I wish this post to be a continuously updated list of my math interests in 2024 with proper citations to literatures, as I keep wandering in the math wonderland and I don’t want to be lost in it without breadcrumbs.

Some interests that have older origins will gradually moved to corresponding posts for earlier years.

I also hope certain interests will be developed into research projects, and leaving only a brief summary and a link here.

Each interest should have one or few central questions, and one or few references to literatures.

Formalization

This part of interests is about small-scale formalization of mathematical concepts and theorems, for learning and potential PRs to Lean’s Mathlib. Each should focus on one reference which is well organized and convenient to be converted into a blueprint.

Spin groups

The PR to Mathlib #9111 about Spin groups is ready to merge, but there are 2 open questions:

  • what more lemmas about Spin groups are interesting to mathematians?
  • what more should be formalized to formalize Versors and what’s in Section “The contents of a geometric algebra” in ( , (). Geometric algebra. arXiv preprint arXiv:1205.5935. Retrieved from http://arxiv.org/abs/1205.5935 ) , e.g. r-blades, r-vectors?

For the former, I should take a closer look at ( , (). Spin geometry. Lecture Notes. University of Edinburgh. Retrieved from http://mat.uab.cat/~rubio/csa2017/SpinNotes.pdf ) and maybe ( , (). Expository paper on clifford algebras, representations, and the octonion algebra. arXiv preprint arXiv:1906.11460. Retrieved from https://arxiv.org/pdf/1906.11460 ) , ( , (). Probing clifford algebras through spin groups: A standard model perspective. arXiv preprint arXiv:2312.10071. Retrieved from https://arxiv.org/pdf/2312.10071 ) .

For the latter, see the Z-filteration in lean-ga and the prototype.

I also wish to include some latest results presented in ( , & al., , & (). Clifford group equivariant neural networks. arXiv preprint arXiv:2305.11141. ) , with supplements from ( , & al., , , & (). Geometric algebra transformers. arXiv preprint arXiv:2305.18415. ) (lately there is a new paper applying this in HEP ( , & al., , , , , & (). Lorentz-equivariant geometric algebra transformers for high-energy physics. arXiv preprint arXiv:2405.14806. ) , in the same spirit, I should also read ( , & al., , , , & (). Geometry-informed neural networks. arXiv preprint arXiv:2402.14009. ) and possibly ( , & al., , & (). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378. 686–707. ) )), in which some of the results are proven in ( & , & (). Graded symmetry groups: Plane and simple. Advances in Applied Clifford Algebras, 33(3). 30. ) .

See also discussions in the lean-ga blueprint.

I’ve started a Forester experiment about the definitions of Spin groups here. I also need to check citations of On some Lie groups in degenerate Clifford geometric algebras.

Matrix

The Matrix Cookbook (November 15, 2012) ( , & al., , & (). The matrix cookbook. Technical University of Denmark, 7(15). 510. Retrieved from https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf ) covers many useful results about matrices, and Eric Wieser’s project lean-matrix-cookbook aims to give one-liner proofs (with reference to the counter part in Mathlib) to all of them.

The project is far from complete and it would be great to claim a small portion of interested results and contribute to it. I also wish to figure out the GA counterpart of the same portion.

Previous interests about matrices rise from Steven De Keninck’s work on GALM ( & , & (). Geometric algebra levenberg-marquardt. Springer. ) , since the paper I have been interested in GA approaches that has practical advantages over traditional matrix-based methods. Notably the paper also discussed the link between degenerate Clifford algebras and dual numbers / automatic differentiation. A more recent inspiration might be his new work LookMaNoMatrices.

TODO: decide which results are interesting and feasible to be formalized for me.

I wish to pursue further on the topic of Matrix/Tensor, see ( , (). An introduction to graphical tensor notation for mechanistic interpretability. arXiv preprint arXiv:2402.01790. ) and ( , (). Matrix representations of clifford algebras. Retrieved from https://www.cphysics.org/article/86175.pdf ) . The former also led me to Einsums in C++. For the latter, I’m thinking of HepLean.SpaceTime.CliffordAlgebra.

Group Algebra

In a sense, group algebras are the source of all you need to know about representation theory.

The primary reference is ( & , & (). Representations and characters of groups. Cambridge university press. ) for understanding FG-module, Group algebra, the presentation of groups, Clifford theory (which is the standard method of constructing representations and characters of semi-direct products, see ( , & al., , & (). Quantum theory, groups and representations. Springer. ) , and “3.6 Clifford theory” in ( & , & (). Representations of groups: A computational approach. Cambridge University Press. ) ), Schur indices etc. We also need to check ( & , & (). Representations of groups: A computational approach. Cambridge University Press. ) for its introduction to GAP, and we should pay close attention to the progress of GAP-LEAN. ( , (). Computation with finitely presented groups. Cambridge University Press. ) might also be interesting in a similar manner as ( & , & (). Representations of groups: A computational approach. Cambridge University Press. ) but with emphasis on the presentation of groups.

See also group algebra on nlab, particularly that “A group algebra is in particular a Hopf algebra and a $G$-graded algebra.”

The related Zulip thread is here, and I have preliminary explorations and experiments in Lean here.

This interest originates from reading Robert A. Wilson’s work ( , (). A discrete model for gell-mann matrices. arXiv preprint arXiv:2401.13000. Retrieved from https://arxiv.org/pdf/2401.13000 ) . The ultimate goal is to understand the group algebra of the binary tetrahedral group ($Q_8 \rtimes Z_3$), then the three-dimensional complex reflection group ($G_{27} \rtimes Q_8 \rtimes Z_3$), a.k.a. the triple cover of the Hessian group, which can be interpreted as a finite analogue of the complete gauge group $U(1) \times SU(2) \times SU(3)$.

Type Theory

Recently I have read some meta-level dependent type theory (Typst source). It might be time to re-read leantt paper, and start reading lean4lean paper/source.

The author ice1000 has strong interest in QIIT (Quotient Inductive-Inductive Types) and QIIR (Quotient Inductive-Inductive Recursion), he has implemented overlap in Aya with termination check and confluence check.

Aya has a philosophy that the kernel could include pattern matching, but at the cost of no generation and translation of eliminators at present. In principle this is feasible, without overlap, it could be implemented by “theory of signatures”, with overlap, it needs “Coherent and concurrent elimination for initial algebras” which I find fascinating, and have read ( , (). Initial algebra, final coalgebra and datatype. ) .

The author also has a great article on TT & Cat ( , (). Type theories in category theory. arXiv preprint arXiv:2107.13242. ) . His recommendation of ( & , & (). A tutorial implementation of dynamic pattern unification. Unpublished draft. 194. ) is also worth reading.

In the process of learning Topos, I wish to have a better understanding of Logic. Particularly, Curry–Howard–Lambek correspondance, Propositional truncation, and Paraconsistent logic are on the plate.

Although remotely related, I’ll place Introduction to Formal Reasoning (COMP2065) in this section so I won’t lose track of it.

Philosophy

Type theories, logic have their origins in philosophy. Lacan had drawn inspiration from the work of the mathematician and philosopher of science Georges Canguilhem, and the philosopher of mathematics Alain Badiou. If he had lived to see the rise of proof assistants, he would be interested in the formalization of his theories.

There is a project on Github called Lacan-Mathemes which visualized some core concepts of Lacan in TikZ. It would be interesting to start there and look for proper objects to formalize in Lean. Another interesting thing to do is to visualize them in Typst.

Here I would like to quote a generated response from Claude 1.2 Instant, which is quite optimistic:

Here are a few thoughts on formalizing aspects of Lacanian psychoanalytic theory using theorem provers:

  • Lacan’s structural theory of the psyche could potentially be formalized using logic. The tripartite structure of the Real, Symbolic, and Imaginary could be modeled as formal domains or ontologies with defined relationships between them.
  • Key concepts like the mirror stage, the Name-of-the-Father, the objet petit a, etc. could be defined as logical predicates or functions operating within this structural framework. For example, one could define predicates like “inMirrorStage(subject)” or “desiresObjetPetitA(subject, object)”.
  • Relations like the split between desire and drive, the tension between the Symbolic order and the Real, the misrecognition of the Imaginary could be expressed through logical rules and inferences between concepts.
  • The dynamics of psychoanalytic concepts like repression, sublimation, transference could be modeled as state transitions or logical transformations within the structured system.
  • Specific psychoanalytic theories like the stages of psychosexual development or the topology of the fantasy could be axiomatized and theorems deduced from the axioms.

However, capturing the open-ended, non-deterministic nature of unconscious processes and the ambiguity/contingency of signification would be challenging and may require non-classical logics or probabilistic approaches.

So in summary, while difficult, certain aspects of Lacanian theory seem amenable to formalization using tools from logic, ontology, and knowledge representation. Careful design would be needed to address theory’s complexity.

Lately there is a paper formalizing Kant ( & , & (). A formalization of kant’s transcendental logic. The Review of Symbolic Logic, 4(2). 254–289. ) which might be interesting.

A Clifford algebra is a $Z_2$-graded algebra, and a Filtered algebra, the associated graded algebra is the exterior algebra.

It may be thought of as quantizations (cf. quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Specifically, for $V$ an inner product space, the symbol map constitutes an isomorphism of the underlying super vector spaces of the Clifford algebra with the exterior algebra on $V$, and one may understand the Clifford algebra as the quantization Grassmann algebra induced from the inner product regarded as an odd symplectic form.

Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

A Clifford module is a representation of a Clifford algebra.

A Generalized Clifford algebra (GCA) can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms, e.g.

For $q_1, q_2, \ldots, q_m \in \mathbb{k}^*$, the generalized Clifford algebra $C^{(n)}\left(q_1, q_2, \ldots, q_m\right)$ is a unital associative algebra generated by $e_1, e_2, \ldots, e_m$ subject to the relations

$$ e_i^n=q_i \mathbf{1}, \quad e_i e_j=\omega e_j e_i, \quad \forall j \lt i . $$

It is easy to see that $C^{(n)}\left(q_1, q_2, \ldots, q_m\right)$ is $\mathbb{Z}_n$-graded where the degree of $e_i$ is $\overline{1}$, the generator of $\mathbb{Z}_n$. ( , & al., , & (). A new view of generalized clifford algebras. Advances in Applied Clifford Algebras, 29. 1–11. )

In ( , & al., , & (). A new view of generalized clifford algebras. Advances in Applied Clifford Algebras, 29. 1–11. ) , note also that “Clifford algebras are weak Hopf algebras in some symmetric tensor categories.” while “generalized Clifford algebras are weak Hopf algebras in some suitable braided linear categories of graded vector spaces.” as well as that “the Clifford process is a powerful technique to construct larger dimensional Clifford algebras from known ones.”

TODO: add papers linking Hopf algebra and Clifford algebra together learned from the adjoint discord here.

Clifford algebras can be obtained by twisting of group algebras ( & , & (). Clifford algebras obtained by twisting of group algebras. Journal of Pure and Applied Algebra, 171(2-3). 133–148. Retrieved from https://arxiv.org/pdf/math/0011040 ) , where twisted group algebras are studied in ( , (). Twisted group algebras and their representations. Journal of the Australian Mathematical Society, 4(2). 152–173. ) , ( & , & (). Twisted group algebras i. Communications in Mathematical Physics, 13. 119–130. ) , ( & , & (). Twisted group algebras II. Communications in Mathematical Physics, 13. 131–141. ) .

There exists isomorphisms between certain Clifford algebras and NDAs (Normed Division Algebras) over $\mathbb{R}$.

Variants of Clifford algebras whose generators are idempotent or nilpotent can be considered. Zeon algebras (“nil-Clifford algebras”) have proven to be useful in enumeration problems on graphs where certain configurations are forbidden, such as in the enumeration of matchings and self-avoiding walks. The idempotent property of the generators of idem-Clifford algebras can be used to avoid redundant information when enumerating certain graph and hypergraph structures. See ( & , & (). Zeon and idem-clifford formulations of hypergraph problems. Advances in Applied Clifford Algebras, 32(5). 61. Retrieved from https://arxiv.org/pdf/2201.05895.pdf ) .

It’s also closely related to universal enveloping algebra (see ( , (). Spin geometry. Lecture Notes. University of Edinburgh. Retrieved from http://mat.uab.cat/~rubio/csa2017/SpinNotes.pdf ) and “The universal enveloping algebra of a Lie algebra is the analogue of the usual group algebra of a group.” from group algebra on nlab).

Great discussions about the limitations and generalizations of Clifford algebras can be found in John C. Baez’s ( , (). The octonions. Bulletin of the american mathematical society, 39(2). 145–205. Retrieved from https://arxiv.org/pdf/math/0105155.pdf ) . Particularly, note Cayley-Dickson construction, Bott periodicity, matrix algebra, triality, and $\mathbb{R}$ as a real commutative associative nicely normed ∗-algebra. Also see Anthony Lasenby’s work on the embedding of octonions in the Clifford geometric algebra for space-time STA ( $\mathop{\mathcal{C}\ell}(1, 3)$ ) ( , (). Some recent results for su (3) and octonions within the geometric algebra approach to the fundamental forces of nature. Mathematical Methods in the Applied Sciences, 47(3). 1471–1491. Retrieved from https://arxiv.org/pdf/2202.06733 ) .

Note also Kingdon algebras: alternative Clifford-like algebras over vector spaces equipped with a symmetric bilinear form ( , & al., , & (). Octonions as clifford-like algebras. Journal of Algebra, 644. 761–795. Retrieved from https://arxiv.org/pdf/2310.09972 ) .

Categorified Clifford Algebra

A categorical view of Clifford Algebra is discussed in ( , (). Spin geometry. Lecture Notes. University of Edinburgh. Retrieved from http://mat.uab.cat/~rubio/csa2017/SpinNotes.pdf ) .

A Clifford Algebra can be categorified: “An Clifford algebra over a vector space is defined to be the Koszul dual to an abelian fully weak Lie-algebra” where “Fully weak Lie-algebras are Koszul dual to differential graded Clifford algebras.” See also Higher Clifford Algebras.

Sheaves of Clifford Algebras

Sheaves of Clifford Algebras are studied in ( , (). On clifford a-algebras and a framework for localization of a-modules. University of Pretoria (South Africa). ) and its references. See also ( , (). An introduction to categories and sheaves. Retrieved from https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/CatShv.pdf ) , an elementary introduction to ( & , & (). Categories and sheaves (). https://doi.org/10.1007/3-540-27950-4 ) , which presents categories, homological algebra and sheaves in a systematic and exhaustive manner.

DKP Algebra

Duffin-Kemmer-Petiau Algebra is

$$\frac{T(V)}{I(v\otimes w\otimes v - g(v,w)v)}$$

in the same way that Clifford Algebra is

$$\frac{T(V)}{I(v\otimes v - g(v,v))}.$$

See ( , (). Clifford algebraic approach to the de donder–weyl hamiltonian theory. Advances in Applied Clifford Algebras, 32(2). 23. Retrieved from https://arxiv.org/pdf/2112.08483 ) which embeds DKP Algebra in Clifford Algebra with projectors.

Misc

TODO:

Applied mathematics

Topology

I wish to render some pictures in ( & , & (). A topological picturebook. Springer. ) with TikZ, GLSL shader, and by GA with ganja.js & GAmphetamine or their Lean version, inspired by Steven De Keninck’s notebooks, e.g. torus, orbit 1, orbit 2, origami, skinning, slicing etc.

Knots

See ( , (). Knot theory. CRC press. ) and the tracking issue.

For interpreting knots in a sheaf-theoretic perspective, see ( , (). A geometric foundation of virtual knot theory. arXiv preprint arXiv:2301.10318. ) .

See also:

Origami

See ( , (). Origametry: Mathematical methods in paper folding. Cambridge University Press. ) and the tracking issue.

Dynamical Systems: Bifurcation Theory

The preferred reference for me is ( , (). Dynamical systems in neuroscience. MIT press. Retrieved from https://www.izhikevich.org/publications/dsn/ ) for its applications in neuroscience and various excellent diagrams. But it’s not a mathematically rigorous treatment of bifurcation theory.

Sheaves

My first impression of sheaves is that they are useful to local-to-global applications “which ask for global solutions to problems whose hypotheses are local”.

Roughly speaking, a sheaf requires some gluing conditions (axioms “Locality” and “Gluing”) so that local data can be collated compatibly into a global algebraic structure that varies continuously over local covering domains (“sections” of sheaves).

To do so, a sheaf in general, as defined in the category-theoretical language, needs

  • a topological space (or a site in general), denoted $X$ (or $\mathcal{C}$ for a site)
  • a category, sometimes denoted $\mathcal{D}$, meaning “data category”, whose objects are algebraic structures and morphisms are structure-preserving maps

and builds (gluing conditions) on a $\mathcal{D}$-valued presheaf over $X$ (or $\mathcal{C}$), denoted $\mathcal{F}$ (as its French name is “faisceau”), which is essentially a contravariant functor $\mathcal{F}: \mathcal{C}^{op} \to \mathcal{D}$ but a concept with an attitude, and its morphisms are restriction maps between open sets in $X$ (or between objects that satisfy the pretopology $\mathcal{J}$ in $C$, where $\mathcal{J}$ is the pretopology on $\operatorname{Open}(X)$, which specifies when a covering family of open sets exists).

Its latest application to deep learning, Thomas Gebhart’s thesis ( , (). Sheaf representation learning  (PhD thesis). University of Minnesota ) sees a sheaf over a topological space as a data structure “which defines rules for associating data to a space so that local agreements in data assignment align with a coherent global representation”, thus a generalization of both:

  • relational learning, which aims to “combine symbolic, potentially hand-engineered prior knowledge with the tabula rasa representational flexibility of deep learning to achieve a synthetic model family which can be defined with respect to symbolic knowledge priors about the data domain”
  • geometric deep learning, which “provides a group-theoretic approach to reasoning about and encoding domain-symmetry invariance or equivariance within machine learning models”,

“providing a mathematical framework for characterizing the interplay between the topological information embedded within a domain and the representations of data learned by machine learning models”.

My prior interest in geometric deep learning, particularly group-equivariant neural networks, and my believe in symbolism, are the background of my interest in sheaf representation learning.

Notably, this thesis treats the discrete case of sheaves, a cellular sheaf, whose

  • topological space is a cell complex, which is “a topological generalization of a graph, with set inclusion and intersection given by the incidence relationships among cells in the complex”, thus “admitting a computable, linear-algebraic representation”.
  • data category is $\mathtt{FVect}$, the category of finite-dimensional vector spaces over a field $\mathbb{F}$, which is a common choice for the data category in machine learning applications, a model-free approach with massive parameter space, flexible representational capacity, but inherits fundamental limitations, e.g. data inefficiency, generalization failure, and interpretability issues.

For more details, see also Thomas Gebhart’s talk Sheaves for AI: Graph Representation Learning through Sheaf Theory.

Its application to physics has the potential to formulate differential geometry in a more general setting, without assuming the existence of a locally Euclidean space as manifold did. It’s believe that this approach can overcome some difficulties in Quantum field theory even Quantum gravity, because locally there might be no concept of a metric space at all ( & , & (). Differential sheaves and connections: A natural approach to physical geometry. World Scientific. ) .

Note that there are CAS systems that can do sheaf cohomology etc., e.g. Macaulay2, OSCAR.

Synthetic Differential Geometry

For SDG, ( , (). Synthetic differential geometry. Cambridge University Press. Retrieved from https://users-math.au.dk/kock/sdg99.pdf ) is a classic. ( & , & (). Circles in synthetic differential geometry. Retrieved from https://pure.tue.nl/ws/portalfiles/portal/308028780/Thesis_BTW_Rob_Schellingerhout_Report_002_.pdf ) is a concise bachelor thesis on the topic, and has interesting discussions on circles.

Ryszard Paweł Kostecki has very approachable notes on Topos ( , (). An introduction to topos theory. Technial Report. Retrieved from https://www.fuw.edu.pl/~kostecki/ittt.pdf ) and SDG ( , (). Differential geometry in toposes. Citeseer. Retrieved from https://www.fuw.edu.pl/~kostecki/sdg.pdf ) .

We should also read ( , (). Sheaf theory through examples. MIT Press. ) . It also has many diagrams in the way I imagined, for examples of sheaf.

ML

I don’t want efforts in Transformers: from self-attention to performance optimizations to be discontinued, lately there is ( , & al., , , & (). A primer on the inner workings of transformer-based language models. arXiv preprint arXiv:2405.00208. ) on this topic.

I might need to follow on the latest development on the linear attention mechanism ( , & al., , , , , , , , , , & (). Eagle and finch: RWKV with matrix-valued states and dynamic recurrence. arXiv preprint arXiv:2404.05892. ) .

I have almost no understanding of diffusion models, so I should read ( , & al., , , , , , & (). All are worth words: A ViT backbone for diffusion models. ) and related papers.

I should also read ( , & al., , , , , , , , & (). Magnushammer: A transformer-based approach to premise selection. arXiv preprint arXiv:2303.04488. ) and related papers.

Consciousness

Lately I became aware of the work on mathematical models of consciousness, namely Integrated information theory ( , & al., , , , , , , , , , & (). Integrated information theory (IIT) 4.0: Formulating the properties of phenomenal existence in physical terms. PLoS Computational Biology, 19(10). e1011465. ) and The information theory of individuality ( , & al., , , , & (). The information theory of individuality. Theory in Biosciences, 139. 209–223. ) .

Physics

Physics-Based Simulation is out.

So is ( & , & (). String field theory: A review. arXiv preprint arXiv:2405.19421. ) on string theory.

TODOs

Incorporating the following interests into this post:

References

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Albantakis, Barbosa, Findlay, Grasso, Haun, Marshall, Mayner, Zaeemzadeh, Boly, Juel & (2023)
, , , , , , , , , & (). Integrated information theory (IIT) 4.0: Formulating the properties of phenomenal existence in physical terms. PLoS Computational Biology, 19(10). e1011465.
Albuquerque & Majid (2002)
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(). The octonions. Bulletin of the american mathematical society, 39(2). 145–205. Retrieved from https://arxiv.org/pdf/math/0105155.pdf
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, , , , , & (). All are worth words: A ViT backbone for diffusion models.
Berzins, Radler, Sanokowski, Hochreiter & Brandstetter (2024)
, , , & (). Geometry-informed neural networks. arXiv preprint arXiv:2402.14009.
Brehmer, De Haan, Behrends & Cohen (2023)
, , & (). Geometric algebra transformers. arXiv preprint arXiv:2305.18415.
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(). Geometric algebra. arXiv preprint arXiv:1205.5935. Retrieved from http://arxiv.org/abs/1205.5935
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(). A geometric foundation of virtual knot theory. arXiv preprint arXiv:2301.10318.
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(). Twisted group algebras and their representations. Journal of the Australian Mathematical Society, 4(2). 152–173.
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(). Clifford algebraic approach to the de donder–weyl hamiltonian theory. Advances in Applied Clifford Algebras, 32(2). 23. Retrieved from https://arxiv.org/pdf/2112.08483
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(). Origametry: Mathematical methods in paper folding. Cambridge University Press.
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(). Synthetic differential geometry. Cambridge University Press. Retrieved from https://users-math.au.dk/kock/sdg99.pdf
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(). An introduction to topos theory. Technial Report. Retrieved from https://www.fuw.edu.pl/~kostecki/ittt.pdf
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, , , & (). The information theory of individuality. Theory in Biosciences, 139. 209–223.
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(). Some recent results for su (3) and octonions within the geometric algebra approach to the fundamental forces of nature. Mathematical Methods in the Applied Sciences, 47(3). 1471–1491. Retrieved from https://arxiv.org/pdf/2202.06733
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, , , , , , , , , & (). Eagle and finch: RWKV with matrix-valued states and dynamic recurrence. arXiv preprint arXiv:2404.05892.
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, & (). The matrix cookbook. Technical University of Denmark, 7(15). 510. Retrieved from https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf
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, & (). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378. 686–707.
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(). Matrix representations of clifford algebras. Retrieved from https://www.cphysics.org/article/86175.pdf
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(). Probing clifford algebras through spin groups: A standard model perspective. arXiv preprint arXiv:2312.10071. Retrieved from https://arxiv.org/pdf/2312.10071
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(). Sheaf theory through examples. MIT Press.
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, & (). Clifford group equivariant neural networks. arXiv preprint arXiv:2305.11141.
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(). An introduction to categories and sheaves. Retrieved from https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/CatShv.pdf
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& (). Circles in synthetic differential geometry. Retrieved from https://pure.tue.nl/ws/portalfiles/portal/308028780/Thesis_BTW_Rob_Schellingerhout_Report_002_.pdf
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(). Expository paper on clifford algebras, representations, and the octonion algebra. arXiv preprint arXiv:1906.11460. Retrieved from https://arxiv.org/pdf/1906.11460
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(). An introduction to graphical tensor notation for mechanistic interpretability. arXiv preprint arXiv:2402.01790.
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