Chad Grothendieck MOGS beta Faltings | June 3 2026

追加: 2026-06-05

[00:00:02]I'm not doing any work this morning.

[00:00:04]That's not true. I already did some work this morning. Uh but it's office hours, you know. I'm basically just going to like look at this paper. Uh I think that's basically it. That's what I'm going to do. Uh this office hours.

[00:00:19]Just try and stop me. Don't like it? Try and stop me.

[00:00:25]Uh, it would be crazy if one of my viewers could hack me and stop it.

[00:00:37]That's my challenge to you. Go [snorts] ahead, hack my channel. Hello, Anchor.

[00:00:45][sighs] How is it going? How are you doing today?

[00:00:51]Hello, Zafar. You never replied to my last email so far.

[00:01:03]Random crap stream. [laughter] Exactly.

[00:01:08]Exactly. Random crap. Not quite so random crap stream today though, actually, cuz really I have in mind uh staring at this paper for a little bit.

[00:01:19][sighs] [gasps] >> [snorts] >> the Caillou drip again. Yeah, I knew you were going to say that as I was putting on this shirt like 3 seconds ago. I was like, I know Boris is going to call me out for for being dripped out like Caillou today. Uh, so it is so it goes. [snorts] Uh, G's connected reductive algebraic group over the real numbers. Arthur conjectured the existence of certain packets.

[00:01:54]Uh um a longstanding conjecture um is that all Arthur packets consist of unitary representations. Wait, is that true in the piata case? I don't think that's true in the pottic case. I don't know. [snorts] Uh we prove this in general. Wow. Okay.

[00:02:15]God damn. Uh the main new idea is a Jordan decomposition for Arthur packets canonical two-step process for realizing an arbitrary Arthur packet via real parabolic and cohomological induction from a unipotent Arthur packet for a certain Levy subgroup.

[00:02:35]Um [sighs and gasps] crazy we reduce the question of unitarity to the case of unipotent arther packets.

[00:02:47]Um interesting I don't think that will could that well that could work in the picate I don't know [snorts] but the answer is already known uh uh as an application of same methods we also give a proof of Xiang's conjecture for real reductive groups which gives an upper bound on the wavefront set uh of members of arthrop packets [sighs] uh couldn't come up with a feasible price. You um eight $8 Canadian per hour is all you could do.

[00:03:30]Um yeah, that is maybe that is low. That is far too low. You're right, Zafar. Um you're right. [laughter] Uh [snorts] I have studied class 12 algebra calculus. Good morning quarterty and YouTube land calculus level heavily computational uh um I don't quite so you're you're asking anchor you're asking how to prepare for a math entrance exam what real analysis book should you start with um I don't Um yeah, maybe Tao's book on analysis or Abbott's understanding analysis or uh you know Cummings Jay Cummings book on visual real analysis.

[00:04:35]That's that's a good one I think. Um but the truth is you need to find the right book for you. You know different people have different uh opinions about what books are good and you just need to check out a couple popular books and and see which one uh you connect with personally. Um but also I mean uh math entrance exam right that's a very an entrance exam looks very different at very different universities right so um you should probably look into the specific entrance exams that you will be writing and see what you need to learn for those specific ones. Um oh yeah Zafar recommends the exact same things as me. Nice twinning.

[00:05:24]Wow. Boris uh has a paper in the works.

[00:05:27]Um nice. A random email got you a co-author offer. Damn. I got to email more people.

[00:05:36]That's what I got to do. [snorts] Um yeah. Yeah. I mean, Ruden is great.

[00:05:43]Yeah. I mean, of course, if you could go and just solve all the problems in Ruden, then then you're golden. But like um a lot of people stud struggle to self-study from Ruden Karant and Fritz John's book uh on intro to calculus analysis may be good as an intro. Never even heard of it but interesting suggestion.

[00:06:11]Yeah, let's look at some of this.

[00:06:15][sighs] [gasps] Uh, real points, real connected group, uh, Langland's classification, um, L packets, Arthur packets.

[00:06:30][sighs] Uh-huh. Uh-huh. Uh-huh.

[00:06:36]a set and a function such that dupy dupydoo dupy dupy doo dah trace formula doo dah hey doo d [laughter] [sighs and gasps] all this stuff already right [snorts] you cleared entrance exam joining university in two months.

[00:07:09]So you have two months to start before the semester.

[00:07:15]I'm not sure.

[00:07:17]I'm not sure what you're asking about here. Anchor, so you already wrote the exam and now you just want to be prepared for the first semester.

[00:07:33]Is that the case? Hello. Always bad at math. [snorts] We'll now briefly explain the structure of our argument.

[00:07:45]Uh what is Arthur packing? I don't know.

[00:07:48]Arthur packets. I could tell you a little bit about Arthur packing. I I don't know. It's when you pack spheres into an Arthur packet as densely as possible.

[00:08:00][sighs] Yeah.

[00:08:02]>> [snorts] >> I should go look at I should go look at some of these uh papers.

[00:08:12]Dr. K, I forgot to answer your question yesterday about reading all the references in a paper. Oh yes, yes, yes, yes. Sorry, I did forget about that.

[00:08:20]Yeah. Um, okay. So, yeah. Well, well, this question blew up because on X, you know, does the archive gate, right? And then people, you know, and then people were like somehow somehow the question of of of not uh of making sure you don't have hallucinated references in your paper then t evolved into a debate about whether you read all the references uh all the papers that you reference in a paper. Um and people were like, "No, I don't read all the references." And other people are like, "Oh my god, how could you not read the references? Like that's insane. And uh of course I think part of the problem is that you know on Twitter unless you're paying for more characters which why would you do that uh the 160 character limit or whatever is like you know doesn't exactly lend itself to nuanced careful discussion.

[00:09:20]And so one thing about this is what does it mean to read a reference, right? What does it mean to read a paper? Okay. Um if by [snorts] read a paper you mean you read every single sentence carefully and have a complete 100% understanding of every sentence, every step of the argument of the entire paper start to finish. Right? Uh, no one does this. I don't want to say no one, but like especially look in a field like this, 50 pages and and papers like this are being pumped out every day. It's not happening, right? It's not happening.

[00:10:04]And that's not a good use of your of your time, right? To go through and really deeply understand every single detail of like one paper that you reference. it is it is simply not feasible um to do that anymore and it's and it's just not you don't need to right because I mean if the you you you have some faith that the paper's been peer reviewed and you trust the arguments well enough and you know when and if everything becomes done and lean then we'll have even even greater assurances that it's okay to do this um Uh yeah.

[00:10:48]So of course [laughter] if you site a paper you should open that paper. You know your eyes should look at the paper. Of course if you site a paper you should you know go you should understand like why are you typing why are you citing a paper? Probably because you're quoting a theorem from it. And so undoubtedly undoubtedly uh you should you would have read the statement of that theorem understood what it meant and applied it accurately in your argument in your proof right uh and that I think is good enough right I think that's good enough I don't know I don't know what else to say about it Um, so but but does that count as reading the paper? I mean, most certainly not.

[00:11:44]Like like this paper, okay, it's not inconceivable that I might need to uh uh cite this paper at some point in my life. You think I'm going to read every single uh part of the argument of this 50 p? Absolutely not. Absolutely not. Um so [snorts] yeah the the question what does it mean to read a paper right and and and uh if reading a paper means understanding every single detail then I think few people ever read a paper very few people ever ever read a paper these days [snorts] depends on what field you're in uh in some fields I'd say it's more common to to read a paper in that sense but if but if read a paper just means okay you you have some general understanding of uh of um you know proofs the proofs and the theorems then I don't know uh that was the craziest tweet ever. Uh you were an archive hater but you still found that insane. Why are you an archive hater? Uh you saw this exact argument on LinkedIn. Uh gross. Why are you on LinkedIn? [laughter] [gasps] Uh, and I feel like you can read a good portion of a paper that's referenced yet still don't know whether it is AI generated or not.

[00:13:14]No, complete wait complete disagree.

[00:13:17]Complete disagree. Okay, first off, uh, a paper that's 100% AI generated, no, no one's no one's reading a math paper that's 100% AI generated and not understanding, unless maybe I don't know if you're an undergrad and you're not at like research level yet. Um, yeah. So, that's not my definition, by the way. That's not the def that's not my definition I'm putting forward. Um, but but that's not even what the question's about. The question's the question's not about referencing an AI generated paper. That's not even where this came from, right? The original debate was just about a a a so a reference being hallucinated, meaning that there's not even an actual paper behind it, right? That's the that's where it first came from. But then also number two, yeah, you should avoid um you should avoid you should not be citing at this point in time 100% AI generated paper. And if you can't tell I today as of June 3rd, 2026, if you can't tell that uh a math paper is 100% AI generated or not, you should you're not I mean you're not at the research level yet. I would say that may change a month from now. I would not be surprised if that changed a month from now. You know, we're we're we're not that far away from that being the case. But currently, I mean, 100% AI generated paper. That's just uh you know that we're not at that level yet. We could be soon. [snorts] Uh Leo, you finally have returned, huh?

[00:15:06]After abandoning me for doing your own streams, huh?

[00:15:12]Uh big lock in summer. Oo, big lock in summer. What is Leo up to?

[00:15:20]What is Leo working on? Hello Farza.

[00:15:23]Hello Snack Daddy.

[00:15:27]Uh, why is the coefficient of the quadratic term defined as the curvature of a plane curve? The quad the quadratic term of what?

[00:15:40]Um, I need some more context here so far.

[00:15:46]the quadratic term.

[00:15:50]Uh [groaning] why do we not consider cubic terms?

[00:15:57]Uh yeah, I'm going to need a little more context here.

[00:16:07]It is crazy that it's June 3 already.

[00:16:10]Can you phrase the curvature in terms of something? Oh, yeah.

[00:16:14]It is Wednesday. Yeah, my dude.

[00:16:23]Don't know the context, but that wouldn't stop. Yeah, of course. Uh I wouldn't I wouldn't expect anything other than major push back from Leo on literally anything I say ever is okay to forget a proof. Yes.

[00:16:40]If you have not done it in like a month or longer, like you tried to reconstruct, took a couple hints to get it right. Uh yeah.

[00:16:49]Well, I don't know what's what's the context here. Uh is it is it a proof for a course that you're taking? Uh and is it a very important proof in that course? Well, then you know, do more memory.

[00:17:06]Why is there a drinking bleach emoji on YouTube? Um, well, we didn't we all have a big uh bleach drinking phase during COVID. Wasn't there some bleach drinking meme and then some people actually drank bleach and died? That's probably why. [laughter] Wow. Leo streams are more fun. Fine.

[00:17:33]That's it.

[00:17:35]>> [snorts] >> My channel's done.

[00:17:38]My channel's cooked. Leo streams are more fun. You all heard it here.

[00:17:42]Everyone, all 19 of you watching now, stop. Uh, just stop. Leave my channel.

[00:17:49]I'm going to shut it down.

[00:17:52]Leo's bested me. His streams are more fun.

[00:18:02]I'm waiting for you all to leave.

[00:18:06]I'm waiting for you all to leave.

[00:18:08]There's still 16 of you here. I don't know what's going on.

[00:18:12]Why? Why haven't you left yet?

[00:18:15]The lack of two-minute lag makes it fun.

[00:18:18][laughter] [sighs and snorts] Uh 17. No, we're going in the opposite direction. One more person has joined.

[00:18:26]You guys are supposed to leave. And now there's more people joining. 19. Jesus Christ. What is wrong with you people?

[00:18:32][laughter] >> [gasps] >> Um, yeah, you all better shut your mouths or I'm going to make it three minutes of lag. All right, so don't don't don't push your luck.

[00:18:48]Uh, basically properties of mappings is contextual. What depends on domains and co-domains? Well, sure, but I mean I mean two functions depend on domain or co-domain. Right. Um unlike numbers where an even number is even no matter what.

[00:19:10]Well, no because I mean a mapping the domain and the co-domain are part of the datim of a mapping. So from that perspective actually uh it's it is absolute. It is uh it is not contextual strictly speaking. You should consider the domain and co-domain as part of the datim of a function.

[00:19:35]This would be the most principle thing to do.

[00:19:40]Um, [clears throat] they're still here cuz you aren't live.

[00:19:48]I see. You've graciously granted me but a pittance of of views today. [snorts] It was a random proof not in the course.

[00:20:00]You self-studied it two months ago. I don't know why you would worry about remembering it or not.

[00:20:07]Uh [sighs and gasps] not part of the datim of your mappings.

[00:20:12]Well, how about this? How about this?

[00:20:16]You're going to come to my channel.

[00:20:17]You're going to you're you're going to have better streams than me. And you're going to tell me that uh and you're going to tell me that uh domain and co-domain is not part of your mappings. Banned. Banned for a whole minute.

[00:20:33]How do you like them apples?

[00:20:36]Yeah, I'm going to read your messages anyways. [laughter] >> [snorts] >> Is there a word that describes a mathematical property based in another one but weaker like upper hemi continuity for continuity be weakerized?

[00:21:00]Yeah, I would just say it's a weaker condition. I don't I don't know if it needs to be a single word. Um yeah, a weakening of continuity.

[00:21:13]Yeah.

[00:21:14]>> [snorts] >> Um, let's look at some more of this paper.

[00:21:32]Oh, uh, growth and mogging fault things.

[00:21:34]Okay.

[00:21:49]Is this the paper?

[00:21:59][snorts] Dear Mr. Faldings, many thanks for your quick answer uh and for sending me your reprints.

[00:22:06]Your comments on the so-called theory of motives are of the usual kind and for a large part can be traced to a tradition which is deeply rooted in mathematics.

[00:22:16]Namely, that research, probably long and exacting, and attention, is devoted only to mathematical situations and relations for which one entertains not merely the hope of coming to a provisional, possibly in part conjectural understanding of a hitherto mysterious region.

[00:22:38]Oh, [clears throat] I jumped ahead. as it has indeed and should be the case in the natural sciences, but also at the same time the prospect of a possibility of permanently supporting the newly gained insights by means of conclusive arguments. What, bro? What the [ __ ] are you talking about?

[00:23:00]Uh, wait, is is mogging happening here? I don't see [snorts] on the first page. I don't know what's he saying. This attitude now appears to me as extraordinary extraordinarily strong psychological obstacle to the development of visionary power in mathematics and therefore to the progress of mathematical insight in the usual sense namely the insight which is sufficiently penetrating or comprehending to finally lead to a proof.

[00:23:29]Um what in my experience of mathematical work has taught me again and again is that the proof always springs from the insight and not the other way around and that the insight itself has its source first and foremost in a delicate obstinate feeling of relevant entities and concepts and their mutual relations.

[00:23:49]The guiding thread is the inner coherence of the image which gradually emerges from the mist as well as its uh continence with what is known or foreshadowed from other sources. And it guides all the more surely as the exigence of coherence is stronger and more delicate.

[00:24:15]Uh okay. To return to motives, there exists to my knowledge no theory of motives for the simple reason that nobody has taken the trouble to work out such a theory. There is an impressive wealth of a var of available material both of known facts and anticipated connections incomparably more. It seems to me that never presented itself for working out physical theory. There [snorts] exists at this time a kind of yoga of motives which is familiar to a handful of initiates and in some situations provides a firm support for guessing precise relations which can then be sometimes uh can sometimes be actually proved in one way or another.

[00:24:57]Okay. So yeah is there is mogging happening here?

[00:25:02]I don't I wouldn't say there's mogging happening. I also don't I mean where is part one of this letter? I would have liked to have seen part one of this letter to under to have a little more um uh context here. But uh I'm guessing that Falting was complaining that his paper was too um you know handwavy and and and uh vague and the theory wasn't quite fully developed you know rigorously let's say and growth and Dee is is saying no no no actually it's okay to sort of be vague about these things and develop the ideas first and then later the proofs.

[00:25:58]Yeah. But I I don't know. I don't know.

[00:26:01]Um I I don't know. I don't know if I agree with your interpretation, Snack Daddy.

[00:26:08]He's saying you focus only on lowhanging fruits to get results.

[00:26:12]Is he saying that? I I I don't I don't see any part of this which looks to me like such an accusation.

[00:26:22]Um and if he made such an accusation that's obviously false.

[00:26:32]So I I don't agree with your interpretation uh of this that clearly he's pushing back. Clearly, clearly fault things, you know, wants to see something more concrete and rigorous and growth and is pushing back that uh, you know, things don't always have to be that way.

[00:26:56]That's what I would say is happening in this paper. Um, I don't think Roth and is making that accusation of faultings.

[00:27:04][sighs and gasps] Um, [clears throat] yeah. Any recommendations on subjects that would be approachable for research during a master's?

[00:27:14]Just study up and pass exams. Yeah, I would well I would say actually so uh it is awesome if you can come into a graduate degree masters or PhD with your own well-formed idea um about a research project that is like awesome. But uh probably the vast majority of people going into a masters are simply not capable of finding a research project that is simultaneously uh feasible and interesting at the mast's level.

[00:27:51]I it's just probably not going to happen. I'm not saying don't try but don't worry that much about of so would be fantastic if you could do such a thing. Um that's the point of having a supervisor, right? I mean your your supervisor should I mean their job the the reason you have a super the the reason you want a supervisor basically is because this is someone who's experienced and so they'll have developed some sort of sense of kinds of questions that um you know should be approachable and interesting for a master's project. Yeah. So I would say with that in mind, I mean do explore things you're interested in. Do try to get generally familiar with the area that you want to focus in and try and learn about what are the interesting problems and ideas and results. But um you don't put too much pressure on yourself to come up with an original research project.

[00:29:01]How would you prove that proof of something can't exist?

[00:29:07]Um, I don't know. Study logic.

[00:29:11]Go get a book on girdles incompleteness serums. [sighs and gasps] Um, [snorts] he's crashing out cuz he hasn't won a single one of your starting soon DVD logo games. How many of these games have you had? You've never even invited me to play in one. That's why I'm actually crashing out.

[00:29:32]Um, nice. I want the pointer stick the size of a spear. [snorts] Anchor is asking a good question. There is one way to do this that should be guessable.

[00:29:47]Uh, good morning Ken K on learning formal logic for math. Is it better to learn logical equivalents or laws of logic and jump to set theory?

[00:29:59]Huh?

[00:30:01][snorts] I'm not sure I understand the question.

[00:30:06]To learn logical equivalent, I mean it takes like two minutes to learn logical. I mean is if and only if, right? That's all logical equivalence is.

[00:30:20]Uh, but are you asking to if you should spend a bunch of time on formal logic before set theory or jump or or go into set the I mean probably uh you know probably it would be nice to do some formal logic first but I don't know you don't have to. What do you want to do?

[00:30:44]What a French ass running sentence.

[00:30:46]Yeah. [sighs and gasps] Uh yeah, it actually reminds me there was this this YouTuber this kind of this this this whole first page actually reminds me there's this YouTuber Strange Eons uh and I watch her content sometimes and she was you know deeply embedded in in Tumblr uh during its heydays and anyway she was talking about you know being like 14 and you know desperately wanting to sound like an intellectual and just you know taking But, you know, taking like a a four-word sentence and stretching it out to like three sentences just to sound like you're an intellectual. Uh, and this kind of reads like that. But, you know, it also could just be because of the way that things are translated.

[00:31:35]Um, you know, maybe would have been much more clear in in to be read in the, uh, original French. [snorts] Why does this letter matter? Why does any letter matter?

[00:31:48]Uh, why does anything matter? Mo, answer me that. That's my response to everything. [gasps] Um, [snorts] uh, what if Anchor is constructivist? Good question.

[00:32:14]Um, what is a reprint? Probably something that got printed again, I would say.

[00:32:22][sighs] Uh, yeah, I did look at the lead in declaration about AI and math and I read a couple sentences and I got bored. I don't care that much. I don't really care.

[00:32:34][snorts] Um, if you could start a PhD having written your dis dissertation.

[00:32:42]Yeah, you would be ahead one. Yeah, I I would I would agree with that.

[00:32:54]Have I felt like advanced physics gave you new ideas in mathematics?

[00:33:00]Uh, well, I've haven't. I don't know advanced physics, so no, I wouldn't say I do. I mean, I know some basic stuff about advanced physics. So is that advanced physics? I don't know. Uh yeah, depends on your field. Of course, a ton of a ton of math has come out of string theory, for example.

[00:33:23]Um do I have a math mast's only course in Canada? No. Uh or sorry, yes.

[00:33:30][laughter] Yes. No, it's it's it's not part of a page. Yeah. In Canada, we have masters as its own separate thing. Uh masters is its own separate thing. You don't fail out of a PhD with a master's in Canada.

[00:33:43][snorts] I didn't sign the leaden pledge. No. No.

[00:33:50]What is the pledge anyways?

[00:33:53]Uh, your American math friends look at you a bit weird for doing a master's instead of PhD directly. Yeah. Are you Canadian too, Sergio Polar Bear or European?

[00:34:05]Uh, why do your eyes start twitching because of clabby yella manifolds? Your PI is big on this mirror symmetry and string stuff. H, you never understood what he's talking about.

[00:34:19]Yeah. Yeah. Lots of math from there.

[00:34:26]You exaggerated. You refer to the part where he says the deeply rooted tradition only devoting research to situations where you expect to gain understanding from proving results.

[00:34:38]Yeah, I I don't know. I just don't I don't really read that as much of a disc.

[00:34:44]Um [snorts] and yeah, in Growth and De's opinion, it's an obstacle.

[00:34:54]Well, attitude appears to be strong psychological obso development of visionary power in mathematics and therefore also the progress of mathematical. Yeah. I mean, I guess maybe you could read it as a dig to fault things.

[00:35:27]I don't know. I I I'd like to see the letter that preceded this from Faultings.

[00:35:33]Um Isn't the leader in leading declaration kind of what most people do already?

[00:35:50]Yeah. I don't know. What is it even? I don't know.

[00:35:54]Um, Langland's is understood by people you can count on your fingers. No, that's false. 14 people. No. False. What do you mean? Uh, completely false.

[00:36:09]I mean, okay. What do you mean by understand Langland's? Do you mean understand like every proof in every version of Langland's in every paper? Well, then the number zero.

[00:36:23]Do you mean understand the proof of geometric Langland's? Well, first off, [gasps] can you count 14 people on your fingers?

[00:36:33][panting] you've you've if you can you've been blessed with a rare gift. Uh mind you, you could do base two, right? So actually you could count I don't know what you can count two to the power of um to the^ of 11 minus one. I guess you could count if you do base 2 arithmetic.

[00:36:52]Okay, fair enough. Um yeah.

[00:37:03]Uh You're Puerto Rican. I think you mentioned that before. You know, sorry, I forgot.

[00:37:14]Um, somehow K's stream summons people from every country. Good. Yeah, we're a multicultural. We're a mixing pod in this chat here.

[00:37:26]Oh, I'm not even reading that message out loud. How about how about another uh how about another one minute ban for putting this propag for pledging to anything but my channel in here. How about that? [laughter] [gasps and sighs] Not even going to read the other messages.

[00:37:47]How much formal logic really for math?

[00:37:50]Just asking will be learning logical operations be enough? I mean, I feel like in order to be sufficiently cultured in math, it would be good to learn a little bit more and understand a little bit more about the inner workings of like the incompleteness theorem. But no, I mean for real analysis, for being a working mathematician, you could be you could be very ignorant of a lot of formal logic. Um, so I don't know actually Langland is only understood by 13 people. [laughter] Uh, yeah.

[00:38:38]A statue made by Charlotte Langland's.

[00:38:41]Who's Charlotte Langland's?

[00:38:44]You took leaden to be be ethical. If you use AI and site your sources, use models that are ethical as possible.

[00:38:52]Yeah, I don't know what a boring milk toast. Like, who cares? Okay, [laughter] you need to sign a pledge to do that. I don't know.

[00:39:02]Um, anyways, uh, we've done enough random crap for today in my opinion. Uh, I'm going to go do less random crap maybe.

[00:39:16]Uh, who knows what I'll do. Who knows what I'm up to when I'm offline. I could be lying about my entire life, you know?

[00:39:23]Uh, everything you know about me could be a lie. And it probably is. So, think about that as the as the sound of these three letters reverberate through your ears for the next 24 hours.

[00:39:39]QED.