#combinatorics — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #combinatorics, aggregated by home.social.
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Alright, future engineers!
**Combinations:** Ways to *choose* items from a set where the order *doesn't* matter.
Ex: Picking 3 teammates from 5 friends: `C(5,3) = 10` ways.
Pro-Tip: If swapping items doesn't create a new outcome, it's a combination!
#DiscreteMath #Combinatorics #STEM #StudyNotes -
A while ago, I wrote about Welter's game: a simple combinatorial game where two players take turns moving coins down a line until no moves are possible. It has some curious connections to coding theory, but for our purposes, it's just a little game to play over the holiday.
However, the game setup implicitly discriminates against people who don't have any coins, and we can't have that, so here's a web version: https://welter.fuglede.dk/
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https://arxiv.org/abs/2603.12358
Here is the *third* manuscript coming out of the "Topics in Ramsey theory" online-only problem-solving session (https://sparse-graphs.mimuw.edu.pl/doku.php?id=sessions:2025sessions:2025session1) of the Sparse (Graphs) Coalition, which took place less than a year ago.
It is still surprising to realise what one can make of such events, if they are set up well.
#combinatorics #remoteconferences #graphtheory #extremalcombinatorics #openscience
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As part of the CWI thematic research semester programme Phase Transitions in Combinatorics, Algorithms and Probability (PhaseCAP), we organise a series of three colloquia in Amsterdam. Please register via the links below if you want to attend. Registration closes a week before the meeting or when capacity is reached.
Thursday, 2 April — PhaseP colloquium
13:30–14:00 Coffee reception
14:00–15:00 Christina Goldschmidt (Oxford): Stable trees
15:00–16:00 Tom Bohman (Carnegie Mellon): Notes on two-point concentration in random graphs
16:00–17:30 Drinks receptionFriday, 17 April — PhaseC colloquium
13:30–14:00 Coffee reception
14:00–15:00 Penny Haxell (Waterloo): Algorithms for Independent Transversals and Reconfiguration
15:00–16:00 Rob Morris (IMPA): Recent results in Ramsey theory
16:00–17:30 Drinks receptionFriday, 29 May — PhaseA colloquium
14:00–15:00 Leslie Goldberg (Oxford): Fundamental Instability of Backoff Protocols
15:00–16:00 Amin Coja-Oghlan (Dortmund): The cavity method
16:00–17:30 Drinks receptionPlease feel free to share this announcement with colleagues who may be interested.
#combinatorics #probability #algorithms #amsterdam #phasetransitions #CWI
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📑 A new paper by CPC-CG members introduces the first method that can predict how many relatives of any kind a person is likely to have at different points in their life, and how likely each outcome is:
https://www.demographic-research.org/articles/volume/54/9#demography #kinship #mathematicaldemography #populationstudies #lifeCourse #mortality #fertility #probability #matrixalgebra #combinatorics #convolution #kin #familyStructure #analyticModel #populationResearch #population #family #familystructures #demographicforecasting
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📑 A new paper by CPC-CG members introduces the first method that can predict how many relatives of any kind a person is likely to have at different points in their life, and how likely each outcome is:
https://www.demographic-research.org/articles/volume/54/9#demography #kinship #mathematicaldemography #populationstudies #lifeCourse #mortality #fertility #probability #matrixalgebra #combinatorics #convolution #kin #familyStructure #analyticModel #populationResearch #population #family #familystructures #demographicforecasting
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📑 A new paper by CPC-CG members introduces the first method that can predict how many relatives of any kind a person is likely to have at different points in their life, and how likely each outcome is:
https://www.demographic-research.org/articles/volume/54/9#demography #kinship #mathematicaldemography #populationstudies #lifeCourse #mortality #fertility #probability #matrixalgebra #combinatorics #convolution #kin #familyStructure #analyticModel #populationResearch #population #family #familystructures #demographicforecasting
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📑 A new paper by CPC-CG members introduces the first method that can predict how many relatives of any kind a person is likely to have at different points in their life, and how likely each outcome is:
https://www.demographic-research.org/articles/volume/54/9#demography #kinship #mathematicaldemography #populationstudies #lifeCourse #mortality #fertility #probability #matrixalgebra #combinatorics #convolution #kin #familyStructure #analyticModel #populationResearch #population #family #familystructures #demographicforecasting
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📑 A new paper by CPC-CG members introduces the first method that can predict how many relatives of any kind a person is likely to have at different points in their life, and how likely each outcome is:
https://www.demographic-research.org/articles/volume/54/9#demography #kinship #mathematicaldemography #populationstudies #lifeCourse #mortality #fertility #probability #matrixalgebra #combinatorics #convolution #kin #familyStructure #analyticModel #populationResearch #population #family #familystructures #demographicforecasting
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This 2-min piece was composed on a musical dodecahedron at dodecahedron.newritual.com — 12 faces, 12 pitch classes, 7 modes. Each face holds a unique key. How many possible dodecahedra are there? Burnside's lemma meets music theory: ~110 quadrillion distinct configurations, after accounting for rotational symmetry. This is just one. #musictheory #dodecahedron #combinatorics #math #generativemusic #composition
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BlueSky’s Solution To Moderating Is Moderating Without Moderating via Social Proximity
I have noticed a lot of people are confused about why some posts don’t show up on threads, though they are not labeled by the moderation layer. Bluesky has begun using what it calls social neighborhoods (or network proximity) as a ranking signal for replies in threads. Replies from people who are closer to you in the social graph, accounts you follow, interact with, or share mutual connections with, are prioritized and shown more prominently. Replies from accounts that are farther away in that network are down-ranked. They are pushed far down the thread or placed behind “hidden replies.”
Each person gets their own unique view of a thread based on their social graph. It creates the impression that replies from distant users simply don’t exist. This is true even though they’re still technically public and viewable if you expand the thread or adjust filters. Bluesky is explicitly using features of subgraphs to moderate without moderating. Their reasoning is that if you can’t see each other, you can’t harass each other. Ergo, there is nothing to moderate.
Bluesky mentions that here:
https://bsky.social/about/blog/10-31-2025-building-healthier-social-media-update
As a digression, I’m not going to lie: I really enjoyed working on software built on the AT protocol, but their fucking users are so goddamn weird. It’s sort of like enjoying building houses, but hating every single person who moves into them. But, you don’t have to deal with them because you’re just the contractor. That is how I feel about Bluesky. I hate the people. I really like the protocol and infrastructure.
I sort of am a sadist who does enjoy drama, so I do get schadenfreude from people with social media addictions and parasocial fixations who reply to random people on Bluesky, because they don’t realize their replies are disconnected from the author’s thread unless that person is within their network. They aren’t part of the conversation they think they are. They’re algorithmically isolated from everyone else. Their replies aren’t viewable from the author’s thread because of how Bluesky handles social neighborhoods.
Bluesky’s idea of social neighborhoods is about grouping users into overlapping clusters based on real interaction patterns rather than just the follow graph. Unlike Twitter, it does not treat the network as one big public square. Instead, it models networks of “social neighborhoods” made up of people you follow, people who follow you, people you frequently interact with, and people who are closely connected to those groups. They’re soft, probabilistic groupings rather than strict labels.
Everyone does not see the same replies. Bluesky is being a bit vague with “hidden.” Hidden means your reply is still anchored to the thread and can be expanded. There is another way Bluesky can handle this. Bluesky uses social neighborhoods to judge contextual relevance. Replies from people inside or near your social neighborhood are more likely to be shown inline with a thread, expanded by default, or served in feeds. Replies from outside your neighborhood are still public and still indexed, but they’re treated as lower-context contributions.
Basically, if you reply to a thread, you will see it anchored to the conversation, and everyone will see it in search results, as a hashtag, or from your profile, but it will not be accessible via the thread of the person you were replying to. It is like shadow-banning people from threads unless they are strongly networked.
Because people have not been working with the AT Protocol like I have, they assume they are shadow-banned across the entire Bluesky app view. No—everyone is automatically shadow-banned from everyone else unless they are within the same social neighborhood. In other words, you are not part of the conversation you think you are joining because you are not part of their social group.
Your replies will appear in profiles, hashtag feeds, or search results without being visually anchored to the full thread. Discovery impressions are neighborhood-agnostic: they serve content because it matches a query, tag, or activity stream. Once the reply is shown, the app then decides whether it’s worth pulling in the rest of the conversation for you. If the original author and most participants fall outside your neighborhood, Bluesky often chooses not to expand that context automatically.
Bluesky really is trying to avoid having to moderate, so this is their solution. Instead of banning or issuing takedown labels to DIDs, the system lets replies exist everywhere, but not in that particular instance of the thread.
I find this ironic because a large reason why many people are staying on Bluesky and not moving to the fediverse—thank God, because I do not want them there—is discoverability, virality, and engagement.
In case anyone is asking how I know so much about how these algorithms work: I was a consultant on a lot of these types of algorithms, so I certainly hope I’d know how they work, lol. No, you get no more details about the work I’ve done. I have no hand in the algorithm Bluesky is using, but I have proposed and implemented that type of algorithm before.
I have an interest in noetics and the noosphere. A large amount of my ontological work is an extension of my attempts to model domains that have no spatial or temporal coordinates. The question is how do you generalize a metric space that has no physically, spatial properties. I went to school to try to formalize those ideas. Turns out they’re rather useful for digital social networks, too. The ontological analog to spatial distance, when you have no space, is a graph of similarities.
This can be modeled by representing each item as a node in a weighted graph, where edges are weighted by dissimilarity rather than similarity. Highly similar items are connected by low-weight edges, while less similar items are connected by higher-weight edges. Distances in the graph, computed using standard shortest-path algorithms, then correspond to degrees of similarity. Closely related items are separated by short path lengths, while increasingly dissimilar items require longer paths through the graph. It turns out that attempts to generalize metric spaces for noetic domains—to model noetic/psychic spaces—are actually pretty useful for social media algorithms, lol.
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BlueSky’s Solution To Moderating Is Moderating Without Moderating via Social Proximity
I have noticed a lot of people are confused about why some posts don’t show up on threads, though they are not labeled by the moderation layer. Bluesky has begun using what it calls social neighborhoods (or network proximity) as a ranking signal for replies in threads. Replies from people who are closer to you in the social graph, accounts you follow, interact with, or share mutual connections with, are prioritized and shown more prominently. Replies from accounts that are farther away in that network are down-ranked. They are pushed far down the thread or placed behind “hidden replies.”
Each person gets their own unique view of a thread based on their social graph. It creates the impression that replies from distant users simply don’t exist. This is true even though they’re still technically public and viewable if you expand the thread or adjust filters. Bluesky is explicitly using features of subgraphs to moderate without moderating. Their reasoning is that if you can’t see each other, you can’t harass each other. Ergo, there is nothing to moderate.
Bluesky mentions that here:
https://bsky.social/about/blog/10-31-2025-building-healthier-social-media-update
As a digression, I’m not going to lie: I really enjoyed working on software built on the AT protocol, but their fucking users are so goddamn weird. It’s sort of like enjoying building houses, but hating every single person who moves into them. But, you don’t have to deal with them because you’re just the contractor. That is how I feel about Bluesky. I hate the people. I really like the protocol and infrastructure.
I sort of am a sadist who does enjoy drama, so I do get schadenfreude from people with social media addictions and parasocial fixations who reply to random people on Bluesky, because they don’t realize their replies are disconnected from the author’s thread unless that person is within their network. They aren’t part of the conversation they think they are. They’re algorithmically isolated from everyone else. Their replies aren’t viewable from the author’s thread because of how Bluesky handles social neighborhoods.
Bluesky’s idea of social neighborhoods is about grouping users into overlapping clusters based on real interaction patterns rather than just the follow graph. Unlike Twitter, it does not treat the network as one big public square. Instead, it models networks of “social neighborhoods” made up of people you follow, people who follow you, people you frequently interact with, and people who are closely connected to those groups. They’re soft, probabilistic groupings rather than strict labels.
Everyone does not see the same replies. Bluesky is being a bit vague with “hidden.” Hidden means your reply is still anchored to the thread and can be expanded. There is another way Bluesky can handle this. Bluesky uses social neighborhoods to judge contextual relevance. Replies from people inside or near your social neighborhood are more likely to be shown inline with a thread, expanded by default, or served in feeds. Replies from outside your neighborhood are still public and still indexed, but they’re treated as lower-context contributions.
Basically, if you reply to a thread, you will see it anchored to the conversation, and everyone will see it in search results, as a hashtag, or from your profile, but it will not be accessible via the thread of the person you were replying to. It is like shadow-banning people from threads unless they are strongly networked.
Because people have not been working with the AT Protocol like I have, they assume they are shadow-banned across the entire Bluesky app view. No—everyone is automatically shadow-banned from everyone else unless they are within the same social neighborhood. In other words, you are not part of the conversation you think you are joining because you are not part of their social group.
Your replies will appear in profiles, hashtag feeds, or search results without being visually anchored to the full thread. Discovery impressions are neighborhood-agnostic: they serve content because it matches a query, tag, or activity stream. Once the reply is shown, the app then decides whether it’s worth pulling in the rest of the conversation for you. If the original author and most participants fall outside your neighborhood, Bluesky often chooses not to expand that context automatically.
Bluesky really is trying to avoid having to moderate, so this is their solution. Instead of banning or issuing takedown labels to DIDs, the system lets replies exist everywhere, but not in that particular instance of the thread.
I find this ironic because a large reason why many people are staying on Bluesky and not moving to the fediverse—thank God, because I do not want them there—is discoverability, virality, and engagement.
In case anyone is asking how I know so much about how these algorithms work: I was a consultant on a lot of these types of algorithms, so I certainly hope I’d know how they work, lol. No, you get no more details about the work I’ve done. I have no hand in the algorithm Bluesky is using, but I have proposed and implemented that type of algorithm before.
I have an interest in noetics and the noosphere. A large amount of my ontological work is an extension of my attempts to model domains that have no spatial or temporal coordinates. The question is how do you generalize a metric space that has no physically, spatial properties. I went to school to try to formalize those ideas. Turns out they’re rather useful for digital social networks, too. The ontological analog to spatial distance, when you have no space, is a graph of similarities.
-
BlueSky’s Solution To Moderating Is Moderating Without Moderating via Social Proximity
I have noticed a lot of people are confused about why some posts don’t show up on threads, though they are not labeled by the moderation layer. Bluesky has begun using what it calls social neighborhoods (or network proximity) as a ranking signal for replies in threads. Replies from people who are closer to you in the social graph, accounts you follow, interact with, or share mutual connections with, are prioritized and shown more prominently. Replies from accounts that are farther away in that network are down-ranked. They are pushed far down the thread or placed behind “hidden replies.”
Each person gets their own unique view of a thread based on their social graph. It creates the impression that replies from distant users simply don’t exist. This is true even though they’re still technically public and viewable if you expand the thread or adjust filters. Bluesky is explicitly using features of subgraphs to moderate without moderating. Their reasoning is that if you can’t see each other, you can’t harass each other. Ergo, there is nothing to moderate.
Bluesky mentions that here:
https://bsky.social/about/blog/10-31-2025-building-healthier-social-media-update
As a digression, I’m not going to lie: I really enjoyed working on software built on the AT protocol, but their fucking users are so goddamn weird. It’s sort of like enjoying building houses, but hating every single person who moves into them. But, you don’t have to deal with them because you’re just the contractor. That is how I feel about Bluesky. I hate the people. I really like the protocol and infrastructure.
I sort of am a sadist who does enjoy drama, so I do get schadenfreude from people with social media addictions and parasocial fixations who reply to random people on Bluesky, because they don’t realize their replies are disconnected from the author’s thread unless that person is within their network. They aren’t part of the conversation they think they are. They’re algorithmically isolated from everyone else. Their replies aren’t viewable from the author’s thread because of how Bluesky handles social neighborhoods.
Bluesky’s idea of social neighborhoods is about grouping users into overlapping clusters based on real interaction patterns rather than just the follow graph. Unlike Twitter, it does not treat the network as one big public square. Instead, it models networks of “social neighborhoods” made up of people you follow, people who follow you, people you frequently interact with, and people who are closely connected to those groups. They’re soft, probabilistic groupings rather than strict labels.
Everyone does not see the same replies. Bluesky is being a bit vague with “hidden.” Hidden means your reply is still anchored to the thread and can be expanded. There is another way Bluesky can handle this. Bluesky uses social neighborhoods to judge contextual relevance. Replies from people inside or near your social neighborhood are more likely to be shown inline with a thread, expanded by default, or served in feeds. Replies from outside your neighborhood are still public and still indexed, but they’re treated as lower-context contributions.
Basically, if you reply to a thread, you will see it anchored to the conversation, and everyone will see it in search results, as a hashtag, or from your profile, but it will not be accessible via the thread of the person you were replying to. It is like shadow-banning people from threads unless they are strongly networked.
Because people have not been working with the AT Protocol like I have, they assume they are shadow-banned across the entire Bluesky app view. No—everyone is automatically shadow-banned from everyone else unless they are within the same social neighborhood. In other words, you are not part of the conversation you think you are joining because you are not part of their social group.
Your replies will appear in profiles, hashtag feeds, or search results without being visually anchored to the full thread. Discovery impressions are neighborhood-agnostic: they serve content because it matches a query, tag, or activity stream. Once the reply is shown, the app then decides whether it’s worth pulling in the rest of the conversation for you. If the original author and most participants fall outside your neighborhood, Bluesky often chooses not to expand that context automatically.
Bluesky really is trying to avoid having to moderate, so this is their solution. Instead of banning or issuing takedown labels to DIDs, the system lets replies exist everywhere, but not in that particular instance of the thread.
I find this ironic because a large reason why many people are staying on Bluesky and not moving to the fediverse—thank God, because I do not want them there—is discoverability, virality, and engagement.
In case anyone is asking how I know so much about how these algorithms work: I was a consultant on a lot of these types of algorithms, so I certainly hope I’d know how they work, lol. No, you get no more details about the work I’ve done. I have no hand in the algorithm Bluesky is using, but I have proposed and implemented that type of algorithm before.
I have an interest in noetics and the noosphere. A large amount of my ontological work is an extension of my attempts to model domains that have no spatial or temporal coordinates. The question is how do you generalize a metric space that has no physically, spatial properties. I went to school to try to formalize those ideas. Turns out they’re rather useful for digital social networks, too. The ontological analog to spatial distance, when you have no space, is a graph of similarities.
This can be modeled by representing each item as a node in a weighted graph, where edges are weighted by dissimilarity rather than similarity. Highly similar items are connected by low-weight edges, while less similar items are connected by higher-weight edges. Distances in the graph, computed using standard shortest-path algorithms, then correspond to degrees of similarity. Closely related items are separated by short path lengths, while increasingly dissimilar items require longer paths through the graph. It turns out that attempts to generalize metric spaces for noetic domains—to model noetic/psychic spaces—are actually pretty useful for social media algorithms, lol.
-
BlueSky’s Solution To Moderating Is Moderating Without Moderating via Social Proximity
I have noticed a lot of people are confused about why some posts don’t show up on threads, though they are not labeled by the moderation layer. Bluesky has begun using what it calls social neighborhoods (or network proximity) as a ranking signal for replies in threads. Replies from people who are closer to you in the social graph, accounts you follow, interact with, or share mutual connections with, are prioritized and shown more prominently. Replies from accounts that are farther away in that network are down-ranked. They are pushed far down the thread or placed behind “hidden replies.”
Each person gets their own unique view of a thread based on their social graph. It creates the impression that replies from distant users simply don’t exist. This is true even though they’re still technically public and viewable if you expand the thread or adjust filters. Bluesky is explicitly using features of subgraphs to moderate without moderating. Their reasoning is that if you can’t see each other, you can’t harass each other. Ergo, there is nothing to moderate.
Bluesky mentions that here:
https://bsky.social/about/blog/10-31-2025-building-healthier-social-media-update
As a digression, I’m not going to lie: I really enjoyed working on software built on the AT protocol, but their fucking users are so goddamn weird. It’s sort of like enjoying building houses, but hating every single person who moves into them. But, you don’t have to deal with them because you’re just the contractor. That is how I feel about Bluesky. I hate the people. I really like the protocol and infrastructure.
I sort of am a sadist who does enjoy drama, so I do get schadenfreude from people with social media addictions and parasocial fixations who reply to random people on Bluesky, because they don’t realize their replies are disconnected from the author’s thread unless that person is within their network. They aren’t part of the conversation they think they are. They’re algorithmically isolated from everyone else. Their replies aren’t viewable from the author’s thread because of how Bluesky handles social neighborhoods.
Bluesky’s idea of social neighborhoods is about grouping users into overlapping clusters based on real interaction patterns rather than just the follow graph. Unlike Twitter, it does not treat the network as one big public square. Instead, it models networks of “social neighborhoods” made up of people you follow, people who follow you, people you frequently interact with, and people who are closely connected to those groups. They’re soft, probabilistic groupings rather than strict labels.
Everyone does not see the same replies. Bluesky is being a bit vague with “hidden.” Hidden means your reply is still anchored to the thread and can be expanded. There is another way Bluesky can handle this. Bluesky uses social neighborhoods to judge contextual relevance. Replies from people inside or near your social neighborhood are more likely to be shown inline with a thread, expanded by default, or served in feeds. Replies from outside your neighborhood are still public and still indexed, but they’re treated as lower-context contributions.
Basically, if you reply to a thread, you will see it anchored to the conversation, and everyone will see it in search results, as a hashtag, or from your profile, but it will not be accessible via the thread of the person you were replying to. It is like shadow-banning people from threads unless they are strongly networked.
Because people have not been working with the AT Protocol like I have, they assume they are shadow-banned across the entire Bluesky app view. No—everyone is automatically shadow-banned from everyone else unless they are within the same social neighborhood. In other words, you are not part of the conversation you think you are joining because you are not part of their social group.
Your replies will appear in profiles, hashtag feeds, or search results without being visually anchored to the full thread. Discovery impressions are neighborhood-agnostic: they serve content because it matches a query, tag, or activity stream. Once the reply is shown, the app then decides whether it’s worth pulling in the rest of the conversation for you. If the original author and most participants fall outside your neighborhood, Bluesky often chooses not to expand that context automatically.
Bluesky really is trying to avoid having to moderate, so this is their solution. Instead of banning or issuing takedown labels to DIDs, the system lets replies exist everywhere, but not in that particular instance of the thread.
I find this ironic because a large reason why many people are staying on Bluesky and not moving to the fediverse—thank God, because I do not want them there—is discoverability, virality, and engagement.
In case anyone is asking how I know so much about how these algorithms work: I was a consultant on a lot of these types of algorithms, so I certainly hope I’d know how they work, lol. No, you get no more details about the work I’ve done. I have no hand in the algorithm Bluesky is using, but I have proposed and implemented that type of algorithm before.
I have an interest in noetics and the noosphere. A large amount of my ontological work is an extension of my attempts to model domains that have no spatial or temporal coordinates. The question is how do you generalize a metric space that has no physically, spatial properties. I went to school to try to formalize those ideas. Turns out they’re rather useful for digital social networks, too. The ontological analog to spatial distance, when you have no space, is a graph of similarities.
This can be modeled by representing each item as a node in a weighted graph, where edges are weighted by dissimilarity rather than similarity. Highly similar items are connected by low-weight edges, while less similar items are connected by higher-weight edges. Distances in the graph, computed using standard shortest-path algorithms, then correspond to degrees of similarity. Closely related items are separated by short path lengths, while increasingly dissimilar items require longer paths through the graph. It turns out that attempts to generalize metric spaces for noetic domains—to model noetic/psychic spaces—are actually pretty useful for social media algorithms, lol.
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RE: https://mastoxiv.page/@arXiv_mathCO_bot/115904052353835071
Here is the second manuscript coming out of the "Topics in Ramsey theory" online-only problem-solving session (https://sparse-graphs.mimuw.edu.pl/doku.php?id=sessions:2025sessions:2025session1) of the Sparse (Graphs) Coalition, which took place less than a year ago.
The first manuscript already came out a couple months earlier (https://arxiv.org/abs/2510.17981).
Both have made serious progress in serious Erdős problems.
#combinatorics #remoteconferences #graphtheory #extremalcombinatorics #erdős
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Riffs and Rotes • Happy New Year 2026
• https://inquiryintoinquiry.com/2026/01/01/riffs-and-rotes-happy-new-year-2026/There's a deep mathematical significance I see in the following structures, and I'm hoping one day to find a way to explain all the things I see there. Meanwhile, you may take them as an amusing diversion in recreational maths.
\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \begin{array}{llcl}
\text{Then} & 2026 & = & 2 \cdot 1013
\\
&& = & p_1 p_{170}
\\
&& = & p_1 p_{2 \cdot 5 \cdot 17}
\\
&& = & p_1 p_{p_1 p_3 p_7}
\\
&& = & p_1 p_{p_1 p_{p_2} p_{p_4}}
\\
&& = & p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}}
\end{array} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2026 = p p_{p p_{p_p} p_{p_{p^p}}} \]
The article linked below tells how forms of that order correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”.
The riff and rote for 2026 are shown in the next two Figures.
Riff 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/riff-2026-card.pngRote 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/rote-2026-card.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotescc: https://www.academia.edu/community/VBA6Qz
cc: https://www.researchgate.net/post/Riffs_and_Rotes_Happy_New_Year_2026#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
Riffs and Rotes • Happy New Year 2026
• https://inquiryintoinquiry.com/2026/01/01/riffs-and-rotes-happy-new-year-2026/There's a deep mathematical significance I see in the following structures, and I'm hoping one day to find a way to explain all the things I see there. Meanwhile, you may take them as an amusing diversion in recreational maths.
\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \begin{array}{llcl}
\text{Then} & 2026 & = & 2 \cdot 1013
\\
&& = & p_1 p_{170}
\\
&& = & p_1 p_{2 \cdot 5 \cdot 17}
\\
&& = & p_1 p_{p_1 p_3 p_7}
\\
&& = & p_1 p_{p_1 p_{p_2} p_{p_4}}
\\
&& = & p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}}
\end{array} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2026 = p p_{p p_{p_p} p_{p_{p^p}}} \]
The article linked below tells how forms of that order correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”.
The riff and rote for 2026 are shown in the next two Figures.
Riff 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/riff-2026-card.pngRote 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/rote-2026-card.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotescc: https://www.academia.edu/community/VBA6Qz
cc: https://www.researchgate.net/post/Riffs_and_Rotes_Happy_New_Year_2026#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
Riffs and Rotes • Happy New Year 2026
• https://inquiryintoinquiry.com/2026/01/01/riffs-and-rotes-happy-new-year-2026/There's a deep mathematical significance I see in the following structures, and I'm hoping one day to find a way to explain all the things I see there. Meanwhile, you may take them as an amusing diversion in recreational maths.
\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \begin{array}{llcl}
\text{Then} & 2026 & = & 2 \cdot 1013
\\
&& = & p_1 p_{170}
\\
&& = & p_1 p_{2 \cdot 5 \cdot 17}
\\
&& = & p_1 p_{p_1 p_3 p_7}
\\
&& = & p_1 p_{p_1 p_{p_2} p_{p_4}}
\\
&& = & p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}}
\end{array} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2026 = p p_{p p_{p_p} p_{p_{p^p}}} \]
The article linked below tells how forms of that order correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”.
The riff and rote for 2026 are shown in the next two Figures.
Riff 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/riff-2026-card.pngRote 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/rote-2026-card.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotescc: https://www.academia.edu/community/VBA6Qz
cc: https://www.researchgate.net/post/Riffs_and_Rotes_Happy_New_Year_2026#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
Riffs and Rotes • Happy New Year 2026
• https://inquiryintoinquiry.com/2026/01/01/riffs-and-rotes-happy-new-year-2026/There's a deep mathematical significance I see in the following structures, and I'm hoping one day to find a way to explain all the things I see there. Meanwhile, you may take them as an amusing diversion in recreational maths.
\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \begin{array}{llcl}
\text{Then} & 2026 & = & 2 \cdot 1013
\\
&& = & p_1 p_{170}
\\
&& = & p_1 p_{2 \cdot 5 \cdot 17}
\\
&& = & p_1 p_{p_1 p_3 p_7}
\\
&& = & p_1 p_{p_1 p_{p_2} p_{p_4}}
\\
&& = & p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}}
\end{array} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2026 = p p_{p p_{p_p} p_{p_{p^p}}} \]
The article linked below tells how forms of that order correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”.
The riff and rote for 2026 are shown in the next two Figures.
Riff 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/riff-2026-card.pngRote 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/rote-2026-card.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotescc: https://www.academia.edu/community/VBA6Qz
cc: https://www.researchgate.net/post/Riffs_and_Rotes_Happy_New_Year_2026#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
Riffs and Rotes • Happy New Year 2026
• https://inquiryintoinquiry.com/2026/01/01/riffs-and-rotes-happy-new-year-2026/There's a deep mathematical significance I see in the following structures, and I'm hoping one day to find a way to explain all the things I see there. Meanwhile, you may take them as an amusing diversion in recreational maths.
\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \begin{array}{llcl}
\text{Then} & 2026 & = & 2 \cdot 1013
\\
&& = & p_1 p_{170}
\\
&& = & p_1 p_{2 \cdot 5 \cdot 17}
\\
&& = & p_1 p_{p_1 p_3 p_7}
\\
&& = & p_1 p_{p_1 p_{p_2} p_{p_4}}
\\
&& = & p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}}
\end{array} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2026 = p p_{p p_{p_p} p_{p_{p^p}}} \]
The article linked below tells how forms of that order correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”.
The riff and rote for 2026 are shown in the next two Figures.
Riff 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/riff-2026-card.pngRote 2026
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/rote-2026-card.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotescc: https://www.academia.edu/community/VBA6Qz
cc: https://www.researchgate.net/post/Riffs_and_Rotes_Happy_New_Year_2026#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
Hey Sunday #mathematicians. I could use some help with something.
You have some number of people, say 20. They each sort some number of possible discussion topics, say 40, into four categories: Love to discuss (score of 4) like to discuss (3), sure, why not (2), and no thanks (0). You want to set up discussion groups of four. How do you assign unique people to the topics they most want to discuss?
#mathematician #math #maths #combinatorics #nerds #eggheads #community
Thanks!
-
Think of online anonymity as being one person in a vast crowd. Every piece of personal information you reveal reduces the size of that crowd, the group of people you could plausibly be. For example, revealing your gender cuts the number of potential identities roughly in half.
One way to regain some anonymity is through deliberate disinformation. Suppose you share \(n\) independent yes/no facts about yourself, but intentionally flip \(k\) of them (without the attacker knowing which). In that case, you increase the number of identities consistent with your answers by a factor of \(C(n,k)\).
#OnlinePrivacy #DigitalAnonymity #InformationSecurity #CyberAwareness #PrivacyMatters #DigitalFootprint #DataProtection #InformationTheory #Anonymity #PrivacyEngineering #DataAnonymization #Disinformation #Combinatorics #SecurityResearch #ThinkBeforeYouShare #OnlineIdentity #PrivacyByDesign #DigitalEthics #ProtectYourData #InternetSafety #Privacy #CyberSecurity #Infosec #DataPrivacy #OnlineSafety #SecurityMindset
-
Jax: Fast Combinations Calculation
https://github.com/phoenicyan/combinadics
#HackerNews #Jax #Fast #Combinations #Calculation #combinatorics #JAX #programming #GitHub #open-source
-
“Although mathematics may appear too abstract and detached from real life to most people, everybody has been exposed to hot topics such as digital security or artificial intelligence, which, in fact, rely heavily on progress in mathematics. I therefore strongly believe that maths is invaluable to our society and a field worth pursuing a career in.” - Mihyun Kang
➡️ Find her full story at https://hermathsstory.eu/mihyun-kang/
#Academia #Combinatorics #Mathematics #PhD #Professor #HerMathsStory
-
How many paths of length K are there between A and B? (2021)
#HackerNews #paths #K #length #A #B #walks #combinatorics #graph #theory
-
Day 1 of #Integers2025 just wrapped up! Integers is a #NumberTheory and #Combinatorics conference. There's historically a bit of CGT too. Here are my summaries of the #CombinatorialGames talks: https://combinatorialgametheory.blogspot.com/2025/05/integers-2025-cgt-talks.html
Two of the talks were the result of #UndergraduateResearch!
-
@catselbow
Astonishing! I hadn't come across this before - absolutely fascinating!
#maths #mathematics #polynomial #polynomials #CatalanNumbers #combinatorics #Galois #GaloisTheory
https://www.tandfonline.com/doi/epdf/10.1080/00029890.2025.2460966?needAccess=true -
Decomposing factorial of 300K as the product of 300K factors larger than 100K
http://gus-massa.blogspot.com/2025/04/decomposing-factorial-of-300k-as.html
#HackerNews #Decomposing #Factorial #Math #Combinatorics #LargeNumbers #Factorial300K #Mathematics
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A fundamental result in universal algebra is the Subdirect Representation Theorem, which tells us how to decompose an algebra \(A\) into its "basic parts". Formally, we say that \(A\) is a subdirect product of \(A_1\), \(A_2\), ..., \(A_n\) when \(A\) is a subalgebra of the product
\[
A_1\times A_2\times\cdots\times A_n
\]
and for each index \(1\le i\le n\) we have for the projection \(\pi_i\) that \(\pi_i(A)=A_i\). In other words, a subdirect product "uses each component completely", but may be smaller than the full product.A trivial circumstance is that \(\pi_i:A\to A_i\) is an isomorphism for some \(i\). The remaining components would then be superfluous. If an algebra \(A\) has the property than any way of representing it as a subdirect product is trivial in this sense, we say that \(A\) is "subdirectly irreducible".
Subdirectly irreducible algebras generalize simple algebras. Subdirectly irreducible groups include all simple groups, as well as the cyclic \(p\)-groups \(\mathbb{Z}_{p^n}\) and the Prüfer groups \(\mathbb{Z}_{p^\infty}\).
In the case of lattices, there is no known classification of the finite subdirectly irreducible (or simple) lattices. This page (https://math.chapman.edu/~jipsen/posets/si_lattices92.html) by Peter Jipsen has diagrams showing the 92 different nontrivial subdirectly irreducible lattices of order at most 8. See any patterns?
We know that every finite subdirectly irreducible lattice can be extended to a simple lattice by adding at most two new elements (Lemma 2.3 from Grätzer's "The Congruences of a Finite Lattice", https://arxiv.org/pdf/2104.06539), so there must be oodles of finite simple lattices out there.
#UniversalAlgebra #combinatorics #logic #math #algebra #AbstractAlgebra
-
Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2025 = {p_p}^{p^p} {p_{p_p}}^p \]
The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.
Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.pngRote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2025 = {p_p}^{p^p} {p_{p_p}}^p \]
The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.
Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.pngRote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2025 = {p_p}^{p^p} {p_{p_p}}^p \]
The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.
Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.pngRote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2025 = {p_p}^{p^p} {p_{p_p}}^p \]
The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.
Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.pngRote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)
\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)No information is lost by dropping the terminal 1s. Thus we may write the following form.
\[ 2025 = {p_p}^{p^p} {p_{p_p}}^p \]
The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.
Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.pngRote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.pngReference —
Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes -
New account, new introduction!
I'm Beth. I'm a queer mathematician who loves musical theater, webcomics, teaching math, and my cat.
Favorite areas of math: Topology, geometry, and combinatorics.
Favorite musicals: Chess, Into the Woods, Next to Normal, Sunday in the Park With George, Sweeney Todd
Favorite webcomics: this is long enough to get its own post:
https://transfem.social/notes/a1ryogrga5qu00g9
#Introduction #Queer #Mathematician #Math #Musicals #MusicalTheater #MusicalTheatre #Webcomics #Teaching #TeachingMath #Cat #Cats #SillyGoose #Topology #Geometry #Combinatorics #Chess #ChessTheMusical #IntoTheWoods #NextToNormal #SundayInTheParkWithGeorge #SweeneyTodd #PandorasTaleWiki #RainverseWiki -
I've been on a longer hiatus from livestreaming than I originally intended, but you can see me give a seminar talk this evening at the The New York City Category Theory Seminar:
https://www.sci.brooklyn.cuny.edu/~noson/Seminar/index.html
I'll be talking about the invariant theory part of my thesis (https://arxiv.org/abs/2402.18063) at 7PM, New York time. I'll discuss how I found that every (positive) property of finite structures can be checked by counting small* substructures.
*Terms and conditions may apply. Small is constrained by the logical complexity of a property and may not conform to mundane notions of smallness in bad cases.
#CategoryTheory #combinatorics #logic #Bourbaki #algebra #AbstractAlgebra
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I've found a citation of my own work on Wikipedia for the first time!
https://en.wikipedia.org/wiki/Commutative_magma
Naturally, I read this page before I wrote my rock-paper-scissors paper. It's neat to see that my own work is now the citation for something that was unsourced "original research" on Wikipedia.
#math #research #Wikipedia #algebra #games #RockPaperScissors #AbstractAlgebra #UniversalAlgebra #combinatorics #GameTheory
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The registration to our #combinatorics in #physics workshop is finally open! November 26-29, virtual on zoom hosted by #MITP Mainz.
https://indico.mitp.uni-mainz.de/event/422/
From the intro: "The workshop is designed to build new bridges across adjacent disciplines, encouraging the exchange of ideas and the transfer of advanced mathematical tools from pure combinatorics to physics. By bringing together experts and young researchers, it aims to inspire mathematicians to explore topics relevant to physicists and facilitate interdisciplinary collaboration." -
1434 ways of subdividing a square merging areas from a grid of 9 sub-squares. Code at: https://github.com/villares/sketch-a-day/tree/main/2024/sketch_2024_10_08
More sketch-a-day: https://abav.lugaralgum.com/sketch-a-day
I really need your support to keep going, if you can, donate any amount at: https://www.paypal.com/donate/?hosted_button_id=5B4MZ78C9J724
#shapely #ProgramaçãoCriativa #math #maths #combinatorics #Processing #Python #py5 #CreativeCoding -
My fourteenth Math Research Livestream is now available on YouTube:
https://www.youtube.com/watch?v=pVoFfZAyXzk
I talked about some topics related to my recent preprint (https://arxiv.org/abs/2409.12923) about topological lattices.
I decided to skip streaming today because I wanted to talk about polyhedral products, but I haven't found the old calculation that I wanted to talk about yet. Shocking I couldn't find something I did like six years ago in the ten minutes before I would start streaming. I'll look for it now, so hopefully I'll be ready next week.
#math #topology #algebra #AbstractAlgebra #UniversalAlgebra #combinatorics #LatticeTheory
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I'll be streaming again in 20 minutes at twitch.tv/charlotteaten. I'll be talking about my recent preprint (https://arxiv.org/abs/2409.12923) about topological lattices!
#math #topology #algebra #AbstractAlgebra #UniversalAlgebra #combinatorics #LatticeTheory
-
I posted a new paper on the arXiv!
https://arxiv.org/abs/2409.12923
In "Higher-dimensional book-spaces" I show that for each \(n\) there exists an \(n\)-dimensional compact simplicial complex which is a topological modular lattice but cannot be endowed with the structure of topological distributive lattice. This extends a result of Walter Taylor, who did the \(2\)-dimensional case.
I think this kind of result is interesting because we can see that whether spaces continuously model certain equations is a true topological invariant. All of the spaces that I discuss here are contractible, but only some can have a distributive lattice structure.
A similar phenomenon happens with H-spaces. The \(7\)-sphere is an H-space, and it is even a topological Moufang loop, but it cannot be made into a topological group, even though our homotopical tools tell us that it "looks like a topological group".
This is (a cleaned up version of) something I did during my second year of graduate school. It only took me about six years to post it.
#math #topology #algebra #AbstractAlgebra #UniversalAlgebra #combinatorics #LatticeTheory
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My eleventh Math Research Livestream is now available on YouTube:
This time, I continued my work from the previous week and produced a higher-dimensional version of the formula for the number of Latin squares given in this paper (https://www.sciencedirect.com/science/article/pii/0012365X9290722R). It turned out to be quite similar!
The only real difference in the higher-dimensional case was the need for an analogue of the permanent of a matrix for a rank \(d\) hypermatrix. This can be obtained by summing over all \((d-1)\)-ary quasigroups, which specializes to the usual unary quasigroups (i.e. permutations) in the \(d=2\) case.
#math #livestream #Twitch #YouTube #research #combinatorics #LinearAlgebra #AbstractAlgebra #UniversalAlgebra
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I'll be streaming again in 30 minutes at https://www.twitch.tv/charlotteaten. This week I'm going to continue my work from last week and attempt to produce a higher-dimensional version of the formula for the number of Latin squares given in this paper (https://www.sciencedirect.com/science/article/pii/0012365X9290722R).
#math #livestream #Twitch #research #combinatorics #LinearAlgebra #AbstractAlgebra
