From b366875ddc60528d360ae0afb2615b6422be28d3 Mon Sep 17 00:00:00 2001 From: utensil Date: Mon, 7 Oct 2024 21:26:28 +0800 Subject: [PATCH] Port: My math interests in 2024 --- assets/uts-overrides.xsl | 16 +++++++++-- bun/markdownit.js | 10 ++++++- tex/forest.bib | 7 +++++ trees/macros.tree | 10 +++---- trees/refs/fernandes2022clifford.tree | 6 ++-- trees/refs/wu2024multi.tree | 14 +++++++++ trees/uts-0018.tree | 1 + trees/uts-001R.tree | 23 +++++++++++++++ trees/uts-001S.tree | 11 +++++++ trees/uts-001T.tree | 12 ++++++++ trees/uts-001U.tree | 18 ++++++++++++ trees/uts-001V.tree | 13 +++++++++ trees/uts-001W.tree | 13 +++++++++ trees/uts-001X.tree | 15 ++++++++++ trees/uts-001Y.tree | 23 +++++++++++++++ trees/uts-001Z.tree | 41 +++++++++++++++++++++++++++ trees/uts-0020.tree | 7 +++++ trees/uts-0021.tree | 5 ++++ trees/uts-0022.tree | 14 +++++++++ trees/uts-0023.tree | 10 +++++++ trees/uts-0024.tree | 12 ++++++++ trees/uts-0025.tree | 5 ++++ trees/uts-0026.tree | 12 ++++++++ trees/uts-0027.tree | 5 ++++ trees/uts-0028.tree | 5 ++++ trees/uts-0029.tree | 41 +++++++++++++++++++++++++++ trees/uts-002A.tree | 9 ++++++ trees/uts-002B.tree | 11 +++++++ trees/uts-002C.tree | 5 ++++ trees/uts-002D.tree | 7 +++++ trees/uts-002E.tree | 8 ++++++ 31 files changed, 377 insertions(+), 12 deletions(-) create mode 100644 trees/refs/wu2024multi.tree create mode 100644 trees/uts-001R.tree create mode 100644 trees/uts-001S.tree create mode 100644 trees/uts-001T.tree create mode 100644 trees/uts-001U.tree create mode 100644 trees/uts-001V.tree create mode 100644 trees/uts-001W.tree create mode 100644 trees/uts-001X.tree create mode 100644 trees/uts-001Y.tree create mode 100644 trees/uts-001Z.tree create mode 100644 trees/uts-0020.tree create mode 100644 trees/uts-0021.tree create mode 100644 trees/uts-0022.tree create mode 100644 trees/uts-0023.tree create mode 100644 trees/uts-0024.tree create mode 100644 trees/uts-0025.tree create mode 100644 trees/uts-0026.tree create mode 100644 trees/uts-0027.tree create mode 100644 trees/uts-0028.tree create mode 100644 trees/uts-0029.tree create mode 100644 trees/uts-002A.tree create mode 100644 trees/uts-002B.tree create mode 100644 trees/uts-002C.tree create mode 100644 trees/uts-002D.tree create mode 100644 trees/uts-002E.tree diff --git a/assets/uts-overrides.xsl b/assets/uts-overrides.xsl index f0d5d68..d285641 100644 --- a/assets/uts-overrides.xsl +++ b/assets/uts-overrides.xsl @@ -7,7 +7,7 @@ xmlns:f="http://www.jonmsterling.com/jms-005P.xml" xmlns:html="http://www.w3.org/1999/xhtml" > - + + + + diff --git a/bun/markdownit.js b/bun/markdownit.js index 859202e..9d81fe8 100644 --- a/bun/markdownit.js +++ b/bun/markdownit.js @@ -10,6 +10,14 @@ for (let i = 0; i < markdownit_tags.length; i++) { const markdownit_tag = markdownit_tags[i] const markdown_source = markdownit_tag.innerHTML // console.log(markdown_source); - markdownit_tag.innerHTML = md.render(markdown_source) + const converted_source = markdown_source + .replaceAll(/</g, '<') + // unescape to make quotes work + .replaceAll(/>/g, '>') + // we escape fr:tex tags to avoid conflicts with markdown syntax + // note that we need to use `+?` which is a lazy quantifier, meaning it matches as few characters as possible + .replaceAll(/\\([\[\(])(.+?)\\([\)\]])/g, '\\\\$1$2\\\\$3') + console.log(converted_source) + markdownit_tag.innerHTML = md.render(converted_source) markdownit_tag.classList.remove('grace-loading') } diff --git a/tex/forest.bib b/tex/forest.bib index e7ecce8..5c2f1bd 100644 --- a/tex/forest.bib +++ b/tex/forest.bib @@ -1097,3 +1097,10 @@ @article{paszke2021getting year={2021}, publisher={ACM New York, NY, USA} } + +@article{wu2024multi, + title={A Multi-Level Superoptimizer for Tensor Programs}, + author={Wu, Mengdi and Cheng, Xinhao and Padon, Oded and Jia, Zhihao}, + journal={arXiv preprint arXiv:2405.05751}, + year={2024} +} diff --git a/trees/macros.tree b/trees/macros.tree index c03c394..0548ea3 100644 --- a/trees/macros.tree +++ b/trees/macros.tree @@ -527,13 +527,11 @@ so the text size almost matches the size output by native forester code. This do } } -\def\md[body]{ - \[class]{markdownit grace-loading}{\body} -} +\def\md[body]{\[class]{markdownit grace-loading}{\body}} -\def\mdblock[t][body]{ - \block{\t}{\[class]{markdownit grace-loading}{\body}} -} +\def\mdblock[t][body]{\block{\t}{\md{\body}}} + +\def\mdnote[t][body]{\note{\t}{\md{\body}}} \def\loadjs[src]{ \[type]{module}[src]{\src}{} diff --git a/trees/refs/fernandes2022clifford.tree b/trees/refs/fernandes2022clifford.tree index 2fec0f9..e086465 100644 --- a/trees/refs/fernandes2022clifford.tree +++ b/trees/refs/fernandes2022clifford.tree @@ -1,5 +1,5 @@ -% ["math-2024"] -\title{Clifford algebraic approach to the de donder–weyl hamiltonian theory} +% ["manual", "math-2024"] +\title{Clifford Algebraic Approach to the De Donder-Weyl Hamiltonian Theory} \date{2022} \author{Marco Cezar Barbosa Fernandes} \taxon{reference} @@ -7,7 +7,7 @@ \meta{bibtex}{\startverb @article{fernandes2022clifford, - title = {Clifford Algebraic Approach to the De Donder--Weyl Hamiltonian Theory}, + title = {Clifford Algebraic Approach to the De Donder-Weyl Hamiltonian Theory}, author = {Fernandes, Marco Cezar Barbosa}, year = {2022}, url = {https://arxiv.org/pdf/2112.08483}, diff --git a/trees/refs/wu2024multi.tree b/trees/refs/wu2024multi.tree new file mode 100644 index 0000000..56ba849 --- /dev/null +++ b/trees/refs/wu2024multi.tree @@ -0,0 +1,14 @@ +% ["forest"] +\title{A multi-level superoptimizer for tensor programs} +\date{2024} +\author{Mengdi Wu}\author{Xinhao Cheng}\author{Oded Padon}\author{Zhihao Jia} +\taxon{reference} + +\meta{bibtex}{\startverb +@article{wu2024multi, + title = {A Multi-Level Superoptimizer for Tensor Programs}, + author = {Wu, Mengdi and Cheng, Xinhao and Padon, Oded and Jia, Zhihao}, + year = {2024}, + journal = {arXiv preprint arXiv:2405.05751} +} +\stopverb} diff --git a/trees/uts-0018.tree b/trees/uts-0018.tree index 2764646..039650b 100644 --- a/trees/uts-0018.tree +++ b/trees/uts-0018.tree @@ -17,6 +17,7 @@ \mdblock{10-07}{ - learned about [lobste.rs](https://lobste.rs/), a computing-focused community centered around link aggregation and discussion, a bit like Hacker News but less noise maybe - read [Rewriting Rust](https://lobste.rs/s/29a1eo/rewriting_rust) and [Josh Triplett's comment](https://www.reddit.com/r/rust/comments/1fpomvp/rewriting_rust/lozktuv/) +- read \citek{wu2024multi} and learn about [mirage](https://github.com/mirage-project/mirage) } \mdblock{10-06}{ diff --git a/trees/uts-001R.tree b/trees/uts-001R.tree new file mode 100644 index 0000000..795e638 --- /dev/null +++ b/trees/uts-001R.tree @@ -0,0 +1,23 @@ +\import{macros} +% clifford hopf spin tt ag math draft tech exp notes +\tag{math} +\tag{notes} + +\author{utensil} +\date{2024-03-16} + +\note{My math interests in 2024}{ + +\quote{Initially posted [here](https://utensil.github.io/blog/posts/math-2024/), moved here for updates.} + +\transclude{uts-001S} +\transclude{uts-001T} +\transclude{uts-001Z} +\transclude{uts-0024} +\transclude{uts-002D} +\transclude{uts-002E} +} + +% porting note: +% replace \{\{\}\} to \citek{$1} modulo spaces +% replace \$([^$]+)\$ to #{$1} with manual fixes for ##{} cases diff --git a/trees/uts-001S.tree b/trees/uts-001S.tree new file mode 100644 index 0000000..f6daf57 --- /dev/null +++ b/trees/uts-001S.tree @@ -0,0 +1,11 @@ +\import{macros} + +\mdnote{Rationale}{ +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. +} diff --git a/trees/uts-001T.tree b/trees/uts-001T.tree new file mode 100644 index 0000000..ac01cf2 --- /dev/null +++ b/trees/uts-001T.tree @@ -0,0 +1,12 @@ +\import{macros} + +\note{Formalization}{ +\md{ +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. +} + +\transclude{uts-001U} +\transclude{uts-001V} +\transclude{uts-001W} +\transclude{uts-001X} +} diff --git a/trees/uts-001U.tree b/trees/uts-001U.tree new file mode 100644 index 0000000..0925be4 --- /dev/null +++ b/trees/uts-001U.tree @@ -0,0 +1,18 @@ +\import{macros} + +\mdnote{Spin groups}{ +The PR to Mathlib [#9111](https://github.com/leanprover-community/mathlib4/pull/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 \citek{chisolm2012geometric} , e.g. r-blades, r-vectors? + +For the former, I should take a closer look at \citek{figueroa2010spin} and maybe \citek{suarez2019expository}, \citek{reynoso2023probing}. + +For the latter, see the [Z-filteration in lean-ga](https://github.com/pygae/lean-ga/blob/master/src/geometric_algebra/from_mathlib/versors.lean) and [the prototype](https://github.com/eric-wieser/lftcm2023-clifford_algebra/). + +I also wish to include some latest results presented in \citek{ruhe2023clifford}, with supplements from \citek{brehmer2023geometric} (lately there is a new paper applying this in HEP \citek{spinner2024lorentz}, in the same spirit, I should also read \citek{berzins2024geometry} and possibly \citek{raissi2019physics}), in which some of the results are proven in \citek{roelfs2023graded}. + +See also discussions in the [lean-ga blueprint](https://utensil.github.io/lean-ga/blueprint/sect0007.html). + +I've started a Forester experiment about the definitions of Spin groups [here](https://utensil.github.io/forest/spin-0001.xml). I also need to check citations of [On some Lie groups in degenerate Clifford geometric algebras](https://arxiv.org/abs/2301.06842). +} diff --git a/trees/uts-001V.tree b/trees/uts-001V.tree new file mode 100644 index 0000000..38586d8 --- /dev/null +++ b/trees/uts-001V.tree @@ -0,0 +1,13 @@ +\import{macros} + +\mdnote{Matrix}{ +The Matrix Cookbook (November 15, 2012) \citek{petersen2008matrix} covers many useful results about matrices, and Eric Wieser's project [lean-matrix-cookbook](https://github.com/eric-wieser/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 \citek{de2019geometric}, 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](https://github.com/enkimute/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 \citek{taylor2024introduction} and \citek{randy2023matrix}. The former also led me to [Einsums in C++](https://github.com/Einsums/Einsums). For the latter, I'm thinking of [HepLean.SpaceTime.CliffordAlgebra](https://github.com/HEPLean/HepLean/blob/master/HepLean/SpaceTime/CliffordAlgebra.lean). +} diff --git a/trees/uts-001W.tree b/trees/uts-001W.tree new file mode 100644 index 0000000..afa6dbb --- /dev/null +++ b/trees/uts-001W.tree @@ -0,0 +1,13 @@ +\import{macros} + +\mdnote{Group Algebra}{% +> In a sense, group algebras are the source of all you need to know about representation theory. + +The primary reference is \citek{james2001representations} 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 \citek{woit2017quantum}, and "3.6 Clifford theory" in \citek{lux2010representations}), Schur indices etc. We also need to check \citek{lux2010representations} for its introduction to [GAP](https://www.gap-system.org/), and we should pay close attention to the progress of [GAP-LEAN](https://github.com/opencompl/lean-gap). \citek{sims1994computation} might also be interesting in a similar manner as \citek{lux2010representations} but with emphasis on the presentation of groups. + +See also [group algebra on nlab](https://ncatlab.org/nlab/show/group+algebra), particularly that "A group algebra is in particular a Hopf algebra and a #{G}-graded algebra." + +The related Zulip thread is [here](https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Group.20algebra.20over.20finite.20groups), and I have preliminary explorations and experiments in Lean [here](https://github.com/utensil/lean-playground/blob/master/lean4/examples/FiniteGroup.lean). + +This interest originates from reading Robert A. Wilson's work \citek{wilson2024discrete}. 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)}. +} diff --git a/trees/uts-001X.tree b/trees/uts-001X.tree new file mode 100644 index 0000000..936c0fe --- /dev/null +++ b/trees/uts-001X.tree @@ -0,0 +1,15 @@ +\import{macros} + +\mdnote{Type Theory}{ +Recently I have read [some meta-level dependent type theory](https://ice1000.org/) ([Typst source](https://github.com/ice1000/website/tree/main/dtt-dev)). 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](https://github.com/aya-prover/aya-dev/pull/1042) with [termination check and confluence check](https://github.com/aya-prover/aya-dev/blob/main/cli-impl/src/test/resources/negative/PatCohError.txt). + +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 \citek{yeasin2011initial}. + +The author also has a great article on TT & Cat \citek{zhang2021type}. His recommendation of \citek{gundry2013tutorial} is also worth reading. + +In the process of learning Topos, I wish to have a better understanding of Logic. Particularly, [Curry–Howard–Lambek correspondance](\verb!|https://www.bananaspace.org/wiki/%E7%B1%BB%E5%9E%8B%E8%AE%BA%E2%80%93%E8%8C%83%E7%95%B4%E8%AE%BA%E2%80%93%E9%80%BB%E8%BE%91%E5%AD%A6%E7%B1%BB%E6%AF%94!), [Propositional truncation](https://1lab.dev/1Lab.HIT.Truncation.html#propositional-truncation), and [Paraconsistent logic](https://en.wikipedia.org/wiki/Paraconsistent_logic) are on the plate. + +Although remotely related, I'll place [Introduction to Formal Reasoning (COMP2065)](http://www.cs.nott.ac.uk/~psztxa/comp2065.23-24.ifr-notes/_build/html/index.html#) in this section so I won't lose track of it. +} diff --git a/trees/uts-001Y.tree b/trees/uts-001Y.tree new file mode 100644 index 0000000..b56edd7 --- /dev/null +++ b/trees/uts-001Y.tree @@ -0,0 +1,23 @@ +\import{macros} + +\mdnote{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](https://github.com/gjoncas/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 \citek{achourioti2011formalization} which might be interesting. +} diff --git a/trees/uts-001Z.tree b/trees/uts-001Z.tree new file mode 100644 index 0000000..dca854a --- /dev/null +++ b/trees/uts-001Z.tree @@ -0,0 +1,41 @@ +\import{macros} + +\note{Generalized Clifford Algebra and related mathematical branches}{ +\md{ +A [Clifford algebra](https://en.wikipedia.org/wiki/Clifford_algebra) is a #{Z_2}-[graded algebra](https://en.wikipedia.org/wiki/Graded_algebra), and a [Filtered algebra](https://en.wikipedia.org/wiki/Filtered_algebra), the associated graded algebra is the [exterior algebra](https://en.wikipedia.org/wiki/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](https://ncatlab.org/nlab/show/Clifford+algebra#AsQuantizedExteriorAlgebra), 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](https://ncatlab.org/nlab/show/symplectic+form). + +Weyl algebras and Clifford algebras admit a further structure of a [*-algebra](https://en.wikipedia.org/wiki/*-algebra), and can be unified as even and odd terms of a [superalgebra](https://en.wikipedia.org/wiki/Superalgebra), as discussed in [CCR and CAR algebras](https://en.wikipedia.org/wiki/CCR_and_CAR_algebras). + +A [Clifford module](https://en.wikipedia.org/wiki/Clifford_module) is a representation of a Clifford algebra. + +A [Generalized Clifford algebra (GCA)](https://en.wikipedia.org/wiki/Generalized_Clifford_algebra) 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}. \citek{cheng2019new} + +In \citek{cheng2019new}, note also that "Clifford algebras are weak [Hopf algebras](https://en.wikipedia.org/wiki/Hopf_algebra) 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 \citek{albuquerque2002clifford}, where twisted group algebras are studied in \citek{conlon1964twisted}, \citek{edwards1969twisted}, \citek{edwards1969twisted2}. + +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 \citek{ewing2022zeon}. + +It's also closely related to universal enveloping algebra (see \citek{figueroa2010spin} 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](https://ncatlab.org/nlab/show/group+algebra#RelationToUniversalEnvelopingAlgebra)). + +Great discussions about the limitations and generalizations of Clifford algebras can be found in John C. Baez's \citek{baez2002octonions}. Particularly, note [Cayley-Dickson construction](https://en.wikipedia.org/wiki/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)} ) \citek{lasenby2024some}. + +Note also Kingdon algebras: alternative Clifford-like algebras over vector spaces equipped with a symmetric bilinear form \citek{depies2024octonions}. +} + +\transclude{uts-0020} +\transclude{uts-0021} +\transclude{uts-0022} +\transclude{uts-0023} +} diff --git a/trees/uts-0020.tree b/trees/uts-0020.tree new file mode 100644 index 0000000..b926983 --- /dev/null +++ b/trees/uts-0020.tree @@ -0,0 +1,7 @@ +\import{macros} + +\mdnote{Categorified Clifford Algebra}{ +A categorical view of Clifford Algebra is discussed in \citek{figueroa2010spin}. + +A Clifford Algebra can be [categorified](https://golem.ph.utexas.edu/category/2007/10/categorified_clifford_algebra.html): "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](https://golem.ph.utexas.edu/category/2007/10/higher_clifford_algebras.html). +} diff --git a/trees/uts-0021.tree b/trees/uts-0021.tree new file mode 100644 index 0000000..e402ce8 --- /dev/null +++ b/trees/uts-0021.tree @@ -0,0 +1,5 @@ +\import{macros} + +\mdnote{Sheaves of Clifford Algebras}{ +Sheaves of Clifford Algebras are studied in \citek{yizengaw2015clifford} and its references. See also \citek{schapira2023introduction}, an elementary introduction to \citek{kashiwara2006categories}, which presents categories, homological algebra and sheaves in a systematic and exhaustive manner. +} diff --git a/trees/uts-0022.tree b/trees/uts-0022.tree new file mode 100644 index 0000000..62f1356 --- /dev/null +++ b/trees/uts-0022.tree @@ -0,0 +1,14 @@ +\import{macros} + +\mdnote{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 \citek{fernandes2022clifford} which embeds DKP Algebra in Clifford Algebra with projectors. + +} diff --git a/trees/uts-0023.tree b/trees/uts-0023.tree new file mode 100644 index 0000000..9de96a8 --- /dev/null +++ b/trees/uts-0023.tree @@ -0,0 +1,10 @@ +\import{macros} + +\mdnote{Misc}{ +TODO: + +- [Recovering Composition Algebras from 3D Geometric Algebras](https://arxiv.org/abs/2403.12569) +- Eliminating topological errors in neural network rotation estimation using self-selecting ensembles - Sitao Xiang - SIGGRAPH 2021 +- OSMU24: Quantum Foundations, Particle Physics, and Unification of Forces +- [Bill Baylis' research on Clifford's Geometric Algebra of Physical Space (APS)](https://web.archive.org/web/20201205215239/http://web4.uwindsor.ca/users/b/baylis/main.nsf) +} diff --git a/trees/uts-0024.tree b/trees/uts-0024.tree new file mode 100644 index 0000000..89960e1 --- /dev/null +++ b/trees/uts-0024.tree @@ -0,0 +1,12 @@ +\import{macros} + +\note{Applied mathematics}{ +\transclude{uts-0025} +\transclude{uts-0026} +\transclude{uts-0027} +\transclude{uts-0028} +\transclude{uts-0029} +\transclude{uts-002A} +\transclude{uts-002B} +\transclude{uts-002C} +} diff --git a/trees/uts-0025.tree b/trees/uts-0025.tree new file mode 100644 index 0000000..df35e69 --- /dev/null +++ b/trees/uts-0025.tree @@ -0,0 +1,5 @@ +\import{macros} + +\note{Topology}{ +I wish to render some pictures in \citek{francis1987topological} with [TikZ](https://github.com/kisonecat/tikzjax), GLSL shader, and by GA with [ganja.js](https://github.com/enkimute/ganja.js) & [GAmphetamine](https://enki.ws/GAM/src/GAmphetamine.js) or their Lean version, inspired by Steven De Keninck's notebooks, e.g. [torus](https://observablehq.com/@enkimute/torus-circle-circle), [orbit 1](https://observablehq.com/@enkimute/orbit-of-the-day), [orbit 2](https://observablehq.com/@enkimute/animated-orbits), [origami](https://enki.ws/ganja.js/examples/coffeeshop.html#pga3d_origami), [skinning](https://enki.ws/ganja.js/examples/coffeeshop.html#pga3d_skinning), [slicing](https://enki.ws/ganja.js/examples/coffeeshop.html#pga3d_slicing) etc. +} diff --git a/trees/uts-0026.tree b/trees/uts-0026.tree new file mode 100644 index 0000000..5b3aa0a --- /dev/null +++ b/trees/uts-0026.tree @@ -0,0 +1,12 @@ +\import{macros} + +\mdnote{Knots}{ +See \citek{manturov2018knot} and [the tracking issue](https://github.com/utensil/utensil.github.io/issues/229). + +For interpreting knots in a sheaf-theoretic perspective, see \citek{chrisman2023geometric}. + +See also: + +- [A Table of Virtual Knots](https://www.math.toronto.edu/drorbn/Students/GreenJ/index.html) +- [The homological arrow polynomial for virtual links](https://arxiv.org/abs/2207.02427) ([code](https://github.com/kmill/arrow_poly)) +} diff --git a/trees/uts-0027.tree b/trees/uts-0027.tree new file mode 100644 index 0000000..69c7bfb --- /dev/null +++ b/trees/uts-0027.tree @@ -0,0 +1,5 @@ +\import{macros} + +\note{Origami}{ +See \citek{hull2020origametry} and [the tracking issue](https://github.com/utensil/utensil.github.io/issues/133). +} diff --git a/trees/uts-0028.tree b/trees/uts-0028.tree new file mode 100644 index 0000000..d0d1f5f --- /dev/null +++ b/trees/uts-0028.tree @@ -0,0 +1,5 @@ +\import{macros} + +\note{Dynamical Systems: Bifurcation Theory}{ +The preferred reference for me is \citek{izhikevich2007dynamical} for its applications in neuroscience and various excellent diagrams. But it's not a mathematically rigorous treatment of bifurcation theory. +} diff --git a/trees/uts-0029.tree b/trees/uts-0029.tree new file mode 100644 index 0000000..13a7a47 --- /dev/null +++ b/trees/uts-0029.tree @@ -0,0 +1,41 @@ +\import{macros} + +\mdnote{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](https://ncatlab.org/nlab/show/concept+with+an+attitude#presheaves_and_copresheaves), 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 \citek{gebhart2023sheaf} 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](https://cats.for.ai/assets/slides/sheaves_for_AI.pdf). + + + +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 \citek{mallios2015differential}. + +Note that there are CAS systems that can do sheaf cohomology etc., e.g. [Macaulay2](https://macaulay2.com/), [OSCAR](https://www.oscar-system.org/). +} diff --git a/trees/uts-002A.tree b/trees/uts-002A.tree new file mode 100644 index 0000000..095c94b --- /dev/null +++ b/trees/uts-002A.tree @@ -0,0 +1,9 @@ +\import{macros} + +\mdnote{Synthetic Differential Geometry}{ +For SDG, \citek{kock2006synthetic} is a classic. \citek{schellingerhout2023circles} is a concise bachelor thesis on the topic, and has interesting discussions on circles. + +[Ryszard Paweł Kostecki](https://www.fuw.edu.pl/~kostecki/) has very approachable notes on Topos \citek{kostecki2011introduction} and SDG \citek{kostecki2009differential}. + +We should also read \citek{rosiak2022sheaf}. It also has many diagrams in the way I imagined, for examples of sheaf. +} diff --git a/trees/uts-002B.tree b/trees/uts-002B.tree new file mode 100644 index 0000000..eeb67c1 --- /dev/null +++ b/trees/uts-002B.tree @@ -0,0 +1,11 @@ +\import{macros} + +\mdnote{ML}{ +I don't want efforts in [Transformers: from self-attention to performance optimizations](https://utensil.github.io/blog/posts/transformer/) to be discontinued, lately there is \citek{ferrando2024primer} on this topic. + +I might need to follow on the latest development on the linear attention mechanism \citek{peng2024eagle}. + +I have almost no understanding of diffusion models, so I should read \citek{bao2023all} and related papers. + +I should also read \citek{mikula2023magnushammer} and related papers. +} diff --git a/trees/uts-002C.tree b/trees/uts-002C.tree new file mode 100644 index 0000000..3ea66ff --- /dev/null +++ b/trees/uts-002C.tree @@ -0,0 +1,5 @@ +\import{macros} + +\note{Consciousness}{ +Lately I became aware of the work on mathematical models of consciousness, namely Integrated information theory \citek{albantakis2023integrated} and The information theory of individuality \citek{krakauer2020information}. +} diff --git a/trees/uts-002D.tree b/trees/uts-002D.tree new file mode 100644 index 0000000..4b0bf83 --- /dev/null +++ b/trees/uts-002D.tree @@ -0,0 +1,7 @@ +\import{macros} + +\mdnote{Physics}{ +[Physics-Based Simulation](https://phys-sim-book.github.io/) is out. + +So is \citek{sen2024string} on string theory. +} diff --git a/trees/uts-002E.tree b/trees/uts-002E.tree new file mode 100644 index 0000000..25a3785 --- /dev/null +++ b/trees/uts-002E.tree @@ -0,0 +1,8 @@ +\import{macros} + +\mdnote{TODOs}{ +Incorporating the following interests into this post: + +- [Threads of thought in recent years (in Chinese)](https://www.douban.com/doulist/156001812/) +- Interests logged [here](https://github.com/utensil/utensil.github.io/issues) +}