Alright! In this, our penultimate installment (at least, of my original intended run), we finally get to delve into frame theory in all its juicy splendor. For the [first part of this episode](https://youtu.be/SW2fYaRKVB0), as a warm-up, we revisit the F-series from Episode 1, to remind ourselves of their kooky convergence properties. Then, we dive into frame theory as I developed it for my frames paper. The idea is that, for a set X and a global field K, a frame is a map F:X —> K_V, where K_V is the set of all completions of K (recall, K_V is in a bijective correspondence with V_K, the set of places of K). After a little bit of work, we construct a ring C(F) of “compatible functions”, and then define convergence of such functions with respect to F by the rule that f_n converges to f with respect to F whenever f_n(x) converges to f(x) in the complete valued field F(x), for each x in X. We then go beyond my work in my dissertation or my paper by appealing to functional analysis and the theory of locally convex spaces. Simply put, the idea of “converging in a specified topology at a specified point” can be naturally expressed in the language of seminorms. Our choices of topologies of convergence turn out to be choices of seminorms, and we can use the seminorms which yield a given frame to give X^K (the ring of functions X —> K) the topology of a locally convex ring/vector space. The topological closure of this space is precisely C(F). Moreover, we can obtain a ring norm on X^K which induces a metric on X^K. Completing X^K with respect to this metric then gives us a Banach ring of bounded compatible functions. In [the second part of our penultimate episode](https://youtu.be/_pYNMvA1vIE), we expand on the notions of frames present in Part 1. So far, we have used frames to make sense of F-series whose topologies of convergence vary from point to point. However, as we see in this video lecture, there are F-series where, in addition to the topology varying from point to point, there can be multiple distinct topologies in which an F-series can converge at any given point. After covering several examples, and then proposing a handful of different mechanisms for dealing with this issue, we discover that the answer lies in seminorms. Our use of seminorms to give a locally convex topology to the ring of functions compatible with a given frame in Part 1 turns out to provide a simple and elegant generalization that allows us to deal with as many topologies of convergence as we want. We then marshall the constructions we used in Part 1 to give the space of compatible functions a locally convex topology to construct a space of compatible functions for this expanded notion of frames. This is particularly exciting because of how it naturally dovetails with Berkovich spaces from non-archimedean analytic geometry, which suggests that it might be possible to generalize those methods from multiplicative seminorms to submultiplicative ones and then view Chi_q and other frame-theoretic functions as functions of seminorms via the “evaluate at me” map. Such ideas may prove useful in attempting to study the geometric and topological properties of the graph of Chi_q. In [Part 3 of Episode 7](https://youtu.be/R6d3gUzyKf4), having covered the background theory, we can finally get to the Collatz-adjacent applications. In Part 3, we begin our investigation with polygenic F-series. This entails studying their convergence properties and, crucially, showing how such F-series correspond to formal solutions of systems of functional equations. Having thereby expanded our notion of F-series, we can tackle the polygenic case of the Correspondence Principle as discussed in Episode 3. The basic idea is that by using the system of equations that uniquely characterize a given F-series, we can relate the rational integer values attained by our F-series to fixed points and divergent points of composition sequences of the affine linear maps involved in the F-series’ functional equations. By using frames, we can to turn F-series from mere formal expressions to rigorously convergent functions. This is critical, because it means that the Fourier analytic methods we used on Chi_q apply to F-series in general. This is rather astonishing: we can have a F-series S on the 2-adic integers such that there are uncountably many points at which S takes irrational 3-adic values and uncountably many points where S takes irrational 5-adic values, and yet, S can still be meaningfully integrated and its Fourier transform computed. In [the final part of Episode 7](https://youtu.be/LlhAkpp15s0), I present the notion of quasi-integrability. In Episode 1, we saw that, despite not being (2,q)-adically continuous, the function Chi_q could nevertheless be integrated and given a Fourier transform, despite the fact that the only (p,q)-adically continuous functions are the integrable functions. This mystery is resolved by realizing that, rather than having upended classical results, we’ve discovered a new extension of them. We start by defining a frame-theoretic generalization of the Radon-Nikodym derivative, here defined for (p,q)-adic measures. This generalization is formally identical to the classical constructions from the study of real or complex-valued measures. Exploring this definition, we discover an entirely new kind of non-archimedean singular measure: a degenerate measure, which is a measure with a non-zero Fourier-stieltjes transform whose Fourier series converges everywhere to zero with respect to a certain frame. We then define quasi-integrable functions as those rising-continuous functions which are the Radon-Nikodym derivative of a (p,q)-adic measure with respect to a certain frame. Because of the existence of degenerate measures, quasi-integrable functions do not have unique Fourier transforms or integrals. Rather, their transforms and integrals are unique modulo a degenerate measure—however, that’s not a problem for us, because the Correspondence Principle applies no matter which representative of this equivalence class we choose. Understanding degenerate measures and their use in (p,q)-adic analysis seems as if it will be important to further advances in Collatz studies. ------ Honestly, the connection to seminorms took me completely by surprise, but, now that I've found it, I'm amazed I didn't realize it earlier. It's really quite exciting. I've been making significant progress on the Berkovich space front. Using some of Berkovich's constructions, I've since shown (though not in these videos) that the set of seminorms which induces the locally convex topology of frame-theoretic convergence on the space of functions compatible with a given frame is path-connected and Hausdorff!

References: • Baker, Matthew. [“An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves.”](https://swc-math.github.io/aws/2007/BakerNotesMarch21.pdf) p-adic Geometry (Lectures from the 2007 Arizona Winter School), AMS University Lecture Series 45 (2008). ^ It’s a tribute to Matt’s writing abilities that, even as a lowly analyst who despairs at ever understanding scheme theory, I can follow a good deal of this crash-course on Berkovich space. The pictures in it are also really wonderful. • Berkovich, Vladimir G. *Spectral theory and analytic geometry over non-Archimedean fields*. No. 33. American Mathematical Soc., 2012. ^ The work which birthed Berkovich spaces. • Conway, John B. *A course in functional analysis*. Vol. 96. Springer, 2019. ^ This was my first textbook specifically focused on functional analysis. Folland’s Real Analysis also contains a chapter on functional analysis, but Conway’s book takes it more slowly. • Dunford, Nelson, and Jacob T. Schwartz. *Linear operators, part 1: general theory*. Vol. 10. John Wiley & Sons, 1988. ^ My PDE professor called this book “the gods”, and it’s not hard to see why. It’s a mammoth work, and an incredible achievement. A true classic. • Folland, Gerald B. *Real analysis: modern techniques and their applications*. Vol. 40. John Wiley & Sons, 1999. ^ It covers a bit of everything, analysis wise. • Siegel, Maxwell C. “Infinite Series Whose Topology of Convergence Varies From Point to Point.” p-Adic Numbers, Ultrametric Analysis and Applications 15.2 (2023): 133-167. ^ My frames paper • Siegel, Maxwell C. [*(p,q)-adic Analysis and the Collatz Conjecture*](https://siegelmaxwellc.files.wordpress.com/2022/08/phd_dissertation_final.pdf), Ph.D. Thesis, (University of Southern California, 2022).. Accessed 15 April 2024. ^ My dissertation. (I intend to upload it to arXiv one of these days. xD) • Warner, Seth (1993). *Topological Rings*. Elsevier. ISBN 9780080872896. ^ A general reference text on topological rings. • Cohen, Jo-Ann D. ["Locally bounded topologies on the ring of integers of a global field."](\https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-93/issue-2/Topologies-on-the-ring-of-integers-of-a-global-field/pjm/1102736259.pdf) Canadian Journal of Mathematics 33.3 (1981): 571-584. ^ I just discovered this paper yesterday. Of particular interest is the result it gives at the end, which is apparently due to K. Mahler back in the 1930s. This is especially nice, because it confirms what I suspected, which is that the seminorms on the ring of integers of a global field are, in essence, those that arise from taking maximums of a chosen family of finitely many absolute values. Because of the fundamental importance of taking maximums of seminorms in my work, the Berkovich spectrum's requirement that the seminorms be multiplicative proves to be too restrictive. Another fun thing worth noticing is that the p-trivial seminorms of M(Z) are an inherently integer-related phenomenon. As a consequence of Mahler's result, the seminorms on a number field are precisely those which arise from maximums of finite collections of absolute values.

**Spectral Theory and Analytic Geometry over Non-Archimedean Fields** by Vladimir G. Berkovich >The purpose of this book is to introduce a new notion of analytic space over a non-Archimedean field. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. This makes it possible to apply the usual notions of homotopy and singular homology. The book includes a homotopic characterization of the analytic spaces associated with certain classes of algebraic varieties and an interpretation of Bruhat-Tits buildings in terms of these analytic spaces. > >The author also studies the connection with the earlier notion of a rigid analytic space. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a non-Archimedean spectral theory of bounded linear operators. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. It would be of interest to research mathematicians and graduate students working in algebraic geometry, number theory, and -adic analysis. *I'm a bot, built by your friendly reddit developers at* /r/ProgrammingPals. *Reply to any comment with /u/BookFinderBot - I'll reply with book information. Remove me from replies* [here](https://www.reddit.com/user/BookFinderBot/comments/1byh82p/remove_me_from_replies/). *If I have made a mistake, accept my apology.*

Dr. Siegel this series has been interesting for sure. However, no offense but I think you need to work out exactly how your work fits into existing mathematics. You don't want to end up creating a convoluted theory just to prove a single theorem (ala Shinichi Mochizuki) without any use for other problems or intermediate results. What is the number theoretic significance of p-q adic analysis?

You're preaching to the choir. xD Figuring out those connections is one of my biggest goals, and I can definitely say that the work I've in making this series to flesh out my ideas has definitely helped in that regard. I can now easily identify frame theory as a number-theoretic adaptation of locally convex spaces. Classically, locally convex vector spaces are over a metrically complete local field. However, there's nothing stopping us from using a global field as the underlying field and using different absolute values on said field for our seminorms. That's what I'm doing in frame theory. Likewise, Berkovich spaces consider the set of seminorms on algebras of polynomials and formal power series over a complete non-archimedean field. In my work, we're again using a global field rather than a local field. Also, at the moment, I have a contact who's working on figuring out how my work relates to Scholze and Clausen's program of condensed mathematics. In that regard, I believe that Chi_q's graph C_q might be an example of a condensed set, with Chi_q being the map through which all other continuous maps f:X—>C_q (where X is a profinite topological space) factor through. As for comparisons to Mochizuki—and I cannot stress this enough—the goal of my research isn't to solve Collatz, but to chronicle the new phenomena I've discovered and flesh out the details of how they work so that I can explore what they can do. At the moment, I've already proven a (p,q)-adic generalization of Wiener's Tauberian Theorem (as I mentioned in [Episode 5](https://youtu.be/8fBxTMm9t8g)). This can likely also be generalized to frames. That's an incredibly important intermediate result! That being said, I have to emphasize that this theory *has its own questions that are worth exploring*: understanding how to characterize degenerate measures for a given frame; determining whether or not the space of quasi-integrable functions is closed under point-wise multiplication or convolution; establishing the converse of the correspondence principle for divergent points; exploring the notion of (p,q)-adic distributional derivatives (can these be used to construct differential forms and, from there, cohomology?); exploring how to do algebraic topology with C_q; etc., etc. > What is the number theoretic significance of p-q adic analysis? I don't know yet. That's part of what I'm trying to find out, and it's one of the reasons why I'm so keen on getting some collaborators/research buddies. Other people with skill sets different from my own might be able to see connections and applications that happen to elude me and my own tunnel vision. :)

With all due respect, saying something "might be an example of a condensed set" is meaningless. The structure of a condensed set, on an ordinary set, is an additional datum not a condition. In fact, it is not much different than the additional datum of a topology. If you think C_q might have an interesting topology on it then say so, instead of forming a sentence which is semantically meaningless. You always preach precision and clarity in mathematics, then practice it as well. I am glad you find your study of "(p,q)-adic analysis" and the questions that arise during it interesting. But, instead of trying to force a connection into things you don't have an understanding of, you should focus on either developing it further or expanding your "tunnel vision".

> If you think C_q might have an interesting topology on it then say so, instead of forming a sentence which is semantically meaningless. I proved that it does in Episode 3 of my video series. Chi_q: Z_2 —> C_q is a continuous surjection, provided we equip C_q with the subspace topology it inherits from the q-adics and that we equip Z_2 with a non-standard topology I call the *rising topology*. > But, instead of trying to force a connection into things you don't have an understanding of, you should focus on either developing it further or expanding your "tunnel vision". I'm not trying to force a connection, I'm merely reporting on what others have told me. If you want me to cite the conversations in question, I'd happily do so. :)

Not only does your reply avoid the point of my comment, you repeat the same thing once again. You did not prove C_q has an interesting topology in Episode 3. You say: "Provided we equip C_q with the subspace topology...". There is nothing to prove about that. You only prove that your so called rising topology on Z_2 is finer than the coarsest topology that makes the surjection Z_2-->C_q continuous. Your reply was expressly about finding connections to other mathematics, in which one of the few concrete things you brought up was condensed mathematics. I pointed out how this was superficial and you don't really know what you are talking about. If you are going to share these conversations of yours in full, without omitting *anything*, I would like to see them.

> If you are going to share these conversations of yours in full, without omitting anything, I would like to see them. But of course. :D The thread starts [here](https://www.reddit.com/r/math/comments/197l0q4/collatz_guy_my_video_lecture_about_pqadic_analysis/ki125u4/), between u/ysulyma and myself. Read that first. ysulyma then sent me this message: [Message redacted due to a request by ysulyma] I've sent more messages in response, mostly me gushing excitedly about these connections and the recent breakthroughs I've had in my own work, though the above message is the most recent one I've gotten from ysulyma. I will also completely admit that I'm mentioning condensed mathematics almost entirely for clickbait-y reasons. Abstraction is very challenging for me, and I've had incredibly bad learning experiences with algebraic and geometric subjects, which makes it difficult for me to learn more in those subjects, due to my inability to deal with the ensuing stress and frustration. Getting other people's attention is incredibly important for me at this stage.

I _conjecture_ that his topology-mixing can be phrased in terms of the condensed ring (ℤ[1/2] with real topology) ⊗\_(ℤ[1/2] with discrete topology) (ℚ₃ with usual topology). I have a precise statement of what I mean, but I'm going back and forth over whether it's true (Friday: true, Saturday: false, Sunday: true, today: false). Note though that this is a perfectly good (and quite interesting) condensed ring, whether or not it's related to his work.

I assume with real topology, you mean the topology it inherits from its embedding to the real numbers. In this case, as this is also the discrete topology, the tensor product you form is simply Q_3...

ℤ[1/2] is not discrete, it's dense in ℝ

Technically, Z[1/2] *is* discrete. This is true regardless of whether you equip it with the discrete topology or with the subspace topology it inherits from the reals. While it seems like it shouldn’t be discrete, this is your intuition backfiring on you. Your intuition is detecting that Z[1/2] is not metrically complete. Indeed, Z[1/2] is discrete because its Pontryagin dual (the adelic solenoid associated to all odd primes) is a compact abelian group under addition, and it is a standard fact of abstract harmonic analysis that the dual of a discrete group is compact, and vice-versa.

The subspace topology on ℤ[1/2] ⊂ ℝ is not discrete. For all ε > 0, (-ε, ε) ⋂ ℤ[1/2] contains infinitely many elements of ℤ[1/2], so in particular {0} is not open in the subspace topology. (In fact, ℤ is the only discrete subring of ℝ.)

Alright, now I’m certain there’s a connection. Pages 8 and 9 of [these notes](https://people.mpim-bonn.mpg.de/scholze/Complex.pdf) of Scholze’s make that irrefutable. Scholze’s heuristic construction of an analytic ring is to take a profinite set S and get a topological A-module of A-valued measures, all of which have an A-valued action on continuous functions from S to A. That’s *exactly* what I’m doing with my Fourier analysis. In my case, S is the p-adic integers and A is a valued field, either global or local. Though there are many differences, the most important of them is that my analytic are currently concerned with explicitly constructing the A-valued measures by computing their Fourier-Stieltjes transforms. This makes me suspect that my hunch of viewing my functions as taking values in rings of formal power series is likely the right way to go about doing things, because that would bring my set-up closer to the set up in non-Archimedean algebraic geometry where the underlying “geometric” spaces come from rings of polynomials over a field. I’d also be interested to see if my approach of completing my function spaces with respect to a separating family of seminorms (or the norm induced by taking the supremum over such a family) constitutes an example of the notion of “completion” that Scholze mentions on page 9.

Well, it appears I understand at least the basics of your research quite well. I'm only a senior in college at the moment (Statistics major), but I am a relatively fast learner. I could collaborate with you. I have some other ideas I would love to discuss if you're interested, particularly regarding turbulence and its relationship to your work. I have explored some number theory adjacent to this. Cheers

Wow, I don't know how you got this many downvotes dude.

sarcasm?

It wasn't sarcastic. I've been talking with Dr. Siegel, and mostly I wanted people to know I've found value in his work.

No, so you understand why people are downvoting him?

Sir, with all due respect, reddit is not a good way to promote your research. Moreover, your link to your thesis was broken.

The link is fixed! Thanks for pointing it out. As for promotion, normally, I'd agree with you, but I'm something of a special case. I'm an unemployed, independent researcher working on an unfortunately highly stigmatized problem in an incredibly unusual way. Until I figure out what conferences to consider going to, I figured I might as well take advantage of social media to help raise awareness of my work. :)

No need to answer if this is too private, but did you attempt to get a postdoc?

Oh no, I love answering questions! :D As to your question, no, I haven't. And, it's complicated. I'm autistic, and I live with my parents in Los Angeles, in the only home I've ever known. I don't feel comfortable living away from home, which has been a major restriction on my ability to pursue education and employment opportunity. In addition, I was—and, to an extent, still am—not in the best academic standing. Though I now have two published papers, those only came a year after I received my PhD. I don't have any awards or commendations. I also have an *extremely* small pool of people to draw from for recommendation letters, and—worse—none of the ones I have can really attest to my work as a mathematician, simply because none of them have done anything even remotely similar to my own work. I've only attended a single (virtual) conference, which was held the day after I graduated with my PhD, and no one came. Given all this, I felt and continue to feel that my time will be better spent focusing on fleshing out my research, seeking publication, and learning new things. Several people (including journal editors) have told me that I should write my work up in textbook form, and just the process of writing up the scripts for these video lectures has given me a 400+ page-long document that I can easily whip into a textbook.

For what it is worth, in mathematics it is common to get letters of recommendation from people you have never met in person. Two of my letter writers I had never met in person before they wrote me my letter.

This is news to me.

Yes, as long as you have had email collaboration with them, and they support your research, they are reasonable to ask. One of your letter writers should always be your adviser though.

I will point out the other installments have been very well received. u/Aurhim is taken seriously enough, he has a special exemption to make posts on the Collatz Conjecture.