on Calabi-Yau 3-folds

• 2019-2020: postdoc at Institute for Advanced Study 

• 2020-2024: postdoc at MIT (CLE Moore Instructor 2021-2022), sponsored by Clay Mathematics Institute research fellowship.

• 2024-present: Royal Society University Research Fellow at Cambridge University

Research Summary: 

My main research field is differential geometry, which is about the interplay between partial differential equations (PDE) and the underlying geometric spaces (manifolds). My work can be divided into a number of (inevitably overlapping) subfields:

• Calabi-Yau metrics

Calabi-Yau metrics are special solutions to the vaccum Einstein equation, with close ties to algebraic geometry (which studies the zero sets of polynomial equations). In the compact manifold setting, the existence and uniqueness question was definitively answered by Yau, but the abstract existence proof gives little information about what the solution looks like. The central problem for me is to describe the limiting behaviour of the solution when the parameters vary (eg. when you vary the coefficients of the polynomial equations), especially the situation where the geometric limit drops dimension. This means that the number of independent variables decreases in the limit, so something drastic happens to the PDE!

My works revealed some unexpected non-compact Calabi-Yau metrics which model the microscopic behaviour of certain degenerate solutions, and I drew techniques from complex pluripotential theory to describe the limits in some very singular settings.

• SYZ conjecture

The SYZ conjecture is about the emergence of a new structure (known as `special Lagrangian fibration') when Calabi-Yau metrics approach a very singular type of limit (the `large complex structure limit'). The conjecture is a central prediction of mirror symmetry.

My works proved a version of the conjecture for the first examples beyond abelian variety and K3 surfaces (the `Fermat family'), and my latest results proved this for many examples. In the general case I reduced the problem from PDE to non-archimedean geometry, and I proved that the potential of the Calabi-Yau metrics converge in the hybrid topology to the non-archimedean Calabi-Yau metric on the Berkovich space.

• Special Lagrangian

In minimal surface theory, special Lagrangians are certain real n-dimensional volume minimizing submanifolds inside complex n-dimensional Calabi-Yau manifolds. Currently, we lack genuinely nonlinear techniques to construct examples. In symplectic topology, Lagrangians with suitable brane structures define objects in the Fukaya category. The conjectural picture of Thomas-Yau and the update by Joyce, suggest that the existence question of special Lagrangians should be controlled by some stability condition on the Fukaya category, and the construction of the stability condition should be related to the Lagrangian mean curvature flow. 

Part of my work seeks to clarify the conjectural picture, and I introduced a variational approach to attack the Thomas-Yau conjecture. I am also interested in the regularity question for special Lagrangians and Lagrangian mean curvature flows, such as uniqueness of tangent cones, and type II limits of Lagrangian mean curvature flow. The techniques involve a mixture of geometric measure theory, PDE, and sometime ideas from symplectic geometry.

On the constructive side, I generalised the pair of pants surface to higher dimensional special Lagrangians, and jointly with Saman Esfahani I solved a local version of the Donaldson-Scaduto conjecture on the existence of certain special Lagrangians with prescribed asymptotically cylindrical behaviour. 

• Gauge theory

The paradigm of gauge theory is Donaldson's theory of ASD instantons and its applications to 4-manifold topology. This has a few less understood cousins in higher dimension: Hermitian-Yang-Mills connections, G2-instantons and Spin(7) instantons, which are all first order reductions of the Yang-Mills equation, namely the nonlinear generalisations of the Maxwell equation. The main goal of the subject is to understand analytic compactifications of the space of solutions, and potentially use the counting of the solutions to define geometric invariants.

In the same vein, the monopole equation has a natural generalisation in dimension 6 and 7, known as Calabi-Yau monopoles and G2 monopoles, and the Donaldson-Segal programme aims to relate these monopole equations to calibrated cycles, in the same spirit that the Seiberg-Witten equation is related to holomorphic curves. 

My work produced some exotic examples of Hermitian-Yang-Mills equation, which in particular shows that under a uniform energy bound, the higher codimenional singularities can be arbitrarily complicated. 

I developed a structure theory for the SU(2) Calabi-Yau monopoles (resp. G2 monopoles) on asymptotically conical Calabi-Yau 3-folds (resp. G2 manifolds) when the mass parameter tends to infinity, and showed how to extract an abelian Calabi-Yau monopole (resp. G2 monopole) with Dirac singularity along a special Lagrangian cycle (resp. coassociative cycle), and proved that monopole bubbling phenomenon accounts for almost all the L2 energy, thus establishing part of the Donaldson-Segal programme.  In joint work with Saman Esfahani, I analysed the compactness problem for the Fueter sections with target in an Atiyah-Hitchin bundle, which is a nonlinear Dirac type equation arising in the context of the Donaldson-Segal programme, and I showed that Z/2 harmonic 1-forms naturally arise as scaling limits. 

In a completely different vein,  I found certain analogues of the Nahm transform (morally like a nonlinear Fourier transform),  in the context of adiabatic G2 instantons.

• G2 metrics

G2 metrics are the only known mechanism to construct nontrivial compact Ricci-flat metrics in odd dimensions (and the magic dimension here is seven). The main motivating picture for me is Donaldson's proposal of adiabatic coassociative K3 fibrations. Very roughly, the aim is to construct the 7-dimensional manifold as a fibration over a 3-dimensional base, with fibres being very small 4-dimensional spaces known as K3 surfaces (`7=3+4'). The G2 metric is expected to be encoded into an equation on the 3-dimensional manifold, known as the `maximal submanifold equation', which is like the minimal surface system, except that there is an ambient space of Lorentzian type, coming from the second cohomology of K3 surfaces.

My work provided a satisfactory solution for the local Dirichlet problem of the maximal submanifold equation. Another work introduced an iterative fibration mechanism analogous to Donaldson's proposal, where the dimension count is 7=1+4+2. My thesis work on Calabi-Yau 3-fold metrics with Lefschetz K3 fibration is a dimensionally reduced version of Donaldson's proposal. The joint work with Saman Esfahani and Gorapada Bera on the Donaldson-Scaduto conjecture concerns the existence and uniqueness of certain associative submanifolds, which are expected local models in Donaldson's adiabatic fibration. 

• Interdisciplinary (AI related):

Jointly with my high school friend Feng Ruan, we developed a theory of `kernel learning', which is about learning the optimal matrix parameters among a family of kernel ridge regression problems, or in other words optimizing among a family of reproducing kernel Hilbert spaces. This has strong analogy with learning the inner layer parameters within 2-layer neural networks. We discover that the local minima in the parameter space is intimately connected to variable selection and scale detection tasks, and is sensitive to the underlying discrete structure or clustering behaviour within the data. We also design a canonical Riemannian gradient flow on the parameter space, so that in the presence of Gaussian noise variables, this flow has the remarkable property of possessing infinitely many monotone quantities, so that the noise effect decays away automatically.

Introductory note: the papers `On the kernel learning problem' and `Gradient flow in the kernel learning problem' are two companion papers written in the pure maths style. They should be accessible to any mathematician with a strong background in analysis and linear algebra, and some basic familiarity with probability and Riemannian geometry, but no prior knowledge of machine learning is required. The paper `layered models can automatically regularize and discover low dimensional structures via feature learning' has a more statistics flavour. For readers interested in the connection with 2-layer neural networks, we recommend reading the postscript in the paper `Gradient flow in the kernel learning', immediately after one has read the introduction and understood the setup.

Publications and preprints:

Calabi-Yau metrics and SYZ conjecture: 

• A new complete Calabi-Yau metric on C3 , Inventiones mathematicae, July 2019, Volume 217, Issue 1, pp 1-34. 

• A gluing construction of collapsing Calabi-Yau metrics on K3 fibred 3-folds, Geometric and Functional Analysis, August 2019, Volume 29, Issue 4, pp 1002-1047.

• On collapsing Calabi-Yau fibrations, J. Differential Geom. 117(3): 451-483 (March 2021). 

• Diameter bounds for degenerating Calabi-Yau metrics, joint with Valentino Tosatti, J. Differential Geom. 127 (2024), no. 2, 603--614. 

• On the collapsing of Calabi-Yau manifolds and Kaehler-Ricci flows, joint with Valentino Tosatti, J. Reine Angew. Math. 800 (2023), 155--192.

• Collapsing Calabi-Yau fibrations and uniform diameter bounds, Geom. Topol. 27 (2023), no. 1, 397--415. 

• Complete Calabi-Yau metrics in the complement of two divisors, joint with Tristan Collins,  Duke Math. J. 173 (2024), no. 18, 3559--3604.

•Special Kahler geometry and holomorphic Lagrangian fibrations, joint with Valentino Tosatti, C. R. Math. Acad. Sci. Paris 362 (2024), Special issue, 171--196. 

• SYZ conjecture for Calabi-Yau hypersurfaces in the Fermat family, Acta Math. 229 (2022), no. 1, 1–53. 

• SYZ geometry for Calabi-Yau 3-folds: Taub-NUT and Ooguri-Vafa type metrics. Mem. Amer. Math. Soc. 292 (2023), no. 1453, v+126 pp. ISBN: 978-1-4704-6782-1; 978-1-4704-7698-4

• Uniform Skoda integrability and Calabi-Yau degeneration, Anal. PDE17 (2024), no. 7, 2247--2256. 

• Metric SYZ conjecture and non-archimedean geometry, Duke Math. J.172 (2023), no. 17, 3227--3255.

• Survey on the metric SYZ conjecture and non-archimedean geometry, Internat. J. Modern Phys. A 37 (2022), no. 17, Paper No. 2230009, 44 pp.

• Metric SYZ conjecture for certain toric Fano hypersurfaces, Camb. J. Math. 12 (2024), no. 1, 223--252.  

• Intermediate complex structure limit for Calabi-Yau metrics, Inventiones Mathematicae. (2025).

• Degeneration of Calabi-Yau metrics and canonical basis, https://arxiv.org/abs/2505.11087

Special Lagrangians and Thomas-Yau conjecture:

• Thomas-Yau conjecture and holomorphic curves, EMS Surv. Math. Sci. 12 (2025), no. 2, 323--475.

• Quantitative Thomas-Yau uniqueness, Geometry & Topology 29:5 (2025)

• Uniqueness of some cylindrical tangent cones to special Lagrangians, Geom. Funct. Anal. 33 (2023), no. 2, 376--420.

• Special Lagrangian pair of pants, Comm. Pure Appl. Math. 78 (2025), no. 7, 1320--1356.

• On the Donaldson-Scaduto conjecture, joint with Saman Esfahani, https://arxiv.org/abs/2401.15432.

• Uniqueness in the local Donaldson-Scaduto conjecture, joint with Gorapada Bera and Saman Esfahani,  Int. Math. Res. Not. IMRN 2025, no. 16, rnaf245.

• Singularity formations in Lagrangian mean curvature flow, joint with Gábor Székelyhidi, https://arxiv.org/abs/2410.22172.

Gauge theory (Nahm transform and analytic aspects):

• Local Nahm transform and singularity formation of ASD connections, Communications in Mathematical Physics, pp 1-38. 

• Mukai duality on K3 surfaces from the differential geometric perspective, arXiv:1908.05017. 

• Mukai duality on adiabatic coassociative fibrations, Pure Appl. Math. Q. 20 (2024), no. 6, 2533--2599.

• Bubbling phenomenon for Hermitian Yang-Mills connections, International Mathematics Research Notices, Volume 2021, Issue 6, March 2021, Pages 4657-4678. 

• A note on singular Hermitian Yang-Mills connections,  Math. Res. Lett.30 (2023), no. 1, 167--184.

•Fueter sections and $Z_2$-harmonic 1-forms, joint with Saman Esfahani, https://arxiv.org/abs/2410.06367

•The large mass limit of $G_2$ and Calabi-Yau monopoles, https://arxiv.org/abs/2503.12075.

G2 metrics (special holonomy):

• Dirichlet problem for maximal graphs of higher codimension, Int. Math. Res. Not. IMRN 2022, no. 3, 2159–2179.

• Iterated collapsing phenomenon on G2-manifolds, Pure Appl. Math. Q. 18 (2022), no. 3, 971–1036.

Interdisciplinary (AI):

• Layered Models can "Automatically" Regularize and Discover Low-Dimensional Structures via Feature Learning, joint with Yunlu Chen, Keli Liu and Feng Ruan, https://arxiv.org/abs/2310.11736

• On the kernel learning problem, joint with Feng Ruan, https://arxiv.org/abs/2502.11665

• Gradient flow in the kernel learning problem, joint with Feng Ruan, https://arxiv.org/abs/2506.08550