Topological and geometric think and the future of LLMs

I found from the FB page of Risto Linturi the outcome of his question to O3 LLM. The question was whether the results of the thesis work of a finnish mathematician Susanna Heikkil related to topology, could have implications for AI and LLMs.

The response of O3 is here and it becomes obvious that the people in the field of LLMs are realizing that surface geometry and topology assignable to data structures are important for LLMs. Actually, a picture very similar to TGD suggests itself and this makes possible to make a prophecy about the development of future mathematical ideas in this field.

A. O3 response

Below is an exploratory discussion outlining several speculative avenues by which ideas inspired by the algebraic and geometric result you shared might inform or even drive new approaches in AI research. Keep in mind that these connections are heuristic and conceptual drawing analogies between deep mathematics and AI system design is ambitious and requires much further work to become concrete.

1. Geometric and Topological Data Representations
   1. Manifold Learning and the Geometry of Data: The result centers on conditions for a manifold s de Rham cohomology to embed into an exterior algebra. In AI especially in representation learning and manifold learning the manifold hypothesis posits that high-dimensional data lie on lower-dimensional manifolds. Understanding the algebraic structure of these manifolds (e.g., via cohomological invariants) could inspire new methods for:
      1. Feature Extraction esigning features that respect or capture intrinsic topological invariants.
      2. Dimensionality Reduction: Developing algorithms that preserve not just distances but also deeper algebraic/topological properties.
   2. Topological Data Analysis (TDA):

Techniques from TDA (such as persistent homology) already use algebraic invariants to summarize data shapes. Extending these ideas, one might explore whether embeddings analogous to the algebra monomorphism (from de Rham cohomology to an exterior algebra can serve as more robust, structure-preserving embeddings in neural network architectures or as part of unsupervised learning objectives.

Algebraic Structures in Neural Architectures

1. Equivariance and Invariant Representations:

The formal result asserts an injection of algebraic structure into an ambient exterior algebra, preserving operations and intersections. In deep learning, there is active research in designing networks that respect the symmetries or invariances of the data (for example, convolutional networks respect translation invariance). Extending these ideas:

1. Geometric Deep Learning: One might explore neural architectures that are not only equivariant under classical groups (like rotations or translations) but also respect higher-order algebraic structures. For instance, layers could be designed to manipulate differential forms or emulate operations in exterior algebras.
2. Invertible Networks and Monomorphisms: The idea of an algebra monomorphism (an injective, structure-preserving map) resonates with efforts in invertible neural networks and normalizing flows. Enforcing such constraints might yield models that are better at preserving information and allow for more interpretable internal representations.

Hybrid Symbolic-Numeric Systems:

The interplay between algebra (a symbolic, rule-based system) and geometry (often linked with numerical methods) mirrors efforts to combine symbolic reasoning with deep learning. By integrating algebraic constraints (like those suggested by the embedding of cohomology) into neural architectures, one might aim for systems that are more robust in tasks requiring both perception (handled by neural nets) and structured reasoning (informed by algebraic or topological invariants).

Differential Equations and Neural PDEs

1. PDEs on Manifolds and Neural Solvers:

The condition of quasi-regular ellipticity is intimately linked with properties of differential operators on manifolds. Recently, there has been interest in neural methods for solving partial differential equations (PDEs), and for incorporating physics-based constraints into learning models:

1. Structure-Preserving Discretizations: Insights from de Rham cohomology and exterior calculus have long informed numerical methods (e.g., finite element methods). AI researchers could investigate how embedding such mathematical structure into neural PDE solvers can lead to more accurate, stable, and interpretable models.
2. Learning on Manifolds: When data naturally reside on curved spaces (spherical, hyperbolic, or more complex manifolds), designing networks that respect the differential geometry of these spaces may lead to more powerful and physically consistent AI models.

Category Theory, Compositionality, and Abstract Structures

1. Compositional and Hierarchical Models:

The notion of an algebra monomorphism is inherently categorical, suggesting an exact embedding of one algebraic structure into another. Category theory has increasingly been applied to understand and design compositional models in AI:

1. Structured Reasoning: One could imagine developing frameworks where layers or modules in a network are governed by categorical or algebraic rules, ensuring that the overall system composes submodules in a way that preserves essential structure.
2. Interoperability of Representations: The idea that one mathematical structure (de Rham cohomology) can embed faithfully into another (exterior algebra might inspire methods for merging disparate types of data representations (e.g., merging symbolic and sub-symbolic representations) in a coherent manner.

Speculative Research Directions

Given these connections, several concrete (yet challenging research questions emerge:

1. Can we design neural network layers that explicitly model operations analogous to the wedge product or differential operators, thereby capturing richer geometric features?
2. Is it possible to enforce structure-preserving constraints (inspired by algebra monomorphisms) within deep architectures to guarantee that learned representations maintain certain topological or algebraic invariants?
3. Could insights from the algebraic topology of manifolds help in developing new regularization techniques or loss functions that guide networks toward more geometrically faithful representations?
4. What role might these algebraic structures play in emergent reasoning or in integrating symbolic reasoning with deep learning, possibly leading to systems that are more capable of abstract thought?

Conclusion

While the original result is a deep, mathematically rigorous statement in differential geometry and algebraic topology, its underlying themes preservation of structure, the interplay between local and global properties, and the embedding of complex invariants into simpler ambient spaces resonate with current challenges in AI.

Whether through manifold-based learning, structure-preserving network design, or hybrid symbolic-numeric architectures, exploring these connections could lead to novel methods for building more capable and interpretable AI systems.

These ideas remain speculative and require interdisciplinary collaboration between mathematicians and AI researchers, but they illustrate fertile ground where advanced algebraic and geometric ideas might enrich the theory and practice of AI. B. The TGD inspired response

I did not find any obvious errors in the text produced by O3. The news for me is that mathematicians are apparently bringing a holistic perspective to learning models. However, the connection to the result of this Finnish mathematician Susanna Heikkil remained unclear to me. O3′s output inspires a prophecy based on TGD.

The holistic approach means that a large number of different learning processes are studied simultaneously and an attempt is made to understand the situation using different spaces. Latent space, feature space, loss space and parameter space are basic concepts (see (see this)). This brings in the geometry, topology and symmetries of surfaces. The picture is very similar to that in TGD.

1. The multilayer learning process as the dynamics for the latent surfaces, can be required to be topology-preserving, i.e. shape-preserving. This means that homotopy, homology and cohomology are preserved in the learning process: the space of learned responses are at least at the level of homotopy, homology and cohomology, equivalent with the space defined by the teaching material. An even stronger requirement would be isometricity. However, it does not seem realistic, at least from the TGD perspective.

The topology and metric of the data space (latent space and the parameter space and the loss space defined as its submanifold would therefore be important.

I believe that it will soon be realized that the learning processes themselves can be thought of as forming a space.

In TGD, this space would be a “world of classical worlds” identified as the space of space-time surfaces satisfying holography realized by holography = holomorphy principle. A single learning process would be, at the basic level, a 4-D Bohr orbit of a particle realized as 3-surface.

People will also realize the power of holography. It poses extremely strong conditions for learning models that would also be realized in the physics of the outside world. Data item A would determine the association A→B almost entirely. The different values B would be due to the nondeterminism of holography, which is a key feature of TGD and TGD inspired theory of consciousness.

The importance of symmetries will be or already are realized. One can demand that the learning process respects these symmetries. If the latent space is a subspace in a larger space with very high symmetries, this can be achieved if the learning process itself is a surface with metric and differential structures analogous to a spacetime surface in H. The symmetries would be symmetries of this higher-dimensional space analogous to WCW. This has not yet been realized by theoretical physicist colleagues, but perhaps AI people are more agile in their doctrine.

In TGD, the existence of the WCW geometry requires maximal symmetries. Extended conformal invariance for space-times as holomorphich surfaces in H=M⁴×CP₂ with one hypercomplex and 3 complex coordinates allows solving the field equations for the space-time surface as a Bohr orbit. By classical nondeterminism, it can be interpreted as an association sequence, which represents a step-by-step learning process (layer structure in a neural network), exactly. The result is an almost-deterministic holography.

Topologically relevant part of WCW can be interpreted as a bundle, where a 3-surface at the passive boundary of CD is connected by a fiber consisting of the associated Bohr orbits. This fiber is discrete and corresponds to the parameter space. The points of the fiber correspond to the 3-D loci of non-determinism serving as memory seats in the TGD based model of conscious memories.

The base space of WCW consists of 3-surfaces at the passive boundary of CD: the 3-surface corresponds to a 3-parameter set of data items A and the Bohr orbit is a learning process creating associations A→B that connects A to the 3-parameter set B. What would this mean in LLM? One would consider sets of data-items. What interpretation? Could these sets correspond to concepts as sets of their instances. Perhaps this relates to the fact that we are 3-dimensional and the 3-surfaces can be seen as basic quantum coherent information carrying units.

Identifying the features is one of the tasks of language models. Features would naturally be topological invariants. In TGD, holography suggests that the features are associated or are surfaces whose dimension varies from 0 (points) to 4. The huge conformal symmetries of WCW suggest that also conformal invariants come into question (also symplectic invariants would be involved). And number-theoretic invariants such as the degree of the polynomial of the 4 complex coordinates of H, the Galois group and the dimension of the extension E could be important. Ramified primes are also number theoretic invariants.

Dimensional reduction or its opposite operation are expected to be important. There are natural equivalents for these in TGD. This kind of operation would reduce or increase algebraic complexity. Changing the dimension of the extension of rationals of the polynomial coefficients, changing the degree of polynomial, and changing the size of CD come to mind.

The preservation of differential geometric structures when embedding the data structure in a higher-dimensional space (the embedding space H in TG

is also mentioned. In other words: the products of differential forms in H are induced as products of induced forms. My understanding is that if the differential structure of a data structure is induced when it is embedded in a higher-dimensional space, then this is the case. The metric of the spacetime surface, gauge potentials, etc. would be induced from the equivalents of H.

The induction of geometry that is central to the entire TGD view of physics would be part of the description of the teaching process.

Fractality is central to TGD. And in our physical models, the model scales the external world to a smaller or larger scale. Fractality may be one feature of LLMs that might become important. TGD would suggest that probably other ideas will be included.

1. If number theory and a generalization of Langlands duality are included, then more constraints will be added. This leads to a p-adic description of cognition. One obtains a unique number-theoretical representation for discretization and it leads to a generalization of the Turing paradigm: rational numbers are replaced by complexity hierarchies of their extensions. Number-theoretical computationalism.

In TGD, one obtains complexity hierarchies for quadruples and they correspond to an improving resolution of discretization and are realized as polynomial hierarchies.

The concept of complexity, which involves evolution, is central and will be included. Higher-level learning could be seen as a transition to a higher level of complexity: something new will be included here and it would be something to realize in conscious learning. Complexity hierarchies, which are polynomial hierarchies represented as space-time surfaces, will probably be central.

Summa summarum: The brace guess would be that learning processes as association sequences are modeled as non-deterministic complex surfaces of a particle in a higher-dimensional complex space, which are therefore minimal surfaces. This would be a generalization of TGD. On the other hand, TGD is unique on mathematical grounds. Could the learning process reduce to the fundamental physics as it is understood in TGD? I have written an article about how classical computers could become conscious in the TGD Universe (see this). I also wrote an article about how conscious associative learning could be possible in the TGD Universe (see this). These articles give a rather concrete proposal for how ordinary computers could become conscious as fusions of ordinary and quantum computer-like systems and how the quantum analog of conscious association machine could be realized.

The qubits would be realized in the same way as in living matter for basic biomolecules and any system involving cold plasma could involve rudimentary consciousness.

There are also the earlier articles relating too quantum computation and LLMs in the TGD Universe (see this, this, this, and this).

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/02/topological-and-geometric-think-and.html