Research — Tolga Birdal

Research

My work studies representations and algorithms for data with geometry, symmetry, dynamics and higher-order relations. My resaearch trajectory has gradually explored and pivoted across several fields: applied 3D/4D computer vision, machine learning theory, topology/geometry in deep learning, and quantum computing. Future directions involve translating these foundations into AI4Physics and AI4Biology.

3D and 4D vision

How can a machine recover shape, motion, and interaction from observations that are incomplete, noisy or ambiguous?

This work spans rigid and non-rigid registration, reconstruction, articulated motion, human–object interaction, scene flow and generative priors for dynamic three-dimensional worlds.

3D reconstruction and registration [PPFNet, 3DCapsuleNets, SyNoRiM]
Dynamic scenes, humans, and articulated objects [HuMoR, NRMF, Dyn-HaMR, MoLingo]
Generative models of shape, motion, and texture [UV3TeD, SaMoR, HyperSDFusion]

Three-dimensional torus and simplicial complex

Geometry and topology

What mathematical structure should be retained when data do not live in Euclidean space or when relations are richer than pairs?

I work with manifolds, groups, graphs, simplicial and cellular complexes, flag manifolds, optimal transport and discrete differential operators.

Non-Euclidean representation and optimisation [DSG, NRMF, NRDF, LatentRFM]
Higher-order Laplacians and Operators [Root2Leaf, EffOps, TDL-Book]
Symmetry, equivariance & synchronisation [MeasureSync, BirkhoffSync, MultiBodySync]

Neural learning on a curved triangulated surface

Geometric and topological deep learning

How should neural architectures change when the domain itself carries geometry or topology?

The group develops models and software for learning on structured domains, including neural operators, cellular and simplicial networks, higher-order generative models and differentiable geometric computation.

Topological neural networks and operators [TDL, TNO, CoPreSheaves, TDL-Book ]
Geometric / topological generative models [HOG-Diff, UV3TeD, HyperSDFusion]
Open implementations through TopoX and DiGeo

Foundations of learning

Which properties of data, optimisation and representation explain when neural systems generalise?

Projects connect intrinsic dimension, persistent homology, compressibility, feature superposition, robustness, stochastic dynamics and the geometry of learned representations.

Topology of optimisation dynamics and generalisation [GATES, TopoComplex]
Stochastic dynamics [FractioNSDE, GFDM, FracDBM]
Mechanistic questions in representation learning [Grokking, SuperCorr, SuperAdv]

Quantum and physical AI

Can alternative computational primitives expose useful structure in vision and optimisation problems?

This line includes quantum permutation synchronisation, hybrid classical–quantum optimisation and workshops that bring together quantum computing, vision and machine learning.

Quantum computer vision / machine learning [QeCV, QSync, QB-KMeans]
Hybrid optimisation [QFW, QP-SBGD]

Abstract molecular and scientific data drawing

AI for physics and structural biology

How can structure-aware machine learning support scientific modelling when observations are relational, multiscale, and governed by physical constraints?

Current work includes topological deep learning for molecules and proteins, geometric modelling of dynamical systems, differentiable operators on meshes and collaborations across computational science.

Molecular and protein representation [HGR, HOG-Diff]
Learning physical dynamical systems [TNO, SG-NCDE, LIBS, Lévy-VI]

Open source

Software

Reusable tools developed alongside the research.

🌐 TopoX 🍩

TopoX

A unified software ecosystem for computing and learning on topological domains.

Documentation ↗

DiGeo

Differentiable geometry for triangular meshes, including geodesics, transport and Riemannian optimisation.

Documentation ↗

Outputs

Papers, code and datasets

Browse my publications most of which provide accompanying open source software and data.

Browse publications