I don’t need sleep! I need answers!

Added: 2026-06-04

[00:00:00]The summer 2026 semester is upon us and I'm feeling pretty good. In the last video, I was at a conference in Canada and that conference was extremely fun. I loved it so much being there. And I don't remember what day of the week it was when I recorded that video. But I remember on the Tuesday of that week, I had a job interview with a community college. And on Friday of that week, I gave my talk about my research that I've been doing for the past year and a half, 2 years or so. And I was pretty I wasn't nervous during the week. I only really get nervous when I'm about to do something.

[00:00:42]Like right before I'm about to do it, like I'm talking like a minute or two is when I get nervous.

[00:00:49]But it was a very serious conference. I mean, everyone there is like friends with everyone. I'm kind of the new kid on the block, so they don't have a whole lot of I mean, they were nice to me, don't get me wrong. I'm not trying to like paint a bad picture. I really liked those people, but I am the new guy, so it's like you have more to prove when you when you're the new guy. So, I wanted to do a really good job and present my research and and, you know, hang out with them. And I think I did a pretty good job.

[00:01:19]My presentation didn't go 100% the way I wanted it to, but I can't complain because it was like 95% of the way I wanted it to go. And a lot of those mathematicians at the conference were pretty serious mathematicians. I mean, like they've been doing analysis much longer than I have. And they probably did Olympiad competitions when they were in high school. I mean, I'm not I can't I I don't really know if they did or not, but it's it seems like there it wasn't zero. We'll put it this way. Um, but yeah, there's like they were like really talented mathematicians. They get up there, they talk about their research, and you know, they're they're definitely beyond me at this given moment. And remember remember I came from the world of environmental science and I had to switch into mathematics when I was getting my masters.

[00:02:15]But um but what I'm saying is is that during my talk I said okay um what do I have control over? They may be way more comfortable with this subject than I am. But my what I wanted to do differently is I wanted to give a talk that was I wanted to pick and choose my details because you only have 30 minutes to speak. This paper that I'm working on is way more complicated than 30 minutes worth of worth. Okay, let me I'm not saying it very well. There's no way that I can give every detail of my result that I've been working on for the past two years in 30 minutes. It's just not feasible. And I am of the, you know, belief that less is more. Don't be afraid of negative space. In fact, look at these notes that I've been taking here from Sheldon Axel's measure theory book. You can see here that not every line is filled in and there's plenty of like dead space on the page and that dead space is pretty, you know, you don't have there's a reason why margins exist in uh in books and you know I I believe that you might want to say that oh you've wasted all this space but not really that negative space is serving a purpose there and this is only something that I learned maybe like two or three years ago. I can't remember when I when I started using it, but um I try to philosophically or metaphorically include negative space during my presentation.

[00:03:58]Every single word is chosen um everything every single word is chosen specifically for like every word serves a purpose. They're not like these videos where I'm just kind of talking off the cuff. Every word is like everything is slow. I slowed down. So that's two points. Point number one, slow everything down. My result is very close to me. I understand it very well, but the people in the audience, they are good at math, don't get me wrong, but they don't know my result the way I know my result. So, I have to slow everything down. And then if I do my job well, then in 30 minutes, they can understand what it is that I did. And then the second place is, you know, less is more. Don't overload the the slides or the chalkboard with all these details because like if you overwhelm me with details, I'm probably going to tune you out.

[00:04:51]So th that those two lessons I learned um over the past 5 years has helped me craft what in my opinion was a good presentation. Again, other people have to tell me if it was good or not. Like I don't want to I don't want to gloat or anything. I'm really trying not to do that. I want to stay humble, but the reason I'm saying it now in this video is because maybe you'll find it useful when you have to present something in front of someone. This is just my own personal experience. If you think I'm speaking nonsense, then by all means, ignore what I'm saying. But like I'm almost 31 years old.

[00:05:31]Slow things down. It's not a race. Less is more. Choose your words carefully and practice.

[00:05:39]it like it'll make like a world of difference. So, my talk definitely did that. I slowed things down. I I explained I think I explained things pretty well and it is out there on the internet somewhere. It's just hard for me to find. In fact, I had a link to the video, but the link wasn't working for me. Maybe if I'm comfortable showing it to showing it to the world, then I'll share the link, but most likely no. And I really want to talk about my result on the channel because I believe I can do a better job on the channel than I did in person because I was nervous when I was presenting.

[00:06:12]But I want the paper to be published before I talk about it on the channel for obvious reasons. And after I gave the presentation in Canada, I realized that I could say things in a much better way than I had them written in the first draft. So, I spent like the last 10 or 12 days since um oh, it's hard to measure. Uh freaking I got back from Canada and as soon as I got back from Canada, I'm like, "Okay, let's all this learned experience I had from presenting my research. Let me incorporate that into my paper." And it was kind of amazing just like how I was able to explain things much better than I had before. So, for like the next 10 to 12 days, I pulled like sleepless nights. My my sleeping schedule is completely screwed up right now. It's 6:00 a.m. and I am wired at 6:00 a.m.

[00:07:06]I've been up since midnight because I sleep in the afternoon. I'm I want to get back to my regular sleeping schedule, but it's just not doable.

[00:07:15]But I I I sacrificed my circadian rhythm to write what I think is a pretty damn good paper. I I I edited everything. I put in citations, which I know you're that's not really something to gloat about. You're supposed to do that anyway. But I wrote it. I like it. I'm comfortable having like professionals read it. And I gave it to my adviser.

[00:07:35]I'm like, he's like, "You you read this and you tell me how good it is." And uh he called me about two days ago and he asked me if I thought about which journal I want to try and get it published with. And I don't really want to talk about that yet because I think it might be rude. and it was also a private conversation. So, but we talked about a bunch of things, me and him.

[00:07:57]But as soon as it's out there and other people can read it, then yeah, for sure I'll I'll talk about it on the channel.

[00:08:04]And I guess one other thing that I mentioned uh that I haven't talked about yet is the job interview. I had a job interview with a community college um many weeks ago. This was the first interview and the second interview was during Canada and we did it virtually and I did not practice less is more. I probably talked way more than I personally would have preferred. I kind of talked to them the way I'm talking into my camera right now. But um but even still like like uh I I kind of just rehashed a bunch of the things I said in the first interview and explained some other things a little bit better. It was more of like a traditional traditional interview. And I think it went well because they sent me another email afterwards saying, "Okay, we want to we want to push you on to round three." But by the time round three came, you know, came up, I was like, it it didn't make sense for me to take on this full-time job and try and finish my PhD at the same time because it would have been a lot of work and it probably would have delayed my graduation.

[00:09:16]So, I emailed them back and I said, "Okay, I would like to respectfully pull my hat out of the ring." They did not offer me the job, by the way. They just moved me on to round three. I said, "I would like to respectfully pull my hat out of the ring. Please consider the other candidates. If things change in the future, I will, you know, I would like to keep in contact with you guys." And I think that they did like me because they they they emailed me back and I guess they maybe wanted to know more information. I can't really remember.

[00:09:50]It's been a while since since since since we had this exchange, but they were very nice to me and I was very nice to them and like the community was really good. So, it did it was bittersweet having to well, I don't know about bittersweet is the best word for that, but you know what I mean. Like I didn't want to pull my hat out of the ring, but it just made you know mathematical sense.

[00:10:12]So, in the fall, I'm going to be teaching at the university again and I'm going to be finishing up this dissertation and then applying for postocs wherever people will take me. Hopefully, they will. I know some people maybe I shouldn't say anything. [laughter] Okay, I'm not going to comment on postocs, but I will be applying for them for sure.

[00:10:38]And um yeah, so life is pretty good right now.

[00:10:46]I don't know if I should mention this, but I guess I will because the likelihood that she will see this video is like 0%. I may or may not have a girlfriend right now, but um I don't know. We'll see where this we'll see where this goes. I don't know why I'm mentioning it because she could stop texting me tomorrow, but we'll see what happens.

[00:11:11]But um why am I looking at these notes? Why am I looking at Sheldon Axler and reading his stuff? The reason is because I keep coming back to this measure theory nonsense. This measure theory review that I've been wanting to do for like two years. Really, it goes back farther than that.

[00:11:29]And I made a bunch of videos, but I did not like the way they came out. And I ended up deleting them because I wanted to do something more than just what I normally do here where I sit in front of this table and, you know, just talk. So, I made a video just as like a trial video to see if I liked it. And it has some humor interjected in it. I don't know if the humor will come across as, you know, in bad taste or if it was it's maybe not, you know, everyone's cup of tea, but I made a video that I thought was experimental. It's experimental is what it is. And if you keep watching this video, that video will play and then you can leave some feedback.

[00:12:15]Constructive criticism, you know, I'm, you know, PhD process. I am used to constructive criticism at this point. So feel free to comment down below what you think of it. And I'm not done with the Sheldon Axler book. I mentioned Sheldon Axler because I've chosen his book to do the measure theory review. He like just like the content that he has in his book and the way he's organized the content and the way he presents the content is very comfortable for undergrads but also graduate students. So I've chosen his book. He has really nice exercises. We live in the world of AI chat tpt and quad. So if you get stuck on a homework problem, you know, you can always look it up. I mean, you know, obviously I encourage when I when I even do this, I encourage myself and others to try it, you know, by yourself. But if you're just not getting and if it's, you know, outside the realm of your capability, you know, there are resources to help get you unstuck. Just make sure that you internalize the the problem so that you know you can apply it in the future is what I'm trying to say.

[00:13:27]Anyway, so with the with the politics out of the way, um the video that's about to play is an experimental measure theory review. It covers Marov's inequality. So that's why I'm I am uh in this part of the textbook.

[00:13:45]Uh the chapters one through five have already been taken care of with this guy. This one is blank and it'll start with chapter six from Sheldon Axel's measure theory and hopefully I'll be able to get through the rest of it. But anyway, so now we'll start the experimental video. See if you see if you like it. Um, and some of the humor might be a little bit crude, but it's okay. I think it's kind of funny cuz it's my type of humor anyway. Thank you. You know, it's kind of funny, really. Every calculus course in America starts by developing the derivative first and then the integral. But measure theorist walked in and said, "Nah, screw all that, man. You've got it backwards.

[00:14:25]It's the integral that should come first, not the derivative." My personal experience with this suggests that maybe calculus does it this way because it's somehow easier to differentiate a function as opposed to integrate it. But with measure theory, my experience has always been from uh just absolute torment and tears from top to bottom. Do you want to cry now or later? Actually, how about we cry now and later?

[00:14:49]I want you to think of this question for a few seconds before proceeding with the video. Does there exist a league measurable set that fills up exactly half of every interval? Wait, what?

[00:14:59]Yeah, exactly. Half of every interval.

[00:15:02]Think of the real number line and pick a random interval. Does there exist a set E such that the measure of E intersected with this interval [music] is 1/2 the length of the interval? It's important that this set E does not depend upon the interval because that's not the point. We got to be creative when we're doing measure theory or else we're going to suffer.

[00:15:22]Okay, so whatever you want an example where you can draw some things. I know how to draw two. I like doing it. So pick the interval from 0 to 1. Divide it into eight sub intervals all of equal length. Fill in every other sub interval and call this set E. If B is equal to 0, 1/4, 1/2, 3/4 or 1, then the measure of E intersected with the interval from 0 to B exact is exactly half of B. This is sort of what we want, but not really because this only works for a finite number of B's. If we divide the interval into smaller and smaller sub intervals, then we'll get more and more B values.

[00:16:01]But again, it only works for finitely many B's. [music] We want to set E such that for any B between 0 and 1, this equation is true. Is it even possible or are we wasting our time? Is this some sort of pie in the sky nonsense that we're wasting [music] all this precious time with when we could be playing GTA 6? [music] I assume GTA 6 is out by the time of this video is uploaded to YouTube [music] because if it's not, then damn, we really got struggling grad student measure theory review before GTA 6. The investigation into the answering this question will start with developing with some very famous inequalities from analysis. One of which is the weak type bound, which involves a cool ass function called the Hardy Littlewood maximal function.

[00:16:51][laughter] First time I saw Hardy and Littlewood together, I'm like, are you kidding me?

[00:16:57]This is not a re these are not real guys that came together and said, "Let's make an inequality to make every juvenile man in America [music] laugh." Anyway, what are we talking about? Oh, yeah. The weak type bound. I don't want to talk about it right now in this video because it's kind of not really the point. So, we're going to talk about a less funny but maybe even more well-known inequality called Marov's inequality. Take a measure space x s and mu. Let h be an l1 function and c a positive number.

[00:17:29]Consider the set of all x and big x such that the absolute value of h of x is bounded from below by c. What is the measure of this set? Clearly it depends on C and H. But if H is integraable then for larger and larger values of C the measure of this set should shrink.

[00:17:48]Imagine if the absolute value of H took on the value of infinity on a set of positive measure. Then H would not be in L1. It turns out this set has measure bounded above by 1 / C * the L1 norm of H which is reasonable because the L1 norm is some finite number and as C increases the upper bound decreases to zero. So the measure of the set decreases to zero as well. The proof is short and sweet, much like Littlewood himself. Okay, I'll stop with the Little Wood jokes. Take the measure of the set we want to bound from above. This can be rewritten as the integral of the characteristic function of the set, which we can write like this. Put C on the inside and 1 / C on the outside.

[00:18:30]After all, why not? We can always multiply by one [music] in a creative way. over this set. The absolute value of h is bounded from below by c. So we can replace c in the integral by h or the absolute value of h and get something larger. [music] But this set is a subset of x. So we can make the right hand side even larger by replacing the set with x. And look at that. You have the L1 norm of H. And we have Marov's inequality served up hot and fresh.

[00:19:01]Some books I've read call this chbbeev's inequality, but Axel calls this inequality the chbbyev [music] inequality. Prove this by the way. It's an exercise. Marov was a student of the great and powerful fafnudi chbbyev. I know he's like one of the founding fathers of modern analysis, but we let's just talk for a second about what an absolutely wonderful first name this guy has. I mean, how many path nudies do you know? I know like six, but you probably don't know any. He is a Russian man, but the name Pafnudi comes from the Greek name Pepnudius, which means the man of God, which is awesome. I did a minuscule amount of research to find out if he was religious or not, but I couldn't couldn't find anything. [music] I think it'd be ironic if his if it turned out that he was an atheist, but I don't really know. He's Russian, so I assume he is. Anyway, here's another exercise.

[00:19:52]Vitali covering Lima [music] is next.

[00:19:58]>> [music] [music]