Erdos-Gyarfas conjecture and other musings | Office Hours | May 27 2026

本站添加: 2026-05-28

[00:00:00]mathematicians.

[00:00:01]I think we're going to just have a short office hours today.

[00:00:05]Um because I want to go get groceries and I have other stuff to do and chores to do. And um this would be a great day.

[00:00:16]This would be a great uh time for me to do a a nice long pomodoro because I've um giving a talk soon at a conference and I need to make slides for that. But I have other stuff I need to uh take care of first today. So >> short office hour it is. I guess I'll just have to work without the eyes of the internet upon me.

[00:00:45]>> Imagine that. Good morning femal.

[00:00:53]So, I don't know. Maybe we'll just go look at some archive stuff.

[00:01:03]Or I don't know, maybe you guys maybe you guys have something that you want my eyeballs on.

[00:01:12]I'm still waking up. Oh, I haven't had my first sip of coffee yet. That's a problem.

[00:01:21]Hell yeah.

[00:01:23]Love that for me.

[00:01:25]Um, what should we look at? What should we look at? Commutative algebra. Yeah, why don't we start with commutive algebra?

[00:01:34]Actually, I wonder what's happening in general topology. Haven't clicked on that one yet.

[00:01:42]Perfectoid towers. Perfectoid towers.

[00:01:47]Uh, everything's perfectoid towers.

[00:01:49]Okay.

[00:01:53]Art and L functions.

[00:01:56]Um, either say something about them or give you a reference. Yeah. Well, how about uh how about Dietmar? I would say um there we go.

[00:02:22]Automorphic forms by Antoine Demar.

[00:02:25]Anton, sorry. Um and where Where was uh here we go.

[00:02:42]Yeah, I don't actually have any good sources.

[00:02:46]Well, you could go read Nokersh I guess.

[00:02:49]Um but I I don't have recommendations for particular sources on L functions, but there is this very short just a couple pages on art and l functions here.

[00:03:02]Basically you start with a gowa representation on a complex vector space.

[00:03:09]Um you look at for uh a given prime p you look at the um the the subspace fixed by inertia and then you look at the determinant of uh basically forbinius one one minus um forbinius and that is the local factors of your of your L functions. And then the art and conjecture says that if you define your L function in this way, uh then if row is irreducible and not equal to the identity, then uh the L function attached to row is entire. Um and then he talks about the proof here. Basically he goes through the proof um here is that um art and l functions co coincide with deerlay l functions. And so um basically because earlier in the book he proves um the that art and l functions are entire. This this concludes the proof for uh onedimensional um for for for one dimen the proof of Artton's conjecture for onedimensional representations. And then immediately afterwards by by the way in the next paragraph he says well here's why Langland's is like n dimensional class field theory um because we just generalize it. Um yeah great book.

[00:04:48]uh No, no's um no his yeah his number theory book although probably his cohomology of number fields uh his algebraic number theory book though probably cohomology of number fields talks about them too I would I would venture I guess anch uh well I I definitely prefer physical books I have shelves upon shelves upon shelves of physical books. But um yeah, obviously it's just uh very convenient to read things digitally, you know.

[00:05:39]One thing that's interesting, you know, one thing that's uh maybe devastating is if I did leave if I did leave academia to uh to take a more industry position, I would I wouldn't have uh library access to um to like all of these journals and stuff. How would I access these papers?

[00:06:04]Uh, I have enough I have enough friends that are still in academia. I guess I would just uh I guess I would just periodically send them a list and ask them to download these papers for me.

[00:06:17]Uh, you know, f jumping numbers can be irrational.

[00:06:31]Okay.

[00:06:33]You know what? F your jumping numbers. I don't even care.

[00:06:39]Proxy smallness meets T structures.

[00:06:49]Yeah, let's take a look.

[00:06:55]Depending on the company you work for, you could still keep access to That's true. That's true, actually.

[00:07:02]Yeah.

[00:07:07]Uh let he be an F finite an F finite and infinite field.

[00:07:18]F is itself a field.

[00:07:21]Um, we show that there exists infinitely many fite local domains which are not Q Gorenstein.

[00:07:31]This is all okay. I don't know what's going on.

[00:07:36]I'd make enough money to just buy them.

[00:07:41]No. Have you seen the price of journal subscriptions? It's insane.

[00:07:48]I think I'm going to spend my hard earned money on that.

[00:07:53]No, as soon as I start making real money, uh, as soon as I start making real money, I'm just going to start um um savings maxing. I'm going to start S&P 500 maxing.

[00:08:12]Um, have I checked in on Vixra lately? No, no, I don't. Uh, and I don't believe I will.

[00:08:30]I'm sure it's uh I'm sure it's gotten incredible since AI has really blown up.

[00:08:38]I don't know what's happening in this paper.

[00:08:42]Let's see what what people are doing in general topology.

[00:08:48]Co-final types of topological groups.

[00:08:50]That's kind of interesting.

[00:08:54]Hello there. Let's pretend to understand math today for a while. Let's Let's do that. By the way, regarding Vixra, have I seen Jonathan Tucker's article?

[00:09:07]No. Weren't we going on about this guy before? Who is this guy? Why are you Why are you so intrigued by Jonathan Tuker?

[00:09:17]Related to Paragrin Tuk perhaps?

[00:09:26]Oh, what's this?

[00:09:29]Shading a polomials via huge representations of UQSLN.

[00:09:38]Interesting.

[00:09:41]Classical a polomials um define constraints on coordinates uh on SL2C character varieties associated to not compliments.

[00:10:06]What?

[00:10:14]I feel like there's a grammatical error there.

[00:10:23]Quantum A polomials are difference operators annihilating Jones polomials believed to represent wave functions of 3D churn sim Simmons theory with gauge group SU2 on a tooidal pipe surrounding the knot.

[00:10:39]Umhuh.

[00:10:42]Okay. wave functions of 3D transimate transimmons theory with gauge group SU2 uh on the pipe surrounding the knot.

[00:10:53]Okay.

[00:10:55]Uh in this note we suggest a construction of classical shaded a polomials associated to league group uh su n assuming the color n add some extra depth and shade uh to the classical notion of a polomials.

[00:11:17]Huh?

[00:11:18]A formalism of clubs Gordon chords natural interpretation.

[00:11:24]Yeah, I don't know. This is beyond me.

[00:11:25]What can I say?

[00:11:28]Um, you have this book on topological algebra. You've seen some interesting stuff by Teras. Oh, yeah. Wait, is that Terras Banak as one word?

[00:11:41]Terrace. Are you talking about Audrey Terras? Oh, no. Her name is spelled differently. So, that's probably just that person's first name.

[00:11:50]uh the other posts Soviet mathematicians in particular's field has a good reacting in Ukraine.

[00:11:59]It's just that Jonathan Tuker is a crank and he's meme'd on science and math and fortune. Interesting. There's a famous crank that I don't know about.

[00:12:22]Um Oh, pictures. Okay. I was about to click away cuz I'm like, I don't get it. But you know, there's pictures.

[00:12:34]What are these like linking numbers of these knots or something?

[00:12:43]Let's see.

[00:12:49]H Well, let's take a look at something else.

[00:13:11]Uh, let's see what's happening in graph theory. Where is graph theory? Or is it just cominatorics?

[00:13:20]Yeah, I guess it's just com. I guess I could just search graph theory.

[00:13:35]Oh, but the internet's being silly.

[00:13:39]Why?

[00:13:41]What are they hiding from us? What don't they want us to know about graph theory?

[00:13:52]Hm.

[00:13:54]Can you guys still see me? Why is it uh Yeah, I appear to still be streaming.

[00:14:03]Oh, what do Okay. Um change of plan. Oh, okay. Okay.

[00:14:19]right as I was about to give up. That's the That's the key in life. Uh give up or start to give up and then things will change. That's my advice.

[00:14:32]Oh, actually, you know what's something I want to look up? You know what's something I want to look up? Um I want to look up uh recent results on this.

[00:14:55]Uh this is a very interesting problem.

[00:15:00]Uh, maybe if I can find a survey article, survey and strengthening. When's this from What happens if there's a scandal like archive gate with rearch with uh research gate? What do we call it? Cuz it's already called research gate.

[00:15:45]Research gate gate. We can't do that.

[00:15:48]Research gate squared. Maybe we got to think about these things, you know, before we name stuff.

[00:15:58]Yeah. So the conjecture um is that every graph Oh, interesting. They say every graph with minimum degree the originally I heard it of every cubic graph but uh you know whatever um contains a simple cycle whose length is a power of two. In this paper we prove that under what conditions cubic graphs do not possess a cycle.

[00:16:24]Huh?

[00:16:29]In this paper, we shall prove that under what conditions cubic grass wouldn't that contradict uh oh I I I I see I guess they mean they're proving they're they're proving theorems of for you know suppose this is cubic and uh and it doesn't have a cycle of length a power of to then it must satisfy blah blah blah and then we just go and find you know try to show that graphs don't can't have that property or something.

[00:17:16]This paper seems old though. When's this paper from? I kind of want something uh kind of want a more re Okay, 2016.

[00:17:26]That's that's new enough for me for a survey article.

[00:17:30]Although clearly uh this Google search has revealed that um I mean it's it's still popping off. Pe people are working on this actively.

[00:17:50]uh by Erdish minimum degree power two if the conjecture is false a counter example would take the form of graph with minimum degree three um wait is it really it's minimum degree three oh yeah right right but you just investigate it for key I okay I I see. Um uh it is known through computer searches of Gordon Royal um Claus Marstrom that any counter example must have at least 17 vertices and any cubic counter example must have at least 30 vertices.

[00:18:35]Markstrom searches found four graphs on 24 vertices in which the only power of two cycles have 16 vertices. One of these is planer. However, the conjecture is now known to be true for the special case of three connected um cubic planer graphs. Interesting.

[00:18:54]Uh that's interesting.

[00:18:58]I kind of want to go I kind of want to go read that.

[00:19:04]A few of's problems are drunk. Which ones?

[00:19:10]Can you tell me which ones?

[00:19:31]Hello. F.

[00:19:39]How long is this paper? 43 pages. Okay.

[00:19:50]Uh Um following is the main result of this paper. Every three connected cubic planer graph contains a 2m cycle for some m. Wow. Even even they say m has to be between two and seven.

[00:20:07]Huh.

[00:20:11]That's very interesting.

[00:20:14]Let's check out Chevrolet Islandberg coology.

[00:20:19]Why?

[00:20:21]Why AI just solved another bunch of air dish problems? I don't know if I've seen it.

[00:20:34]I can't keep track because there's always uh And are these claimed? Are these verified?

[00:20:43]I I don't know. I don't know. There's a post on Reddit. A cubic graph is a is a graph of degree theory. Wait, is do you just is this just is this just the algebra coalology.

[00:21:04]U post on Reddit about airish problems.

[00:21:14]Oh, the graph reconstruction conjecture.

[00:21:20]Um, right. I did look at this actually.

[00:21:25]Um, yeah, this is a funny comment.

[00:21:34]Uh yeah, the Chevrolet Alenberg algebra of a Lee algebra is a differential graded algebra of elements dual to G whose differential encodes the Lee bracket on G.

[00:21:55]Uh the co-chain cohomology of the underlying complex is le is the le algebra coalology.

[00:22:03]The co-chain cohomology of the underlying complex is the le algebra.

[00:22:09]Huh?

[00:22:12]Huh? What do you mean the co-chain coology?

[00:22:18]You're saying the coalology groups recover the legebra coalology. What is this saying?

[00:22:26]Uh, this generalizes to a notion of Chevrolet Alenberg algebra for G for an L infinity algebra.

[00:22:34]Jesus Christ. Jesus Christ and lab. I'm not even done having my coffee. I'm talking about goddamn L infinity algebbras.

[00:22:43]Uh so V will be sim one blah which is what now?

[00:23:01]Okay.

[00:23:04]Okay.

[00:23:08]Okay.

[00:23:10]So, so this is just Lee algebra cohomology.

[00:23:20]Yeah. The Chevrolet Alenberg algebra of a finite dimensional le algebra is the semifree graded commutative DG algebra whose underlying graded algebra is the grassman algebra. Okay. Yeah. It's just it's just symmetric powers of the duel.

[00:23:39]Um and his differential is defined as the duel of the lee bracket. Uhhuh.

[00:23:53]Extended to a derivation on the whole thing. I see.

[00:23:58]Um.

[00:24:01]Mhm.

[00:24:05]Now if we take a basis and the corresponding structure constants um of the le brackets in that basis and then the action uh of the differential on basis generators does this uh uh Last night you was in the grass, man.

[00:24:34]I'm sure you was. I'm sure you was.

[00:24:38]Um, yeah.

[00:24:41]Okay.

[00:24:45]I mean, it's not uninteresting.

[00:24:50]Ghost of ghosts. Whoa. Hang on. Now we're talking. What the hell is a ghost in physics? Oh god. What is What's the end lab page for physics?

[00:25:02]Well, physics is basically an infinity onetopos of the magnifica category that's uh quantumly uh quantumly toroidal.

[00:25:17]Surprisingly, surprisingly, NL lab isn't saying it's that physics studies the constituting mechanisms of the observable universe. Okay. When do we see infinity? Wow. They don't even say infinity something once. Huh. Okay.

[00:25:34]Well, I'm surprised. I'm surprised.

[00:25:43]What is this? Okay.

[00:25:45]Uh okay. The Chevrolet Alenberg algebra GN of the action of the Lee algebra or Lee infinity algebra of a gauge group on a space N of fields a space N H never called a space N in my that's not true. I called manifolds N.

[00:26:13]uh is called the BRST complex. I've heard of this before.

[00:26:17]Generators are fields.

[00:26:20]The generators Oh, come on. Hey, that's bad. No, that's bad writing. That is bad writing. I would I would say that. That was bad writing. Don't do this. Only villains do this.

[00:26:32]The generators in degree 1 are called ghosts. The generators are called ghosts of ghosts.

[00:26:39]Am I supposed to think of this as infantessimals? Is this some like ghost of departed quantities uh reference or what?

[00:26:55]They forgot to include the unobservable universe. Yeah. Well, you just take the compliment, right?

[00:27:04]Yeah.

[00:27:08]Uh anywh who um maybe this hasn't been I don't know the most exciting episode but uh that's all for today folks. I need to go get groceries. I need to go I need to go pick up a deferred midterms and then I need to or deferred finals and then I need to go start grading said deferred finals.

[00:27:35]Um, and then I need to start writing. I need to start making my slides for my talk in St. John.

[00:27:44]St. John, Canada.

[00:27:47]Um, so maybe I'll see some of you at CMS. Who knows? Uh, I will.

[00:27:53]Yeah, I'll get you a cookio. Uh, send me your address so I can mail it to you.

[00:27:58]And uh end with that QED.