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Andrius Kulikauskas (Dec 06 2023 at 19:21):I am investigating what is meant by "geometry". I think this term represents an intuition that I would like to make more explicit in terms of cognition. So I am gathering evidence from mathematical practice, from elementary math to advanced math.
I want to learn what Category theory has to say about foundations of geometry. I am aware of the articles at nLab on geometry, geometric morphism, cohesive topos and Isbell duality. I have a lot to learn but I do have a basic understanding of adjunctions and adjoint strings. I very much appreciate all manner of insights, intuitions and suggestions.
I also want to learn how geometry is understood by geometers. One perspective that I find helpful is by Tazerenix that geometry is the study of spaces with rigidifying structure. In particular, I suspect that cognitively there are four basic geometries (affine, projective, conformal, symplectic) as defined by classical Lie groups/algebras. I want to understand geometry better overall and then I will know what to think about that. This month I plan to work here on all manner of investigations. I appreciate being able to think out loud and get ideas, suggestions, feedback and grow in my understanding.
Todd Trimble (Dec 06 2023 at 19:45):"Geometry" is too big, too broad. It can mean a million different things. But Klein in his era did some fundamental thinking about describing geometries in terms of groups that preserve structure appropriately, in his Erlangen program, and I've heard it said this can be regarded as an intellectual precursor to Category Theory.
Category theorists talk a lot about Klein geometry.
Morgan Rogers (he/him) (Dec 07 2023 at 13:29):Andrius Kulikauskas said:
In particular, I suspect that cognitively there are four basic geometries (affine, projective, conformal, symplectic) as defined by classical Lie groups/algebras.
What do you mean by "cognitively" and "as defined by" in this sentence?
John Baez (Dec 07 2023 at 15:14):The connection between geometries and Lie groups was comprehensively summarized by Freudenthal in his article in the first issue of Advances in Mathematics, so I'd start there if that's what you're interested in:
Morgan Rogers (he/him) (Dec 07 2023 at 18:56):Already there are words that appear in that article that Andrius doesn't mention, such as "elliptic" and "metasymplectic", but "conformal" doesn't appear, so I was mostly wondering why Andrius settled on these four terms. And what it has to do with "cognitive" anything.
John Baez (Dec 07 2023 at 19:06):I understand. I was just trying to get Andrius to read some stuff that would help him. (I don't really care about "cognitive" aspects of geometry, or why he prefers those 4 kinds of geometry.)
Todd Trimble (Dec 07 2023 at 19:13):But if we are talking about "cognitive" aspects: maybe another word closer to what Andrius has in mind is "philosophical", and the philosophical aspects of the Erlangen program were also taken up by Tarski, in his What are Logical Notions?. This is something we discussed long ago at the n-Category Cafe, I think in a post by David.
Andrius Kulikauskas (Dec 07 2023 at 21:17):@Todd Trimble @John Baez @Morgan Rogers (he/him) thank you for your replies! I was aware of Klein's Erlangen program but not of the nLab page on Klein geometry. Freudenthal and Tarski's articles are new to me. Todd, I have now found your post Concrete Groups and Axiomatic Theories I and David Corfield's on Logic as Invariant Theory Revisited. Thank you for this food for thought.
By understanding geometry and geometries "cognitively" I mean metaphysically, but in other words, discovering the cognitive frameworks that underlie the key notions. Of course, that supposes that we experience life through cognitive frameworks, which I have found fruitful. My presentation "A Geometry of Moods: Evoked by Wǔjué Poems of the Táng Dynasty" provides an example of my investigatory approach.
My understanding, at this point, is that "geometry is the uniformity of choice". Which is to say, if various locations offer the same choices, then we are dealing with geometry, but if they all offer different choices, then we are not dealing with geometry, and there is a wide spectrum of possibilities in between.
Here are my notes. Yesterday I wrote out 36 ways of investigating what is geometry. Today I reorganized my notes, which are quite old and scattered. Tomorrow I hope to overview various ideas on what geometry is, and the notions they involve, and make a diagram relating those notions, and see if that can make more precise what I mean by uniformity of choice, or simply arrive at a better understanding.
Jens Hemelaer (Dec 08 2023 at 09:53):You already give lots of examples in your notes of situations that are 'geometric' in your definition. Maybe you could also include examples of situations that are not geometric? If you have both examples and counter-examples you can begin to see more clearly the boundaries of your definition.
For example, would any topological space count as a geometry, or any subset of ? The broader your definition of geometry, the more difficult it will be to get a nice classification in 4 types, or to put your idea of uniformity of choice in practice.
Andrius Kulikauskas (Dec 08 2023 at 20:56):@Jens Hemelaer Thank you for your ideas on how to investigate this! Last night I wondered similarly, what would be an antigeometry, with (perhaps maximally) nonuniform choice or perhaps no notion of choice. You prompted me to think fresh about this. I will want to study random matrix ensembles, how they ground CPT symmetry (charge conjugation, parity, time reversal), which is profoundly metaphysical.
Currently, my hypothesis is that there are four ways of generically adding a dimension (a choice), which is to say, extending the duality of counting forwards and backwards. The possibilities are encoded by the simple roots of the classical Lie algebras. The two directions can be unlinked . Or they can be linked by an extra zero: -3, -2, -1, 0, 1, 2, 3 as with . Or we can fuse the ends to give an internal zero -3, -2, -1=1, 2, 3 as with . Or we can fold them together -3, -2, -1, 1, 2, 3 as with .
I will try to show how this distinction carries over to different ways of interpreting the binomial theorem (Combinatorial Interpretations Which Distinguish Observer and Observed), infinite families of polytopes, Weyl groups, and invariants. To quote Roger Penrose's Road to Reality, page 293, §14.1.
How do we assign this needed structure? Such a local structure could provide a measure of ‘distance’ between points (in the case of a metric structure), or ‘area’ of a surface (as is specified in the case of a symplectic structure, cf. §13.10), or of ‘angle’ between curves (as with the conformal structure of a Riemann surface; see §8.2), etc. In all the examples just referred to, vector-space notions are what are needed to tell us what this local geometry is, the vector space in question being the -dimensional tangent space of a typical point of the manifold M (where we may think of $$T_p $$as the immediate vicinity of in ‘infinitely stretched out’; see Fig. 12.6).
Basically, I think it may simply be the ways of thinking about a triangle, in terms of three paths or three lines or three angles or as an oriented area. I think it would be fruitful for me to overview triangle geometry in the plane. But then this all fits within a global context. Modern geometry describes how to build up that global context. Topology, in particular, may describe the global possibilities. So I need to familiarize myself with the key concepts, including cohesive toposes, coherent sheaves, ringed spaces, Grothendieck categories, bundles, connections, schemes. I want to develop some insight on what that all is contributing.
John Onstead (Dec 09 2023 at 01:53):I would love to see a diagram relating the 36 ways of investigating geometry. Drawing out such diagrams relating mathematical concepts to each other, is often the first step in realizing the diagram you are drawing is in fact a categorical diagram! Maybe all 36 ways can be thought of as categories of geometric structures defined by that approach to geometry, and the relationships you draw between them are functors connecting the categories?
Morgan Rogers (he/him) (Dec 09 2023 at 10:58):Andrius Kulikauskas said:
I will want to study random matrix ensembles, how they ground CPT symmetry (charge conjugation, parity, time reversal), which is profoundly metaphysical.
This is a rather funny thing to say, given that CPT symmetry is a physical principle (perhaps even 'profoundly physical')...
Currently, my hypothesis is that there are four ways of generically adding a dimension (a choice) ... encoded by the simple roots of the classical Lie algebras.
Where does this hypothesis come from? What does "adding choices" mean and why should it be related to (simple) Lie algebras? If there really is a connection, why does it exclude the exceptional Lie algebras?
You've certainly identified a lot of key words associated with geometry in one way or another, but the way you arbitrarily tie these things together and consistently decide that there should be four basic things (in this case geometries), with no apparent basis besides your personal "cognitive" numerology, prevents me from taking anything you write seriously.
Andrius Kulikauskas (Dec 09 2023 at 19:23):@John Onstead here is a diagram that I made of my ways of investigating geometry.
In my understanding, they fall into three general "categories" - the cultural (specific to the circumstances of our external world), the innate (which we seem endowed with psychologically), and the intrinsic (which suggests that geometry is a world of its own). I am curious, what do you think?
John Onstead (Dec 10 2023 at 22:47):@Andrius Kulikauskas It's very interesting! It looks like a more philosophical take on organizing ways to approach geometry. I look forward to your further insights!
John Onstead (Dec 10 2023 at 22:48):This may not be directly relevant to this post, but it did inspire a question of mine that I've had for a long time, and maybe one of you has an answer? There's a series on a channel called Eigenchris (who releases good differential geometry videos, he helped me learn more about GR for instance) that is currently going through various ways of approaching the concept of a rotation on a manifold, and in a more specific case, the concept of a spinor. The two main ways he covers are through lie theory (lie groups and lie algebras) and geometric (clifford) algebra. However, he hasn't established why such a connection exists, why you can use both geometric algebra and lie theory to explain rotations on various manifolds. It makes sense that one can describe rotations on a manifold in terms of lie theory because lie groups are literally defined directly on those manifolds. However, geometric algebra has no connection to any "actual" geometric structures, it is a purely algebraic structure. So how can we justify using geometric algebra, a purely algebraic structure, to represent geometrical things? Any help is appreciated!
Andrius Kulikauskas (Dec 12 2023 at 17:24):@John Onstead Thank you! I have published my video about this diagram: "What is geometry? How and Why I Explore This". Next, I will try to understand and relate various frameworks for geometry, including cohesive toposes, Isbell duality, coherent sheaves, ringed spaces, Grothendieck toposes, schemes, spans of groupoids, curvature and connections.
Andrius Kulikauskas (Dec 12 2023 at 17:33):@John Onstead I am also a fan of Eigenchris's series Spinors for Beginners. I like your question on the relation of Lie theory and Clifford algebra. Clifford algebra (and geometric algebra) is a generalization of complex numbers and also quaternions, both of which can model rotations. In a Clifford algebra, we can have a generator , which can be thought of as a rotation by .
Andrius Kulikauskas (Dec 12 2023 at 17:43):@John Onstead Do you think vectors are real geometric things? Geometric algebra is describing products of vectors, which yields oriented areas and volumes of however many dimensions. This is relevant for my question, What is geometry? Vector spaces, algebraically, have elements (vectors) which can be expressed in many different ways. When we choose a basis, then we have a coordinate system for a geometry. But, in a sense, that is simply empty space. Adding an inner product adds a spatial property such as distance, angle or oriented area to that space. Then we can start to talk about figures. So geometry is arising here, I think. Clifford algebras (and perhaps geometric algebra) typically assume an explicit choice of generators and a basis given by their products. I wonder how this relates to the abstract frameworks for modern geometry, the related category theory, and also to the classical Lie families.
Todd Trimble (Dec 12 2023 at 21:14):How or where do abstract incidence geometries (axiomatic projective geometry being an example) fit into the framework you envisage?
The theory of Tits buildings is a pretty revolutionary rethinking of what this is about.
John Onstead (Dec 13 2023 at 11:12):@Andrius Kulikauskas I think Eigenchris will sort of address the question I raised in his next video, so I am eagerly awaiting that. As for if vectors are geometric entities, this question reminds me of another youtube channel I watch, 3blue1brown. He has a series on the essence of linear algebra where he brings up a similar question: is a vector a set of numbers, or an arrow pointing in space? My conclusion is that a vector is anything, whether it be an arrow in space or a set of numbers, that satisfies the criteria for a vector space, in which case vectors are at their core algebraic objects. The geometric intuition only comes when you add in extra structure, like what you are saying with how adding inner product structure to a vector space is adding a metric space structure. One can think of an algebraic structure internal to Top as an object where elements have both geometric/topological and algebraic interpretation. These include topological groups (of which Lie groups are a special case) as well as all normed vector spaces (which are examples of topological vector spaces, a vector space internal to Top). My question about geometric algebra stems from how it is not naturally such a structure, and so its elements should only have a purely algebraic interpretation. Thus I'm looking to see if there exist any "natural" ways to add on this structure to a Clifford algebra; IE, if there is a "natural" way to associate a clifford algebra with some algebraic object internal to Top that gives the "correct" corresponding geometric intuition behind the elements of the clifford algebra.
Ralph Sarkis (Dec 13 2023 at 11:45):We have this famous saying: "a vector is any element of a vector space". I don't know if it is credited to someone in particular.
Todd Trimble (Dec 13 2023 at 12:27):I somehow doubt that Paul Halmos was the first to say it, but say it he does, somewhere in his automathography. IIRC, he's railing against an Encyclopedia Brittanica article on vector spaces that a student has cited. He's absolutely right, of course.
Andrius Kulikauskas (Dec 13 2023 at 21:21):@Todd Trimble @John Onstead @Ralph Sarkis Thank you for your deep ideas! I will think about them. I want to say for now that Todd, I really appreciate you pointing me to the incidence geometry and incidence structure. I like that the same structure can be thought of also as a hypergraph and as a block design, which suggests to me that this is a concept for cognitive divergence, where the mind wants to look at the same things in different ways. Thus this may well be an early concept in the unfolding of math. Also, I'm curious how this may relate to Grassmanians, flags, and flag varieties, which I also think are basic. I will be studying this. Thank you!
Todd Trimble (Dec 13 2023 at 21:25):Indeed, flag varieties (and certain quotients of those like Grassmannians) are extremely fundamental to this Tits point of view on incidence geometry, so you're pointing in the right direction.
(I learned much of what I know of this subject through conversation with James Dolan.)
Andrius Kulikauskas (Dec 13 2023 at 21:27):I want to share a quote from William Lawvere in his talk in 2012:
Traditional philosophical notions for thousands of years could be given a mathematical formulation sufficiently general using the most up-to-date technology we have in mathematics, i.e., topos theory, etc., etc., could be given a formulation which in some sense almost fully captures philosophical content, not just some fragment of it, so that philosophical considerations could become open to some degree to calculation as Leibniz, for example, dreamed.
And I see that at nLab I suppose Urs Schreiber writes about Kant and Hegel as in the article on modality. I myself am proceeding from the opposite direction. I have worked out a language of cognitve frameworks, similar to that of Kant and Hegel, and having a PhD in math, I am trying to show where I see those frameworks arise.
John Onstead (Dec 13 2023 at 22:59):@Todd Trimble I'm curious to learn more about incidence geometry. Up until learning about it, I had sort of a narrow definition of geometry following the etymology ("measure the earth", which I interpreted to mean that geometry is the study of distances in space. Thus, it's essentially the study of metric spaces). I usually learn about new things by studying the category they live in. Could you provide details about the category IncStr, the category where objects are incidence structures? Things like what a morphism is precisely, the kind of category it is (topos, finite products, complete/cocomplete, etc.), universal properties and special objects (IE, terminal objects) within it and what they represent, if it is a presheaf category or a subcategory of a presheaf category, if its a category of monad algebras, if any obvious forgetful functors exist, etc. And maybe information about how it connects to TitBuild, the category of Tits Buildings (such as any functors between them, or any other way to go between these concepts)? Thanks :)
Todd Trimble (Dec 14 2023 at 00:01):Sorry, I don't know what the category of Incidence Structures is. I've never heard of it until now. I don't know what an Incidence Structure is supposed to be either. But the general idea of an incidence geometry is that one has a notion of types of "figure" (like a point or a line, and there are many others, including flags, ...) and various incidence relations (like "point P is incident to line L"), and a type of geometry, for example the theory of projective planes, is given by axioms on the incidence relations, for example the axiom "given distinct points P and Q, there exists a unique line L that is incident to both".
Todd Trimble (Dec 14 2023 at 00:01):I'll talk just a little about buildings. It takes a little while to give a sense of what they are, but it's helpful to start with flags which are chains of incidence relations, for example it could be a chain of inclusions between subspaces of a vector space V. Really roughly speaking, a building can be thought of as something like a set of all flags of some type, together with a measure of the degree in which two flags are qualitatively different. For example, maybe you can get from flag A to flag B just by changing A's 1-dimensional subspace part. The instruments that measure "how far apart" flags are turn out to be Coxeter groups, usually arising as Weyl groups W that arise in a certain way from the geometry at hand.
Todd Trimble (Dec 14 2023 at 00:01):So suffice it to say, for now, that for each Coxeter group W, one can describe an incidence geometry associated to W. For example, classical projective geometry for -dimensional projective space turns out to be the incidence geometry corresponding to , when presented as a Coxeter group.
Todd Trimble (Dec 14 2023 at 00:02):I've never really thought about categories of W-buildings as such. Insofar as I've had any contact with this area of mathematics, it's more that I use methods of category theory to help formulate the basic notions. For example, there are certain parallels between buildings and Lawvere metric spaces, and I use enriched category theory to help understand both.
John Onstead (Dec 14 2023 at 00:18):Interesting, thanks for the info! As far as I've seen, an "incidence structure" and an "incidence geometry" are often used interchangeably. I only used "incidence structure" because "incidence geometry" could (potentially confusingly) refer to either the mathematical object or to the field of study of that mathematical object.
It seems a basic component of an incidence geometry are sets of figures (like a set of points P, a set of lines L, etc.) and functions between them that characterize the incidence relations. So in stuff, structure, property format an incidence structure is defined as some sets of figures (stuff), equipped with incidence relations (structure), satisfying incidence axioms (properties). As for buildings, I will look more into them as they sound interesting!
Todd Trimble (Dec 14 2023 at 00:33):Well, in the Tits approach, the underlying set of a building qua incidence geometry is some abstract set whose elements can be called "flags" if you like, together with various types of binary relations on , and these binary relations are indexed or catalogued by the elements of a Coxeter group. You can write down the axioms of the geometry based on the structure of the Coxeter group.
Todd Trimble (Dec 14 2023 at 00:33):So in this view, a building is a set equipped with a certain structure of relational type (the underlying language that I know is a relational language, without any operations as such).
Todd Trimble (Dec 14 2023 at 00:33):There are other ways of describing buildings based on simplicial complexes, but I don't want to go into this now. I would say that Ken Brown's book Buildings is pretty good -- he gives lots of intuition and motivation -- and I was also helped by reading Richard Weiss's book The Structure of Spherical Buildings, which expounds on Tits' fundamental work.
Todd Trimble (Dec 14 2023 at 02:00):As long as you don't mind my yakking away, I can say a little bit more, using the example of projective planes and how they are supposed to be connected with , the symmetry group of a set with three elements.
Todd Trimble (Dec 14 2023 at 02:00):So in the traditional way of speaking about projective planes, there's a set of points and a set of lines, and there's a basic relation between the two sets, "a point P is incident to a line L". Axioms of projective geometry are all sentences that talk about this basic relation, the conditions it should satisfy.
Todd Trimble (Dec 14 2023 at 02:01):But in the newer way developed by Tits, one talks about not two sets (set of points, set of lines), but just one set: the set of flags. To get from the traditional way to the newer way, define a flag to be a pair (P, L) such that P is incident to L. But in the newer way, the basic elements you start with are the flags, and then one posits certain relations between flags, and in this scheme "point" and "line" will become derived notions. It works something like this.
Todd Trimble (Dec 14 2023 at 02:01):In the geometry of the projective plane, there are six basic ways in which one flag can be related to another, corresponding to the six elements of . Here we think of as generated by two elements which I will denote and , subject to the relations and -- think and if you like):
Two flags can differ not at all (corresponding to the identity element);
Two flags can differ in their points, but share the same line (corresponding to the element );
Two flags can differ in their lines, but share the same point (corresponding to the element );
A flag A can be related to a flag B by having the point of A lie on the line of B (corresponding to the element );
A flag A can be related to a flag B by having the line of A pass through the point of B (corresponding to );
Two flags can be as different as you please (corresponding to ).
Todd Trimble (Dec 14 2023 at 02:01):In this scheme, a "point" is then defined as an equivalence class of flags, defined by the third relation. Similarly, one can define a "line" as an equivalence class, defined by the second relation. This is more or less how you get from the newer way to the traditional way of describing projective planes.
Todd Trimble (Dec 14 2023 at 02:01):I'll stop here for now.
Andrius Kulikauskas (Dec 14 2023 at 10:11):@Todd Trimble Thank you! I very much appreciate your intuition, insight, help in learning this and making progress in contemplating the big picture! @John Onstead thank you likewise!
Andrius Kulikauskas (Dec 14 2023 at 21:49):@Todd Trimble Thank you again for encouraging me to learn about buildings. Thank you for your basic example. I drew the diagram I attach to see if I understand you. It seems like these ways of relating flags give a hierarchy of degeneracy, where flags (P1,L1) and (P2,L2) are nondegenerate or completely independent, with respect to each other, if P1 is not on L2 and P2 is not on L1.
S3-Example-Building.png
Andrius Kulikauskas (Dec 14 2023 at 22:00):I am reading about Buildings on Wikipedia and I have downloaded the book Buildings by Kenneth Brown, which includes examples for the general linear group, orthogonal group and symplectic group. The concept seems very relevant to my interests for I am investigating whether eightfold Bott periodicity may model the eight-cycle of divisions of cognitive frameworks (divisions of everything) that I describe here. Bott periodicity classifies symmetric spaces (Grassmannians), which buildings generalize. The symmetric spaces are quotients of Lie groups - orthogonal, unitary, symplectic - and I am trying to understand them intuitively by way of their role in geometry, which buildings should certainly speak to. I have focused on Weyl groups and the related infinite families of polytopes - simplexes, cross polytopes, hypercubes - along with a fourth family that is given by the coordinates of a hypercube. These four "choice frameworks" can be thought of as ways of interpreting the binomial theorem.
Andrius Kulikauskas (Dec 14 2023 at 22:04):I am trying to climb the mountain of Bott periodicity by way of the binomial theorem, through Gaussian binomial coefficients (which encode subspaces of vector spaces over a finite field), through the Grassmannians (which gives subspaces of vector spaces), by way of Clifford algebras (whose generators are like choices in the binomial theorem). Surely I will benefit from learning about buildings. I am grateful for insights you share!
Todd Trimble (Dec 14 2023 at 22:04):Yes, you seem to have the idea.
I'll try to say a little more later about where a lot of this is coming from from the direction of some classical Lie group theory. Flag varieties are certain homogeneous spaces where one is taking spaces of cosets of Borel subgroups, and the sorts of relations that hold between flags (that I sketched in a simple case above) has a lot to do with double cosets.
Your investigations into Bott periodicity sound intriguing...
Andrius Kulikauskas (Dec 15 2023 at 21:05):@Todd Trimble thank you for emphasizing Borel subgroups. My limited understanding is that a key concept is the maximal torus, which is a diagonal matrix that consists of rotations. So everything boils down to rotations. And there are four ways of expressing rotations: even (2x2) matrix, odd (3x3) matrix (with an extra fixed dimension), complex and quaternion subalgebra. And they all fit together into the Bloch sphere, which is the qubit, closely related to SU(2), the building block of Lie theory. So I need to learn more about all of these things.
Andrius Kulikauskas (Dec 15 2023 at 21:40):Currently, here is my understanding of the big picture in geometry. The purpose of geometry is to describe our world view on things in the world. So we have to build up levels.
Andrius Kulikauskas (Dec 15 2023 at 21:45):That gives a sense of my mathematical thinking and also my philosophical thinking and my interest in eightfold Bott periodicity. I do appreciate feedback. One of the consequences of my teasing out geometry this way is that for me it seems that affine and projective geometry are highly ambiguous in that they can be understood as taking place on different levels. It's simply a matter of attitude whether we think of a single coordinate system as a "single" coordinate system (not two or three). Likewise with no coordinate system, which is my perhaps erroneous understanding of affine geometry.
I conclude today with a quote from Roger Penrose's Road to Reality, page 293, §14.1.
How do we assign this needed structure? Such a local structure could provide a measure of ‘distance’ between points (in the case of a metric structure), or ‘area’ of a surface (as is specified in the case of a symplectic structure, cf. §13.10), or of ‘angle’ between curves (as with the conformal structure of a Riemann surface; see §8.2), etc. In all the examples just referred to, vector-space notions are what are needed to tell us what this local geometry is, the vector space in question being the n-dimensional tangent space Tp of a typical point p of the manifold M (where we may think of Tp as the immediate vicinity of p in M ‘infinitely stretched out’; see Fig. 12.6).
Next, I will prepare the transcript of a video that I made on Bott periodicity. And I am studying nLab articles, etc., on the modern view of geometry, such as cohesive toposes, coherent sheafs, Isbell duality.
John Onstead (Dec 16 2023 at 00:01):@Andrius Kulikauskas I really like your thinking! Your "big picture" in geometry seems to be a repeated application of stuff, structure, property where we have some object, add structure and property onto it to get another object, and repeat again. In each case we construct a new category of those objects. Here's my best attempt at a connection (caviat: I'm actually learning a lot of this stuff myself at the moment so may be some errors!). To start, there's a category Top of topological spaces and Locale of locales, the two are related through an adjunction. Through this one can transition from locales to thinking about topology. After this one can identify topography and coordinate systems by equipping a topological space with a structure, what is known as an "atlas". This atlas is a set of cohesive "charts", each one that can act as a coordinate system centered on some point. An equipment of an atlas makes a topological space into a manifold. One can alternatively equip a topological space with a vector space, which can help in defining "direction", and this structure is known as a topological vector space. Equipping that with a norm gives rise to a normed topological vector space such as a banach space. One can use banach spaces in atlases to define manifolds which are locally homeomorphic to them; one example of a topological vector space is the Euclidean topological vector space, which gives the usual notion of manifold when used as the basis for an atlas (since it makes the space locally homeomorphic to Euclidean space). It is ultimately this that gets us to talking about differential geometry and differentiable manifolds and then to Penrose's quote, given an atlas based on Euclidean vector space generates a canonical vector bundle, the tangent bundle to the manifold, which assigns a tangent space to each point on the manifold. I am not sure how to get to logic from here, but homotopy type theory, based on homotopy theory, is a branch of math and potential foundation of math that recognizes connections and analogies between logic and geometry and allows one to study logical operations in terms of paths on topological spaces. In all, I very much enjoyed reading your "big picture" and I look forward to any future updates you may have!
Todd Trimble (Dec 16 2023 at 01:31):Andrius Kulikauskas said:
Todd Trimble thank you for emphasizing Borel subgroups. My limited understanding is that a key concept is the maximal torus, which is a diagonal matrix that consists of rotations. So everything boils down to rotations. And there are four ways of expressing rotations: even (2x2) matrix, odd (3x3) matrix (with an extra fixed dimension), complex and quaternion subalgebra. And they all fit together into the Bloch sphere, which is the qubit, closely related to SU(2), the building block of Lie theory. So I need to learn more about all of these things.
Maximal tori are indeed important here.
Todd Trimble (Dec 16 2023 at 01:32):Tits himself emphasizes the notion of a BN-pair. For "nice" Lie groups , there exists a pair of subgroups where is a Borel subgroup containing a maximal torus and is the normalizer of , satisfying some properties that you can read about in the article. The Weyl group will be the quotient group and will be a Coxeter group, i.e., a reflection group.
Todd Trimble (Dec 16 2023 at 01:32):For , this group acts transitively on the space of flags in . Fix a particular flag, say the standard flag whose -dimensional part is the span of the first standard basis vectors. Then you can take to be the stabilizer subgroup of this fixed flag. It consists of upper triangular matrices with invertible diagonal entries. This is a Borel subgroup. By transitivity, the flag space can be represented as the homogeneous coset space . The maximal torus is the subgroup of invertible triangular matrices.
Todd Trimble (Dec 16 2023 at 01:32):acts transitively on , but it does not act transitively on pairs of flags, . For example, consider flags in , equivalently point-line flags in the projective plane , with incident to . Then, if you have a pair of flags sharing the same line but not the same point, then applying an element , the same will be true of the pair . There's no way you could change that! However, does act transitively on the set of pairs which stand in this relation to each other. A similar statement can be made for the other five relations between flags that we discussed before. Thus, there are 6 orbits of the action of on in this case.
Todd Trimble (Dec 16 2023 at 01:33):These orbits correspond to a partition of into 6 pieces, called double cosets . For general , it turns out that there is a bijection between the Weyl group and the set of double cosets; in our example this is . Anyway, the -equivariant incidence relations that can hold between pairs of flags (where flags are elements of ) correspond bijectively to elements in this Coxeter group .
Todd Trimble (Dec 16 2023 at 01:33):So this is still a very sketchy description, but anyway I did want to follow up a bit on what I said earlier.
Andrius Kulikauskas (Dec 19 2023 at 21:01):@Todd Trimble Thank you very much for encouraging me to study the BN-pairs, which I will do, and for explaining to me the main idea, which is very helpful. I see that the flags are central and also, as you note, how the (lack of) transitivity works, yielding the orbits, which illuminate the Weyl group. The four choice frameworks that I think in terms of are very much related to the Weyl groups and the symmetries of the infinite families of polytopes. Thank you for opening these doors for me!
Andrius Kulikauskas (Dec 19 2023 at 21:07):@John Onstead I am glad for your fellowship in thinking about these layers. It reminds me of music, where we may lay down tracks, perhaps starting with a rhythm, and then a chord progression for harmony, and then a melody, and then perhaps some improvisation. Or a dramatic play may perhaps likewise be built up from scenes and acts. Or perhaps even a meal, although I may be going astray. I appreciate your describing differential geometry. What I think I will do now is consider how those eight layers get expressed in various foundations of geometry, as with cohesive toposes, coherent sheaves, Isbell duality, differential geometry and so on. I will share what I learn!
Posina Venkata Rayudu (Jan 01 2024 at 11:07):Andrius Kulikauskas said:
I am investigating what is meant by "geometry". I think this term represents an intuition that I would like to make more explicit in terms of cognition. So I am gathering evidence from mathematical practice, from elementary math to advanced math.
I want to learn what Category theory has to say about foundations of geometry. I am aware of the articles at nLab on geometry, geometric morphism, cohesive topos and Isbell duality. I have a lot to learn but I do have a basic understanding of adjunctions and adjoint strings. I very much appreciate all manner of insights, intuitions and suggestions.
I also want to learn how geometry is understood by geometers. One perspective that I find helpful is by Tazerenix that geometry is the study of spaces with rigidifying structure. In particular, I suspect that cognitively there are four basic geometries (affine, projective, conformal, symplectic) as defined by classical Lie groups/algebras. I want to understand geometry better overall and then I will know what to think about that. This month I plan to work here on all manner of investigations. I appreciate being able to think out loud and get ideas, suggestions, feedback and grow in my understanding.
@Andrius Kulikauskas Pardon me if you're already familiar with Professor F. William Lawvere's Axiomatization and Education in which he shows how geometry is self-founded; how to isolate a discrete subcategory to serve as a background for interpreting the theory subcategory. Then there is the geometry of figures in Conceptual Mathematics, along with an equivalent discussion in Sets for Mathematics in terms of actions. Professor F. William Lawvere also notes how category theory, with its theory subcategories, goes way beyond Klein's symmetries (CM, p. 180). In response to what cognition got to do with geometry, as Professor F. William Lawvere pointed out all notions of constancy are perceptual/conceptual abstractions. Also, Charles Ehresmann's insights into geometry are certainly worth noting (at least for me :) I thought you might find Linton's and Vicker's insights of some interest. Most important of all is my good ol' friend Grassmann's geometric calculus made more accessible thanks to Professor F. William Lawvere's reviews and explication of Grassmann's dialectics.
Speaking of questions of the form:
What is geometry?
I vaguely remember reading somewhere that the difficulty has more to do with 'is', which reminds me of MacNamara pointing out to Professor F. William Lawvere that the correct mathematical interpretation of 'is' is not 1-1 function/monomorphism, but a function.
I wish you all a Happy New Year :)
Todd Trimble (Jan 01 2024 at 11:59):Out of curiosity, why do you keep writing out "Professor F. William Lawvere" every time?
Posina Venkata Rayudu (Jan 01 2024 at 20:38):Todd Trimble said:
Out of curiosity, why do you keep writing out "Professor F. William Lawvere" every time?
I come from Godavari river region (Andhra Pradesh, India), where we are very respectful with one salutation before and one or more after ... it's my Godavari way of expressing my gratitude to Professor F. William Lawvere :)
p.s. it wouldn't be whole truth if I didn't add that your question made me feel more alive (cf. W. E. B. DuBois double consciousness); thank you!
Posina Venkata Rayudu (Jan 01 2024 at 22:42):Posina Venkata Rayudu said:
Andrius Kulikauskas said:
I am investigating what is meant by "geometry". I think this term represents an intuition that I would like to make more explicit in terms of cognition. So I am gathering evidence from mathematical practice, from elementary math to advanced math.
I want to learn what Category theory has to say about foundations of geometry. I am aware of the articles at nLab on geometry, geometric morphism, cohesive topos and Isbell duality. I have a lot to learn but I do have a basic understanding of adjunctions and adjoint strings. I very much appreciate all manner of insights, intuitions and suggestions.
I also want to learn how geometry is understood by geometers. One perspective that I find helpful is by Tazerenix that geometry is the study of spaces with rigidifying structure. In particular, I suspect that cognitively there are four basic geometries (affine, projective, conformal, symplectic) as defined by classical Lie groups/algebras. I want to understand geometry better overall and then I will know what to think about that. This month I plan to work here on all manner of investigations. I appreciate being able to think out loud and get ideas, suggestions, feedback and grow in my understanding.
Andrius Kulikauskas Pardon me if you're already familiar with Professor F. William Lawvere's Axiomatization and Education in which he shows how geometry is self-founded; how to isolate a discrete subcategory to serve as a background for interpreting the theory subcategory. Then there is the geometry of figures in Conceptual Mathematics, along with an equivalent discussion in Sets for Mathematics in terms of actions. Professor F. William Lawvere also notes how category theory, with its theory subcategories, goes way beyond Klein's symmetries (CM, p. 180). In response to what cognition got to do with geometry, as Professor F. William Lawvere pointed out all notions of constancy are perceptual/conceptual abstractions. Also, Charles Ehresmann's insights into geometry are certainly worth noting (at least for me :) I thought you might find Linton's and Vicker's insights of some interest. Most important of all is my good ol' friend Grassmann's geometric calculus made more accessible thanks to Professor F. William Lawvere's reviews and explication of Grassmann's dialectics.
Speaking of questions of the form:
What is geometry?
I vaguely remember reading somewhere that the difficulty has more to do with 'is', which reminds me of MacNamara pointing out to Professor F. William Lawvere that the correct mathematical interpretation of 'is' is not 1-1 function/monomorphism, but a function.
I wish you all a Happy New Year :)
@Andrius Kulikauskas With reference to cognition vis-a-vis geometry, the following may be of some interest:
Cognitive Roots of Mathematical Concepts
Geometries and Cognition
Cognitive Foundations of Mathematics
Along these lines, those exalted (scientific) experiments are planned perceptions. Then there are perceived elements. and sober spaces, where we are spared of seeing doubles ;)
None of the above is, as I write it, championing the cause of cognitive science nor am I its self-anointed brand ambassador. Full Disclosure: I went to war against cognitive science to get Mathematics included in its logo (https://cognitivesciencesociety.org/); as usual there was nobody at the battlefield ;) Mathematical knowing (cf. solving for unknowns) is too special to inform knowing/cognition in general, say Pinker et al., which only shows how one can get to be a professor at THE MIT with wholesome ignorance as added qualification: it is the tooooooooooooo special motion of a dropped thing that gave birth to the science of motion (cf. acceleration) and not twerking ;)
Todd Trimble (Jan 01 2024 at 22:46):Posina Venkata Rayudu said:
Todd Trimble said:
Out of curiosity, why do you keep writing out "Professor F. William Lawvere" every time?
I come from Godavari river region (Andhra Pradesh, India), where we are very respectful with one salutation before and one or more after ... it's my Godavari way of expressing my gratitude to Professor F. William Lawvere :)
p.s. it wouldn't be whole truth if I didn't add that your question made me feel more alive (cf. W. E. B. DuBois double consciousness); thank you!
Thank you for saying so!
Andrius Kulikauskas (Jan 02 2024 at 19:25):@Posina Venkata Rayudu I am glad to hear from you! Thank you for your many recommendations. I appreciate your knowledge of Lawvere's ideas. I will ask for your help to understand him. Thank you also for the links to the papers on math and cognition! I am not familiar with them and they are very relevant. My plan now is to consider a working hypothesis whereby geometry has eight layers which I sketched above and see how that may or may not fit with the mathematical structures thought through by various thinkers we have mentioned here. That will give me a way to try to understand what they are saying and how they relate to each other. I imagine that will bring out details that I am missing but also clarify central themes. But I think I will push this off for a month or so because I am currenlty engrossed in another mathematical adventure which I hope to pursue in a new thread! Thank you, Posina!
Andrius Kulikauskas (Jan 02 2024 at 19:34):@Todd Trimble thank you again for encouraging me to learn about buildings and apartments. I was excited to realize how central they are to the idea of the mythical field with one element. I have some basic intuition about that and I definitely want to master that concept. I have the hope that it may be useful for modeling God, where I think of God as a state of contradiction, in which all things are true, from which by a process of division, self-differentiation, there arises a tentative state of noncontradiction. I write about that in my presentation, Imagining God's Mind As a Question: Is God Necessary? My plan is to focus on the transition between the binomial coefficients (subsets of sets) and the Gaussian binomial coefficients (subspaces of vector spaces). I think I will fit in some exploration of that when I study more about buildings and apartments. Thank you for your advice and fellowship! I want to add that in studying Bott periodicity and Clifford algebras, I found very helpful your note, The Super Brauer Group and Super Division Algebras, and I hope in these coming weeks to share more about my studies of that.
Todd Trimble (Jan 02 2024 at 19:39):I was excited to realize how central they are to the idea of the mythical field with one element.
Yes!! This "field of one element" business: it all started with Tits and these investigations of his. In some sense, the Coxeter group (or a closely related structure) plays a central role of "building" over .
It's a metaphor that has taken on a life of its own... but a still-in-many-ways mysterious life.
Andrius Kulikauskas (Jan 02 2024 at 19:47):@Todd Trimble Great! I forgot to add that in my mind it suggests a trinity whereby . Also, in the study that I have done with simplexes and q-analogues, setting q=1 yields the situation where we are choosing 1 out of 1, which is a choice, but a degenerate one. But I have to revisit that and make slides and a video about it!
Todd Trimble (Jan 02 2024 at 21:48):I don't know about such "poetic" thoughts (I'm more comfortable communicating my thoughts in terms of precise mathematics), but this reminds me of something James Dolan used to discuss: there is, in this discussion, a family of -deformations of the group algebra of the Coxeter group , called Hecke algebras, and they tend to show up a lot in this area.
Todd Trimble (Jan 02 2024 at 21:48):To anchor the discussion, let's take for example , where the standard Coxeter group presentation has two generators , subject to involution equations and the braiding relation
.
Todd Trimble (Jan 02 2024 at 21:48):The group algebra fits into a family of -algebras , with 6 basis elements and , so with a multiplication table like the Coxeter group, except that involution equations are replaced by quadratic relations
which depend on a parameter . In the case , you retrieve the involution equations , hence retrieve the group algebra. At an "opposite" end of the spectrum, , you can interpret the equation as giving not involution equations, but idempotent equations . Thus, you get in this case not a group algebra but a monoid algebra, where the monoid is presented by two generators , with the idempotency relations and the braiding relation. James Dolan used to call this "the Murphy monoid", for fanciful reasons I won't go into here.
Todd Trimble (Jan 02 2024 at 21:49):A lot of the theory of Coxeter groups has counterparts for this Murphy monoid. For example, the class of "reduced words", which looms large in the study of buildings, is the same for the Murphy monoid as for the Coxeter group. Anyway, for us the Murphy monoid (which is actually a -quantale) assumed a lot of importance in the theory we were working out.
Todd Trimble (Jan 02 2024 at 21:49):For "in between", the Hecke algebras crop up in number theory and modular representation theory, as for example power of a prime. I think people also study what happens when is a root of unity, although I forget what exactly. (Maybe to do with quiver representations?)
Todd Trimble (Jan 02 2024 at 21:49):Now, I don't think we ever thought about . Here the inclination would perhaps be to rewrite the quadratic relation as , and then set , to force the degenerate case where is forced. I don't know what that might be good for.
Andrius Kulikauskas (Jan 03 2024 at 08:24):@Todd Trimble Thank you! I will stay on the look out for this structure, this interpolation and this Murphy monoid. One place where comes up is in the anharmonic group, which are six Mobius transformations, which happen to be isomorphic to , indeed the quotient group where is the stabilizer of the cross-ratio. Two elements of the anharmonic group and look just like and . I wonder what that could mean here.
Todd Trimble (Jan 04 2024 at 00:52):I don't see a compelling connection, but I'm not ruling out there could be one.
Todd Trimble (Jan 04 2024 at 00:52):I can never be sure that I remember the standard order of the expressions in the cross-product, but it's something like this: the cross-product of the 4-tuple -- let's assume are distinct -- is the value that gets sent to under the unique projective transformation that takes to , to , and to . (Some of these may be switched around from how it goes standardly, but my way is easy for me to remember.) That's given by the Moebius transformation
(that second quotient is inserted to ensure ). And so "my" cross-ratio is . That's invariant under the permutations given by , , , and the identity.
Todd Trimble (Jan 04 2024 at 00:53):So I see: the Moebius transformations corresponding to the 6 permutations on form the anharmonic group. These would be generated by (a 2-cycle ) and (a 3-cycle ). I notice that the determinant of the matrix corresponding to is -1, so the anharmonic group is a subgroup of but not of .
Todd Trimble (Jan 04 2024 at 00:53):Anyway, I'll take this opportunity to share how Jim Dolan motivated for me the quadratic relations appearing in the presentation of the Hecke algebra. Here we will be working with BN-pairs associated with . So Here the B is a Borel subgroup that stabilizes a given fixed flag, say the standard flag whose -dimensional part is the span of the first standard basis elements; the B here consists of invertible upper triangular matrices. For simplicity, we'll take for example, as we did earlier.
Todd Trimble (Jan 04 2024 at 00:53):So the space of flags is the set of point-line incidence pairs in the projective plane over , and you'll recall there were 6 different -equivariant binary relations on flags, corresponding to the orbits of under the diagonal action of . These orbits are called double cosets; the set of double cosets is denoted by .
Todd Trimble (Jan 04 2024 at 00:53):Another way you can think of this: by taking inverses, there is a bijection between left cosets and right cosets , in such a way that the left action of on gets transferred to the right action of on . The quotient of by the diagonal left action of then corresponds to the quotient of , obtained by "tensoring" the right -set with the left -set ; in other words, the set of double cosets is obtained as a coequalizer as shown here,
.
Todd Trimble (Jan 04 2024 at 00:54):This might make the notation somewhat more understandable or intuitive, at least for category theorists.
Todd Trimble (Jan 04 2024 at 00:54):Anyway, Jim explained to me how each double coset , or equivariant relation, induces an equivariant operator between -modules, namely (remember, this is finite-dimensional, since we're working over ). Take for example the relation "flags and have the same point but differ by a line". The corresponding operator assigns to a flag , as a basis element of , the linear combination obtained by taking the average of flags which lie in this relation to . There are projective lines through a point, so there are summands in this linear combination.
Todd Trimble (Jan 04 2024 at 00:54):Jim called this operator "the line-changer", and thought of it stochastically: change the flag by randomly changing its line. Now consider , which is "change the line, and change it again". Out of the possibilities of changing the line a second time, there's a out of chance that you switch back to the original flag, but there's a out of chance that after the second switch, the flag still differs from the original flag. So we see that this equivariant operator satisfies the equation
which is the quadratic relation in the Hecke algebra.
Todd Trimble (Jan 04 2024 at 00:55):On a more abstract level, this process can be understood as follows. If are two subgroups of , then the set of double cosets still makes sense; it is again constructed as the "tensor product" of the right -space with the left -space . Applying the free vector space functor to the defining coequalizer for this tensor product, the coequalizer is preserved (since free functors, being left adjoints, preserve colimits). Now is a right -module, and the preservation of the coequalizer is saying
.
Todd Trimble (Jan 04 2024 at 00:55):Next, we can identify the right module with the linear dual (the linear dual of a left module naturally acquires a right module structure), just by identifying right cosets with the dual basis of left cosets . Anyway, this gives
and finally there is a canonical isomorphism
Todd Trimble (Jan 04 2024 at 00:55):Putting this together, linear combinations of double cosets correspond precisely to -equivariant operators:
and if you work carefully through this, you wind up with Jim's description in the special case we were just discussing.
Andrius Kulikauskas (Jan 04 2024 at 10:42):@Todd Trimble Thank you very much for your analysis of the anharmonic group and for sharing Jim Dolan's explanation. I am glad to learn of these connections between these deep ideas. Thank you for teaching me, helping me learn these ideas, which are important to me.
Andrius Kulikauskas (Jan 04 2024 at 15:42):@John Onstead I learned today of the work of Garret Sobczyk which might also interest you. He is a coauthor with David Hestenes of the 1985 book “Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics” and this related paper. Here is an interesting interview. I am personally interested in his paper Periodic Table of Geometric Numbers, where he explains Bott periodicity, which he calls the Budinich/Trautman Clifford Clock. Also, he has a book New Foundations in Mathematics: The Geometric Concept of Number.
The book begins with a discusson of modular numbers (clock arithmetic) and modular polynomials. This leads to the idea of a spectral basis, the complex and hyperbolic numbers, and finally to geometric algebra, which lays the groundwork for the remainder of the text. Many topics are presented in a new light, including:
• vector spaces and matrices
• structure of linear operators and quadratic forms
• Hermitian inner product spaces
• geometry of moving planes
• spacetime of special relativity
• classical integration theorems
• differential geometry of curves and smooth surfaces
• projective geometry
• Lie groups and Lie algebras
Here is a related presentation. A quote from the interview:
In this book, I emphasize the development of linear algebra and basic ideas of differential geometry in terms of GA, including the basic concept of the spectral basis. The idea of a spectral basis is what really connects the ideas of a geometric algebra to the traditional approaches to linear algebra and ubiquitous matrices. When GA is properly understood as providing a geometric basis for matrices, the well-established consistency of matrix algebra, in turn, proves the mathematical soundness of GA, without the need to go into more complicated arguments involving tensor analysis and the higher level abstract concepts of advanced mathematics.
Andrius Kulikauskas (Jan 04 2024 at 15:45):Also his presentation: The Spectral Basis of a Linear Operator.
Andrius Kulikauskas (Jan 07 2024 at 15:17):@Todd Trimble Thank you again! I have read your exposition and I will need to study it more. I like Jim Dolan's stochastic interpretation of that equation. It makes a lot of sense, has a lot of intuitive meaning and I will keep it in mind as I study and . Can his formula for the equalizer be understood as a tensor-hom adjunction? Also, is that true of your more general exposition? I am studying adjunction, working to classify them cognitively, and so this will be time well spent for me to understand your exposition, if not right away.
Todd Trimble (Jan 07 2024 at 18:26):Can his formula for the equalizer be understood as a tensor-hom adjunction?
That one is not his (Jim's), nor is it anyone's really, including me, although I had shown this to him at some point and have used it to help explain some of this or related stuff to others (e.g. @John Baez and @Joe Moeller in private conversation).
Todd Trimble (Jan 07 2024 at 18:26):(Nitpick: the construction uses what is called a coequalizer, dual to an equalizer. It's a certain kind of colimit which comes up a lot in practice.)
Todd Trimble (Jan 07 2024 at 18:26):Anyway, yes, it's definitely related to tensor-hom adjunctions. I'll try to explain quickly what's going on.
Todd Trimble (Jan 07 2024 at 18:26):We are in the happy situation where we have a finite group , and we are working for now over a field of characteristic zero (), and we are considering right or left modules over the group algebra . The nice thing about this situation is that every short exact sequence of (say) left modules splits, by Maschke's theorem, and so every -module is projective. Moreover, something like is finitely generated as a left -module.
Todd Trimble (Jan 07 2024 at 18:27):All this might sound very technical so far, but the point is that if you have a -algebra (like ), and if is a finitely generated projective module over , then this condition on is basically saying that the functor itself preserves arbitrary colimits.
Todd Trimble (Jan 07 2024 at 18:27):And it's a fact that preserves colimits iff it is isomorphic to a functor of the form for some right -module (that preserves colimits is because it is a left adjoint, by a tensor-hom adjunction). In that case, notice that we have
so this tells us what this right module is going to be: it's . So in our case, it's , called the dual module of . Or really in our case, it's , considered as a right module over .
Todd Trimble (Jan 07 2024 at 18:27):I'll skip for now the demonstration that we have
as right -modules, although I can explain that some time if you'd like.
Todd Trimble (Jan 07 2024 at 18:27):I am studying adjunction, working to classify them cognitively, and so this will be time well spent for me to understand your exposition, if not right away.
Of course! I don't mean to overwhelm anybody with information, and these things take time. But adjunctions are arguably at least in the top five of important concepts in category theory, and possibly number one on many people's list.
Andrius Kulikauskas (Jan 14 2024 at 15:43):@Todd Trimble Thank you for teaching me so much! I will have to keep returning to this, as I go along, in order to absorb this. I appreciate that you are helping me understand why these puzzle pieces are interesting and how they fit together.
Certainly, this will be important when I focus again on classifying adjunctions. Here are my scattered notes, collecting examples. I attach a diagram of my findings so far, which I am happy with because it matches nicely with a cognitive framework of six conceptions that I am very familiar with, by which we comprehend two perspectives (more precisely, a perspective on a perspective) as related by increasing or decreasing slack (the extent of their difference) or by one of four scopes: everything, anything, something, nothing (the extent of their commonality, the width of the channel that connects them).
ClassificationOfAdjunctions2.png
Andrius Kulikauskas (Jan 14 2024 at 16:45):I realized today that I should explore what Tim Gower's (et al) Princeton Companion to Mathematics has to say. It is a truly marvelous book. Geometry is the second of seven sections on the origins of mathematics. There are 99 sections on concepts in mathematics, 26 sections on branches of mathematics, 35 sections on theorems and problems, 96 sections on mathematicians, and 14 sections on areas that math has impacted.
Andrius Kulikauskas (Jan 14 2024 at 17:10):I attach a diagram that I made in 2020 by assigning each of the 99 concepts to one of the 26 branches of math and noting also the other branches that the concepts were related to. (Here is my data.) And then using the shared concepts as a way to consider which branches should be adjacent to which branches in the diagram. I don't think it was enlightening yet but I do think it is a good time to go back to that.
MathCompanion2.png
Andrius Kulikauskas (Jan 14 2024 at 17:14):I attach a diagram where I simply tried to organize the purposes of various branches of mathematics along with sundry mathematical activities. This came out a little more interesting.
MathAnswers.png
Andrius Kulikauskas (Jan 14 2024 at 17:19):Today I looked at the article in the Companion on Buildings, written by Mark Ronan. I find it helpful. He writes, among other things:
A geometric theory underlying the classical groups has been developed by the first half of the twentieth century. It used projective space and various subgeometries of projective space, which made it possible to provide analogues for the classical groups, but it did not provide analogues for the groups of type . For this reason, Jacques Tits looked for a geometric theory that would embrace all families, and ended up creating the theory of buildings.
I can quote more or simply show the article.
Todd Trimble (Jan 14 2024 at 18:07):Ronan wrote a book (The Structure of Spherical Buildings, I think it is) that I found helpful when I was trying to learn about buildings. Ken Brown (Buildings) is another. For any of these Dynkin diagrams you mention, there's a corresponding Coxeter group, and for each Coxeter group there's a corresponding notion of -building, as I think I said before, with its own special brand of geometry.
Todd Trimble (Jan 14 2024 at 18:07):(My interest in this has been renewed recently in separate conversations with @fosco and @Simon Burton, about categorified derivatives and categorified Pascal triangles of various sorts. But it would be a little premature to say much more about this now. I'll say briefly that Simon and I have been thinking about symplectic groups as a sort of case study.)
Todd Trimble (Jan 14 2024 at 18:08):My own understanding of variables was much clarified by studying the old work of Lawvere, for example on algebraic theories and on hyperdoctrines. If you don't know about the latter, I'd recommend it for your attention.
Andrius Kulikauskas (Jan 14 2024 at 19:38):@Todd Trimble Thank you very much as ever for your recommendations of books and what to read, such as Lawvere's work on variables. I will need to find that. I have given a presentation of my own "A Structural Semiotic Study of How We Use Variables in Mathematics and Logic". I need to transcribe my audio. Here are my slides. I need to return to this and Lawvere's work will be helpful.
Interesting to learn of your conversations about categorified derivatives and categorified Pascal triangles. Currently, I have been working intensely on the combinatorics of the orthogonal Sheffer polynomials, which come up in solutions to the Schroedinger equation, such as the Hermite polynomials (for the quantum harmonic oscillator) and the Laguerre polynomials (for the hydrogen atom). I am trying to understand the physics in terms of combinatorial objects. The orthogonal Sheffer polynomials have a fivefold classification (Meixner, Charlier, Laguerre, Hermite, Meixner-Pollaczek) which I think reflects the division of everything into five perspectives for decision making in space and time, whereby "Every effect has had its cause; but not every cause has had its effects; so there is a critical point for deciding." Which means that causality needs to be comprehended in both directions. For example, at CERN they measure the effects and then they work backwards to figure out which of the many particles participated, which means that they go from effects to causes. I believe that for a particle interaction I will be able to describe a quantum narrative in five zones: asympotically free (Meixner), arisal of the wave function (Charlier), bound state (Laguerre), collapse of the wave function (Hermite), subsequent entanglement (Meixner-Pollaczek). I talk about that in my recent video "Quantum Narrative as Moments, Poles and Distributions of Orthogonal Sheffer Polynomials". I intend to write in a separate thread about the possibility of categorifying Schroedinger's equation. One very encouraging thing is that the combinatorics of the Hermite polynomials in terms of involutions is basically the same as for Wick's theorem in quantum field theory, which suggests that the whole combinatorial theory may lift up to quantum field theory.
Andrius Kulikauskas (Jan 14 2024 at 19:40):@Todd Trimble I found the book "Lectures on Buildings" by Mark Ronan. Whereas there is a book "The Structure of Spherical Buildings" by Richard M. Weiss. So I suppose you have both books?!
Todd Trimble (Jan 14 2024 at 20:42):Oh damn. I meant the second book. That was a slip on my part.
Todd Trimble (Jan 14 2024 at 20:47):The stuff about the combinatorics of Hermite polynomials and the like sounds very interesting. It's the sort of thing @John Baez knows about in connection with species (I have in mind things like his paper with Dolan, From Finite Sets to Finite Diagrams).
Andrius Kulikauskas (Apr 10 2024 at 10:19):@Todd Trimble Thank you, belatedly, for your post. Yes, I should study their paper From Finite Sets to Feynman Diagrams. I have been listening to @John Baez 's videos from This Week's Finds 15: combinatorics, groupoid cardinality and species to This Week's Finds 18: categorifying the quantum harmonic oscillator. In January, I was making good progress on thinking through a combinatorial alternative to the wave function. I realized that the probability distribution for the Meixner polynomials is a discrete step function, which makes it straightforward to interpret combinatorially, and that the Hermite polynomials (for the quantum harmonic oscillator) and the Laguerre polynomials (for the radial component of the hydrogen atom) have continuous distributions that are limiting cases of that step function. So I was able to calculate the probability density for finding a system in a state: But I need to work further to interpret it combinatorially from first principles, and then make sense of it physically.
But then I got sidetracked and now I want to focus on mastering Bott periodicity. I will start a thread about that. Some time back I found helpful your notes on The Super Brauer Group and Super Division Algebras. In November, I made a video about my current understanding, and recently finished preparing the transcript with slides, Bott Periodicity Models Consciousness? Preliminary Exploration. I hope you could take a look, maybe it will interest you, and I value your help to understand it all better.
My current plan is to get clear on how to present the Clifford algebras as matrix algebras for $$C_{0,k}$} for {k=0,\dots,8$$ and make a video about that. Then I want to calculate the groups and make a video about that. Then I want to do the calculations of the Lie group embeddings in the very helpful paper Michael Stone, Ching-Kai Chiu, Abhishek Roy. Symmetries, Dimensions, and Topological Insulators: the mechanism behind the face of the Bott clock. It's basically a concrete tutorial for physicists. I'll make a video about that. Then I want to understand their explanation of the CPT symmetry in that paper and make a video about that. That is where the metaphysics pops out of the math and I want to see how it may relate to the twosome, threesome, foursome that are central to my metaphysics. After I do all that, which seems straightforward and most illuminating for me about my "divisions of everything", then I want to push through the combinatorial alternative to the wave function, and also return to this question, What is geometry?
I started a new thread: Mastering Bott Periodicity.
Ibrahim Tencer (Apr 10 2024 at 18:53):By understanding geometry and geometries "cognitively" I mean metaphysically, but in other words, discovering the cognitive frameworks that underlie the key notions
@Andrius Kulikauskas You may be interested in my research on socionics. Socionics is traditionally thought of as a cognitive personality typology but it also has deep metaphysical structure.
Some thoughts: there is an analogy between space in geometry and a space of states (as in some kind of physical theory). So stasis is represented by a point, change by a line, and each dimension is a dichotomy (which can be formalized directly to model socionics types using cubes or hypercubes). The fundamental geometric distinction appears as here/there but there are many theres and only one here.
High/low is a distinction of power where the higher one dominates the lower one, and horizontal dimensions are akin to possibilities within a single framework. Projective geometry gives a way to understand dualities between concepts like introversion (which is static and pointlike) and extroversion (which is dynamic and linelike).
You can find some further information about this here and here but it requires some background in socionics to understand (for which see here and here).
I haven't read the whole thread yet but there are some very interesting comments that could be connected to this framework.
Andrius Kulikauskas (Apr 10 2024 at 19:37):@Ibrahim Tencer I am glad to learn of you and your research. Thank you for the links. This is the first time that I have heard of socionics and its founder, Aušra Augustinavičiūtė, who lived in Lithuania, where I do! Your metaphysical thinking may well be expressible in Wondrous Wisdom, my conceptual language, which I am identifying in advanced mathematics. With this in mind, I lead Math 4 Wisdom, an investigatory community for absolute truth, and you are welcome to join our email group.
My personality type (my category?) is that I wish to ignore (reject, deny?) personality types. I identify with Jesus's personality type as in the teaching Love your neighbor as yourself. I do like your analysis of the ways that people deal with conflicts and I have a keen interest in peacemaking. In general, I think we could connect on metaphysical explorations, including in geometry. I like your landscape of geometrical relationships.
I am a big fan of self-determination and self-definition. As an organizer, I find it very helpful to ask participants, What is your deepest value in life, which includes all of your other values? What are questions that you don't know the answer to, but wish to answer? What is your relationship with truth? I have collected answers from almost 1,000 people and they are surprisingly unique. A table of deepest values and relationships with truth further suggests that they are very much related. So I am interested to map out the landscape of truth, where people seek to find accord between form and content, between the questions asked by the Conscious and the answers given by the Unconscious. Your thinking may be very helpful. Let's explore!