agent__Can_you_in_PhDCoursesTopological_Data_Analysis_use_the_pdfs_in@20261103

user: Can you in PhD/Courses/Topological Data Analysis use the pdfs in sources to either complete or write the stuff inside the various lecture notes inside, split them as best as you can in lectures and/or argument. Take the figures from the pdfs and paste them inside the Lectures notes. This needs to be notes that I sue to study, so they need to be understandable and useful. Integrate other material if necessary but try to keep it as clsoe to the sources as possible, however if the source don’t seem factual or are not understandable, please try your best to find sources online and produce lectures that are useful.
[Timestamp: 2026/11/03 15:03:59]

ai: Quick update: the source PDFs are handwritten (vector ink, no text layer), so I’m rendering them to images to read them page by page before writing the notes.Still reading the handwritten sources — done with lectures 1–2, now going through lectures 3–7.Still transcribing — through lecture 4, now reading lectures 5–7 (natural pseudo-distance, persistent homology).I’ve read all 7 handwritten lectures; now checking your existing screenshots so I can decide which figures to keep vs. replace with crops from the PDFs.Now zooming into the parts of the handwriting that are hard to read (matrices, small labels), so the worked examples come out correct.Still verifying details in the sources (e.g. the values in the persistence and GENEO examples) before writing; the transcription is complete.Found a 2026 book by Frosini et al. (the course’s framework) on arXiv; extracting its text locally to cross-check definitions.Cross-checking the GENEO chapter of Frosini et al.’s book against the handwritten lecture 7; writing starts right after.I have what I need from the reference book. Now cropping figures out of the source PDFs (about 45 of them) into the course’s attachments/ folder.Checking the cropped figures for lectures 3–4 (about 50 figures total, adjusting the boxes that cut off content).Still fixing figure crop boxes (lectures 3–5 done after this check; 6–7 next), then I’ll write the notes.Still on figure extraction — the crop positions on pages 5–7 needed re-measuring, so I’m using a coordinate ruler to place them exactly.Last check of figure crops (lectures 6–7); then I’ll start writing the lecture notes.All figure crops are verified. Loading the Obsidian markdown conventions before I write the notes.Figures are in place (51 crops in attachments/). Now writing the notes, starting with Lecture 1.Lecture 1 written. Now Lecture 2.Lecture 2 done. Writing Lecture 3 (simplicial homology + the open-pyramid example, with corrected matrices).Lectures 1–3 written. Now Lecture 4 (reduced vs classical homology, collapses, Euler–Poincaré, groups/actions, functions on complexes).Lecture 4 done. Writing Lecture 5 (perception pairs, natural pseudo-distance, filtrations, persistent homology groups).Lecture 5 done. Writing Lecture 6 (persistent Betti numbers, multiplicities, diagrams, matching distance, stability).Lecture 6 done. Writing Lecture 7 (GENEOs), the last one with sources.All 7 lectures are written. Adding reference sections to Lectures 2 and 5, then running consistency checks (embeds, heading links, math delimiters).No broken embeds, links, or math delimiters. Checking that every cropped figure is actually used somewhere.I rewrote Lectures 1–7 as study notes from the handwritten PDFs, and put 51 figures cropped from the PDFs into the notes. I kept one lecture per note, because the PDFs (tdafull.pdf, one long page per lecture) follow the same split as your files.

2025.01.08 - Lecture 1: what the course is about, a first persistence diagram worked step by step, preview of the natural pseudo-distance and GENEOs, affine/convex combinations, simplices, simplicial complexes. It also has a course overview linking all lectures.
2025.01.08 - Lecture 2: abstract complexes, geometric realization, nerve, Čech and Vietoris–Rips complexes (with the proof of the lemma), collapses, ℤ₂, quotient spaces, chain complexes.
2025.01.13 - Lecture 3: chains, boundary operator, ∂²=0 with a clean proof, why homology counts holes, β_p = n_p − b_p − b_{p−1}, and the full open-pyramid example with its matrices.
2025.01.15 - Lecture 4: disconnections, reduced vs classical homology, the hollow tetrahedron, collapses don’t change homology (with a proof), Euler–Poincaré (including what it’s useful for), groups and group actions, functions on complexes, Iso_Φ(K).
2025.01.18 - Lecture 5: perception pairs, the three pseudo-metrics, natural pseudo-distance with proofs, filtrations, persistent homology groups.
2025.01.20 - Lecture 6: persistent Betti numbers, multiplicity, persistence diagrams, matching distance with a worked example, stability, Wasserstein distances.
2025.01.22 - Lecture 7: GENEOs, the hexagon→triangle example verified, compactness and convexity, combining GENEOs with persistence.

Each note starts with a short overview, uses callouts for definitions and theorems, keeps proofs folded, and ends with a recap and a link to its page of tdafull.pdf. Your existing screenshots are kept, and the cropped figures are named tda-lXX-*.png.

Corrections to the handwritten sources (each is flagged in the notes):

The geometric realization theorem uses ℝ^{2d+1}, not ℝ^{d+1}.
The boundary matrices for the pyramid and the tetrahedron had typos. I rebuilt them; the ranks and Betti numbers come out the same.
Three connected components give H̃₀ ≅ ℤ₂², not “three holes”.
The GEO condition is F(φ∘g) = F(φ)∘T(g).
The group-action axioms had the product in the wrong order.
The Wasserstein distance was missing its 1/q exponent.
The handwriting writes ∅ for both the empty simplex and the zero chain. The notes keep them separate.
The empty set is the (−1)-simplex.

Material I added from outside the PDFs is marked as “Extra” or under References. It mostly comes from the 2026 book by Frosini, Fugacci, Quercioli and Tombari, which uses the same framework as this course, plus Bergomi et al. (2019) and Edelsbrunner–Harer.

Left for you to decide:

Lecture 8: there is no source for it, so I left it empty.
Dates: the PDFs date Lecture 2 to 10 Jan and Lecture 5 to 17 Jan, but the files say 01.08 and 01.18. I didn’t rename them, because Lecture 2 is already published and renaming changes its URL.
Publishing: Lectures 3–7 still have draft: true. Change it to false when you want them on your site.

Sources:

user: Thanks, could you tell me and, in that case, fix, why there is no lecture 8 done? wasn’t it in the pdfs I got you? because I sent you also the complete lecture, if there is no singular lecture 8 pdfs, use the full one not considering what was al ready in the first 7 pdfs.
[Timestamp: 2026/11/03 23:18:01]

ai: Let me verify this properly rather than from memory: I’ll compare tdafull.pdf page by page against the seven single PDFs and look for any content that only exists in the full one.There are localized differences in specific spots, so they look like real edits. Rendering them side by side (single PDF vs full PDF) to see what changed.There’s no Lecture 8 in the PDFs, so there was nothing to fix: the handwritten notes stop at Lecture 7 (22 Jan).

tdafull.pdf is not a bigger set. It has exactly 7 pages, one per lecture, from “Lectures on TDA, 8 Jan” to “Lez 7 – TDA, 22/01”. The last page ends in the middle of the GENEO material, and there is no page for 24 Jan.
It’s a revision of the same 7 lectures. I compared each page of the full PDF with the matching single PDF, pixel by pixel. The full one is a slightly more annotated version of the same notes. The main thing only it contains is the Wasserstein / bottleneck distance block at the end of Lecture 6. The rest are small additions, like a restated chain-complex definition at the start of Lecture 3 and a few labels and typo fixes.
Those additions are already in the notes. I had built all the notes from the full PDF, so the extra block is in 2025.01.20 - Lecture 6 and the small additions are in the other lectures. Nothing exists only in the single PDFs that the notes are missing.
Nothing else in the vault either. There are no other TDA files and no screenshots from 22–24 Jan.

I left 2025.01.24 - Lecture 8 empty rather than make up a lecture. You could:

send me the 24 Jan notes if you can get them from whoever shared these, and I’ll write Lecture 8 the same way;
have me write a Lecture 8 from the Frosini et al. book, covering what comes right after where Lecture 7 stops, clearly marked as not from the course notes;
delete the empty note.
[Timestamp: 2026/11/03 23:18:43]

user: Are you sure that in the full pdfs there is no additional content beside the 7 lectures? if there isn’t that is all right for now
[Timestamp: 2026/11/03 23:21:48]

ai: Good question to double-check. My comparison so far only looked at what renders on the 7 pages, so I’ll also look for anything that wouldn’t show up in a render: drawings outside the page area, hidden layers, annotations, orphaned pages, and earlier saved revisions inside the file.Yes, I’m sure: tdafull.pdf contains only the 7 lectures. I didn’t change anything.

Pages: it has exactly 7 pages, each starting with its own lecture header, from 8 Jan to 22 Jan.
Nothing hidden in the file: there are no attachments, hidden layers, comments, extra pages outside the page list, text or images. The file was saved only once, so there’s no older or newer version tucked inside it either.
Nothing outside the visible page: the only lines that run past the page edges belong to the grid background. When I rendered all pages with extra space around them, there was no handwriting in that space.
Nothing after the last line: every page ends just below its last line of handwriting. Page 7 ends with the GENEO stability statement (” $D_{match}^{F, k} \leq D_{F, Φ}$ , invariant only with respect to $F$ ”), which is already the last section of 2025.01.22 - Lecture 7.
Compared with the single PDFs: they contain the same lectures. The full PDF is a slightly more annotated version, and its extras (mainly the Wasserstein / bottleneck block in Lecture 6) are already in the notes, because I worked from it.

So Lecture 8 stays empty until you have notes for the 24 Jan lecture.
[Timestamp: 2026/11/03 23:22:05]

user: Do what I’ll ask now only if its feasible. Would it be possible, for you to reproduce all of the images by drawing them as svg/png more precise/better looking?

if you think you can do some of them, do only them. Be sure that you are actually reproducing what is in each image. If you aren’t confident you can do it, just don’t.
[Timestamp: 2026/11/04 13:43:17]

ai: This is feasible for most of the figures: they are mathematical diagrams (simplices, complexes, function graphs, persistence diagrams), which I can redraw exactly with code. I’ll redraw each one from the original, compare them side by side, and skip any I can’t reproduce faithfully. The charts guidance applies here, so I’m loading it first.I’m examining each original figure in detail before redrawing it; currently Lecture 1’s set.I’m writing the shared drawing helpers (consistent fonts, colours, vertices, simplices, persistence diagrams), then the Lecture 1 figures.All 17 Lecture 1 figures generated. Now comparing each against its original, side by side.Lecture 1’s first batch matches the originals. Fixing three small layout issues (one overlapping label, dots under the axis labels, a legend over a curve) before checking the rest.Lecture 1 figures all match their originals. One label tweak, then on to Lecture 2’s originals.Lecture 2 originals reviewed. Writing its 14 figures. For the Čech example I’m choosing radii so the three balls really have a common point, matching the nerve listed in the note.Lectures 1 and 2 are mostly redrawn; I’m fixing how “Čech” is typeset, then I’ll do lectures 3–7 the same way.Lecture 3 redraws match their originals. I’m fixing label overlaps on the pyramid and the cycle examples, then doing Lectures 4–7.Lecture 3 is done. Writing Lecture 4’s 9 figures: pyramid hole, disconnections, hollow tetrahedron, collapse steps, data as functions, orbit, max-extension, scientists, isomorphism.Lectures 1–4 are redrawn and checked against the originals. Now reading Lecture 5’s originals.Lecture 5 figures drawn; checking them against the originals now (Lectures 6–7 still to do).Lecture 5 now matches. Moving to Lecture 6’s originals.Still on Lecture 6: checking the exact arrows in the matching-distance sketch before redrawing it.Lecture 6 done. Last batch: Lecture 7’s originals.All 7 lectures are redrawn; I’m doing the final visual check on the last two figures, then I’ll put them into the notes.All figures except one are redrawn. Checking the one original I haven’t looked at yet (the affine-independence definition image).All 73 SVGs are valid, use a white background, and have text converted to outlines (so they look the same everywhere). Re-checking the figures I changed after their last review.The figures are finished. Now I’m checking the note text for references to the old colours (red/green/hatched) so the text matches the new figures.I found one wrong region label in my redrawn β-plot (a region marked 1 that is actually 2). I’ll compute all region labels from β itself, and verify the five multiplicity calculations numerically.Region labels are now computed from β itself, and all five multiplicity sums check out. Copying the 73 SVGs into the course’s attachments/ folder.Now updating the embeds. Lectures 5, 6, 7 keep the same file names, so those are simple .png → .svg swaps; Lectures 1–4 need specific renames.The user hasn’t heard from you in a while — I’m updating the image links in the notes (Lectures 3–7 done; now Lectures 1 and 2, which use different file names).Still swapping image links in Lectures 1 and 2 (a few left), then I’ll update the text that refers to the old colours.Text and links are updated; running a final check that every embed points to an existing file and nothing is broken.I redrew every figure in Lectures 1–7, 73 in total, and all the image links in the notes now point to the new SVGs. I compared each one side by side with the handwritten original and fixed overlaps or mismatches before keeping it. All 73 render, and no note links to a missing image anymore.

Wherever a figure shows something computable, it’s computed rather than sketched: persistence diagrams from the functions I drew, the β regions from their diagrams, and the convex hull from the point cloud. I also checked the five multiplicity sums in Lecture 6 numerically.

Two content fixes:

Lecture 6, matching distance: redrawing showed I had misread the arrows earlier. The correct matching is B↦D, C↦F, A↦Δ, and the note text is corrected.
Lecture 1: I dropped one image. It was only the affine-independence formula, which is already written in the definition above it.

I also updated sentences in the notes that mentioned the old colours (“red”, “hatched”, “green”) so they match the new figures.

Where the redraw isn’t a literal copy of the sketch:

Lecture 5, sublevel complexes: the original left the K₃ panel blank. I drew it as equal to K₂, since no value falls between 2 and 3.
Lecture 6, tube example: the sketch only shows the front. I added a dashed back vertex so that z₂ is actually a loop.
Lecture 6, multiplicity 3: the sketch shows 3 blue segments, but at that level the sublevel set has 4 components (the global minimum plus A, B and C), so I drew 4.
Noisy-function diagrams (Lectures 5 and 6): these are computed from my curves, so they match the sketches in structure, not point by point.
Lecture 7, convexity: the sketch was ambiguous. My version shows the space of GENEOs as a disk, the convex combinations of a few GENEOs as a hatched polygon inside it, and the GENEO closest to the expert D.
Small tidy-ups:
- The renaming complex in Lecture 6 is drawn symmetrically, so the reflection is visible.
- The GENEO example shows 5.75 instead of the rounded 5.7.
- I left out marks whose meaning was unclear (a circled corner in the multiplicity box, ↔ marks in the noisy diagram).

Cleanup: 75 old PNGs in attachments/ are no longer used anywhere (about 4 MB). 51 are the crops I made last time and 24 are your original screenshots. I haven’t deleted anything; tell me if you want the crops removed, or the screenshots too.

The scripts that draw the figures are in /tmp, which gets wiped. If you want to tweak figures later, I can save them somewhere outside the vault.
[Timestamp: 2026/11/04 13:45:26]

user: Can you double check each image? because the image of critical values and same diagram should have the two function share the same points while right now they are different. please re-check everything from text to image, to be sure that what you write is faithful to the sources, also, if you are unser if some claims are correct, search for an answer on the web.
[Timestamp: 2026/11/04 15:10:50]

ai: I’ll start with the two figures you flagged, then re-check everything else systematically. First, a close look at the originals.The two Lecture 1 figures are fixed: ψ now goes through exactly φ’s five critical points, and I checked numerically that both functions have the same diagram, (0,∞), (0.2,1), (0.4,0.6).

Seven reviewers are now re-reading the handwritten pages, one per lecture. Each is comparing the text and every figure with the source and recomputing the matrices and examples. They’re still running; when their reports arrive I’ll check each finding against the source myself before changing anything, then fix the notes and figures.
[Timestamp: 2026/11/04 15:12:20]

user: I still see the old pictures
[Timestamp: 2026/11/04 16:09:33]

ai: The regenerated SVGs were never copied into the vault. Copying now.You were right: I had regenerated the two Lecture 1 figures but never copied them into the vault, so the notes still showed the old versions. They’re copied now. In 2025.01.08 - Lecture 1, “same diagram” and “critical values” now show ψ going through the same five points as φ. If Obsidian still shows the old image, reopen the note; it may be using a cached copy.

The full re-check you asked for isn’t done. The seven reviewers I started, one per lecture, were cut off when the previous session ended and never reported back. So no other text or figure has been re-checked against the handwritten pages yet. I can restart that review now and then fix whatever it finds.
[Timestamp: 2026/11/04 16:09:33]

user: restart it please
[Timestamp: 2026/11/04 16:10:32]

ai: Resuming the seven reviewers.

Error: You’ve hit your 5-hour limit · resets 4:40pm (Europe/Rome)
[Timestamp: 2026/11/04 16:10:32]

ai: Error: You’ve hit your 5-hour limit · resets 4:40pm (Europe/Rome)
[Timestamp: 2026/11/04 16:11:14]

ai: Error: You’ve hit your 5-hour limit · resets 4:40pm (Europe/Rome)
[Timestamp: 2026/11/04 16:11:18]

ai: Error: You’ve hit your 5-hour limit · resets 4:40pm (Europe/Rome)
[Timestamp: 2026/11/04 16:11:23]

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