I am pretty sure that the algorithm determined that this would make me feel stupid, and that was the entire goal in how it delivered suggested content to me. It was correct.
Temu is the new Wish. As such, you have poetically described how your brain comes from a relevant place, even if it's not perfectly made. Our brains are precious, even though they are grown from little genetic material, dependant on what we have consumed and affected by their environment. So many factors that make the outcome vary a bit, just like a Temu product. TLDR: You're a genius.
Best part of this is me not knowing why this popped up but 3 seconds in went straight to the comments and got exactly what I wanted lol. Merry Xmas everyone 🤙🏽🤙🏽
just learned that this lady gained significant recognition for solving a longstanding problem concerning the Conway knot, a complex structure in knot theory. In 2018, as a graduate student, she demonstrated that the Conway knot is not "slice," resolving a question that had puzzled mathematicians for over 50 years. Congratulations
Right! I remember reading about her back ago, about the knot theory, when she was “just” a student. Pure inspiration, beautiful mind. Glad to watch a lecture of her
@AdamFontenet-g3fevery problem is imaginary until it isn’t, the real problem you refer to wouldn’t have been a problem without war. You lost here. Take the L.
Dumb down Summary: 1. In the flat, 2D world of a piece of paper, we can easily understand different shapes, like circles and squares. But in the 3D world we live in, shapes get much more complicated. 2. Mathematicians are really interested in understanding 4-dimensional shapes, which are even harder to picture. They want to know if there are different types of 4D shapes that look the same on the outside, but are actually different on the inside. 3. Mathematicians have come up with a few different ways to study 4D shapes. They can try to build different 4D shapes and then figure out how to tell them apart. They also use special math tricks called "invariants" to help identify differences between shapes. 4. Over the years, mathematicians have gone through a few different periods of studying 4D shapes. In each period, they've gotten better at both making new 4D shapes and finding new ways to tell them apart. 5. Recently, mathematicians have started using some new, clever tricks to study 4D shapes. They're finding new ways to construct 4D shapes, and they're also finding new "invariants" that can help them figure out if two 4D shapes are really the same or different. 6. One of these new tricks is called the "slic approach." It involves finding a special loop or knot in one 4D shape that doesn't exist in another 4D shape. This can show that the two shapes are different, even if they look the same on the outside. 7. Mathematicians are also using computers to help them find new 4D shapes that might be different. They're making lots of different 4D shapes and then using machine learning to try to figure out if any of them are really different on the inside. 8. One really cool idea is that the differences between 4D shapes can be hidden in a tiny, simple part of the shape. Mathematicians call this part a "cork," and they've shown that this cork is the key to understanding how 4D shapes can be different. 9. Using this idea of the "cork," mathematicians have been able to make some of the simplest possible examples of 4D shapes that are actually different on the inside, even though they look the same on the outside. 10. By understanding these simple, "corky" 4D shapes, mathematicians are hoping to get a better idea of where all the different types of 4D shapes come from, and how they're related to each other. It's like solving a big puzzle, one piece at a time! Example - The amplituhedron. It's a mind-blowing multi dimensional geometric shape that simplifies particle physics. Instead of using Feynman diagrams to calculate particle interactions, you can just study this shape, which encodes all the info you need. What's crazy is it works without space or time! It suggests that space-time might not be fundamental, but instead something that emerges from deeper geometric rules. Think of it as a cheat code for the universe: a single shape that predicts particle behavior.
This is another amazing lecture which also has a lulling quality - you ever see this great physicist, Dr. Witten? ua-cam.com/video/IE_8596AYsk/v-deo.html
I looked up her dissertation, which begins: "Knot traces are elementary 4-manifolds built by attaching a single 2-handle to the 4-ball; these are the canonical examples 4-manifolds with nontrivial middle dimensional homology. In this thesis, we give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, we show that there are knot traces where the minimal genus smooth surface generating homology is not the canonical surface, resolving a question on the 1978 Kirby problem list. We also use knot traces to give a new technique for showing a knot does not bound a smooth disk in the 4-ball, and we show that the Conway knot does not bound a smooth disk in the 4-ball. This resolves a question from the 1960s, completes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot." I stopped there, went outside, and, like an early hominid, gazed up in wonder at the luminous moon.
If I would have known how clever and funny so many of the commenters would be on this channel I would have been here sooner. I don't know if I'm any smarter but I feel like I am. If ever I want to blow up my brain, I'll just watch the video longer. Fascinating and frightening that there are people this smart. But good for her. She found her lane and she's cruising and crushing.
@@thomasknott7432well it’s almost scary to a certain extent. Take this level of complexity. Look at Geometric langlands. People understand this stuff somehow. But we still die of cancer. We haven’t colonized other planets. We haven’t even gone back to the moon. We are either screwed. Or this is all theoretical and bases itself ON itself. And therefore means nothing in the end.
@@Bubbychungs Yes, some humans still die of cancer; and yes humans haven't colonized other planets; and yes humans haven't even returned to the moon. Humans will probably continue to die from cancer or something else. But nuclear physics is founded and advanced utilizing theoretical mathematics isn't it? And isn't it true that from this basis that radiation treatments, medical imaging( X-ray, CT scan, MRI) were developed. I'm quite sure that the astrophysics, computer science, mechanical engineering and electronics utilized to put humans on the moon the first time and subsequently; had origins In theoretical mathematics and other forms of advanced mathematics. If cancers are ever eradicated, or humans ever return to the moon, or humans ever colonize other planets; all of these accomplishments will have been achieved because some mathematician posed the question; "What if ?". All philosophies, from ancient classical to modern classical cannot be proven objectively. They have their origins in one person's opinion. They are by definition; theoretical; including Atheism and Nihilism. Let's all please be respectful of the validity of other people's lived experience. I hope you well.
chat gdp's explanation of this: Imagine you're playing with a Lego set. You build two castles with the same exact pieces, and when you look at them, they look identical. But then you try to take them apart or change how the pieces fit together, and suddenly you realize… they’re not built the same way at all. You can't just rearrange the pieces the same way in both castles. Weird, right? That's what Lisa Piccirillo is talking about in her lecture. She's studying 4-dimensional shapes (manifolds) that look the same from the outside, but if you try to smooth them out or manipulate them, they’re fundamentally different inside. Why is that important? Because this ONLY happens in 4 dimensions, and no one really knows why. It’s one of the biggest mysteries in math, and solving it could change how we understand space, time, and even physics. Key takeaway? Mathematicians thought they had a system to classify all shapes, but 4D breaks the rules. There are “exotic” 4D shapes that mess with their theories. Lisa is trying to figure out what makes these shapes so weird, why they exist, and how to tell them apart. Why care? Because the universe might actually have 4 or more dimensions (hello, sci-fi vibes). Solving this puzzle could help us understand how the world really works, from black holes to quantum physics. Also, Lisa solved a math problem that stumped people for decades (the Conway Knot problem), so she's kind of a badass. 😎
The STATE of Harvard Math: Commentary on Lisa Piccirillo’s (fabulous!) presentation and the reaction of the sexist, egoist judgmental society offered by her would-be peers, as depicted by the Harvard Math dept. This channel has roughly 18,000 subscribers - a niche group. This presentation was posted 2 days prior and ALREADY saw more than 17,900 views, and almost 500 likes. An amazing response given the following TRUTHS Miss Piccirillo faces: Since the field is DOMINATED by egotistical, semi-intelligent males who compete with each other (and go VERY hard against ALL women in the field), we can SAFELY estimate MOST of these views were men. Men! Being so incredibly intelligent, each view represents up to FOUR men, all secretively salivating, huddled around a single screen. Why? So as to NOT allow their desires to be known to Miss Piccirillo nor how badly they wished to view her (excellent, informed, intreprative, state-of-the-art) summations, techniques and conclusions. Gentlemen? The MATH DOESN’T LIE! THE TRUTH BEHIND THE STATISTICS: 17,900 views by Harvard Males = 64,000 (approx) MALE viewers who were too cowardly to allow their ‘view count’ to be added to the total views for this pres-o. A very niche mostly-male group DID manage to vote, ‘awarding’ her almost 500 likes (probably 10 percent female while the remaining are extremely generous male peers or out-right horndogs, as the comments imply). 500 LIKES from 17,900 views? A rare and highly positive response given the aforementioned egotists. Based upon the comments, the ONLY thing that would have given her sexist colleagues MORE reason to LIKE would be if she disrobed and conducted her presentation topless - The comments about her physical appearance dominate the ‘discussions’ and provide further evidence as to our own conclusions regarding the sexism faced by women in academia. 500 LIKES across a field of 18,300 subscribers exceeds and is comparative in equivalence to internet porn. COMMENTARY: The Pee-nut gallery has Only a few comments which are to be taken seriously. In her chosen field, even ONE comment NOT from her team of friends, family and advisors (i.e., even a SINGLE serious questions or comment NOT from her closest collaborators) is an amazing result. Thank you, Miss Piccirillo! From an all-male, non-academic who appreciates intelligence, talent and the ability to communicate at the highest level!
Lifting weights and doing maths as ours big ancestors. Aristotle would be proud edit: this was just an humoristic comment, there's no need of overthink about it. She's a beautiful hard working woman that sures makes exercise and share with us about topology and knot theory, appreciate it and cheers for her, no need to be disrespectful...
Randomly got recommended this video and decided to watch. After ten minutes I HAD to come see the comments and you all did NOT disappoint. I had SO many questions and I have so many answers 😅
I don't understand a thing on this, but I am so extremely fascinated by Mrs. Piccirillo's passion towards her field and her passion for teaching that it gives me inspiration to pursue the field that I want and continue to share knowledge to others as best as I know how to.
Women lecturers should be forced to dress professionally. It's meant to be a catalyst for the greatest young minds, but this will obviously be hindered when they are inevitably distracted.
What really amazes me is that some people (her parents and teachers) must have (I hope) recognized her talent at a young age, nurtured it, encouraged her to be where she is today. When you think about it a little more, you will realize that many, many brilliant people are either born in poverty or die before they can achieve anything significant. But not her. She is unmistakably one of the most brilliant minds in math in the country right now (do check her wikipedia page). This makes me feel an infinite amount of awe and joy, even as I watch (and understand nothing) in this video. The human mind is an amazing thing, but without the right environment, it can't achieve anything of significance.
Right. That were my thoughts too. When I started to learn math at university and had problems with not having money for living expenses mid throw, and I couldn’t find any solution while trying to approach people. They were telling me that not everyone learns math, and go to such university; “relatives” told me to go to work on factory. Not even proposing something specific, but as a metaphor wish for the baddest work. Though I worked lots hard of works in life. Everyone made sure to put me down. Though there are loans for students and there must be some solutions. They didn’t give me it. And no one just was able to talk constructively with good willing, lacking jealousy or bad attitude. And I was very young to be able to deal with such amount of hostility. And even today I couldn’t be able to.
in a way, the fact that this video is public and accesible to everyone was only a dream 30 years ago...still other many human factors need to align (i.e. the Anna Karenina effect they called) but the access to such sources like this can contribute to democratize the knowledge for the ones who are eager to learn more and improve ...
Not to be a jerk, but this theory isn't as brilliant as one might think. Just another language which has been established for decades. Kudos to all who seek knowledge!
The majority of women are in poverty and many are forced to dress and behave certain ways. Unfortunately, being brilliant in math doesn't solve the issue of the majority of women being in poverty and practically enslaved.
@@senexa01exactly she has no fat that is why her muscles show she does not have a gym body but a starvation body too many trips to those other dimensions no time to eat or maybe no food in those dimensions 😂😂😂😂
Well if you've not come across them before you're not like;ly to know what they mean. If you picked up a novel and started reading it half way through then you wouldn't be surprised if you didn't know who a lot of the characters were. If it helps, I'm a mathematician who knows enough of the field of Algebraic Topology to pick up a broad idea of what she's on about - but if a mechanic started talking about the manifold in my car then I'd be confused and go glassy eyed very quickly. We're all good at different things - and that's good.
@@steviebudden3397 I think it's funny that people think they are never going to be capable of something because they have never done it before. That's the point of learning, you don't know something, you do some work to understand it, and then you have learned it.
Are there practical applications to what she’s discussing? Not to say that there “should” be, but I’m curious if there are, and what some of them might be?
Right off the bat, I had to Google search "what are manifolds in mathematics", and now watching "Elf" for the approximately the 500th time seems even more appealing.
@@rachinvocat9587 You can think of it as a shape, that when you zoom into any area it 'looks' flat. It's like being a flat earther -- they see the world as flat from their point of view, but if you were to zoom out, you would see that the earth is actually a sphere (ish) :) This is effeciecly the definition for a manifold, without the extra mathematical things that make it well-posed in a rigorous context :)
i took discrete math with Dr. Piccirillo last year and took an interest in abstract math shortly afterward. one of the best educators and individuals i've been able to meet
That's really cool. I love that these lectures are available online for anyone to access who has a desire to investigate these subjects. Surely nothing beats actually taking the class, but to someone who is interested and might not otherwise have access, this is wonderful.
When she said “The one you’re thinking of… a nice topological space…” made me laugh out loud as the manifold I was thinking of is attached to the exhaust system of my car.
As a math guy I had no idea cars had manifolds. Apparently they have holes in them so they're not simply connected. This means they're more complicated than the objects discussed in this lecture.
I am really high, but i am enjoying this class. It is clear that the professor is a person who has achieved excellence in her field of study and that is something really interesting. We are used to seeing the body at its maximum potential but not the minds, this is also something admirable to see.
00:02 Lisa Piccirillo speaks on exotic phenomena in dimension four 02:28 Dimension four manifolds and their classification 07:40 Smooth 4-dimensional manifolds are still not well-understood and lack classification theorems. 10:39 Classical process of building Exotica in dimension 4 16:23 Manifolds are built from simple surfaces and basic building blocks called handles. 19:02 Challenges in computing gauge theory explicitly 23:50 Development of Exotica in Different Eras 26:16 Recent work in 2021 has resulted in the first example of a pair of exotic manifolds distinguished by the Slic approach. 31:45 Exotic manifolds in dimension 4 with definite forms and their recent progress 34:00 Ske lasagna module introduced for compact exotic manifolds 40:44 The argument may disprove the ponre conjecture using exotic phenomena. 43:39 Research on the P conjecture and candidates in B4 48:49 Explanations on Exotica origins and co-bound products 51:07 Understanding H cobordism and its relation to exotic pair manifolds 56:00 Existence of exotic contractable manifolds 58:35 Handles are building blocks for creating manifolds. 1:03:37 Building exotic manifolds using two handles and carving 1:06:36 Building different manifolds with the same boundary 1:12:12 Constructing pair of manifolds using disc attachments 1:14:53 Exotic phenomena quantification through cork twisting and construction 1:21:58 Existence of contractable pairs with surprising complexity levels 1:24:34 Exotic four manifolds exist with unique properties 1:30:13 Alpha invariant for a four manifold with a B3 1:32:32 Constructing manifolds with desired invariants Crafted by My college degree from GMU.
@@DelFlo A manifold is pretty easy, it's a space that locally looks like Euclidean space. A classic example is a sphere which locally looks like a two dimensional plane. You can see this because the Earth is spherical but near us on a small scale it's flat. So you can think of it as a deformation of some flat thing like a line, plane, or 3d space. Smoothness here means the opposite of jagged. It's related to the existence of derivatives if you have some calculus. At pinched points these won't exist so you get some jagged. What's wild about this is that you don't typically think of something jagged being flat. So these topological spaces that are not smooth but are still locally flat defies intuition.
@@abebuckingham8198 So would a wormhole be an example of a non-smooth manifold? Because it essentially teleports you across the manifold of space-time while locally it still feels like you’re moving through regular 3D space. Thanks for your answer by the way.
Wicked ! I was so entertained by your explanation. Even as a graduate engineer (having studied a 💩-load of math, calculus, matrices and vectors…) I have no idea whether you are texting truth or just replying in a manner that your peers will find entertaining !! Brilliant. I thank you !!! 🫡
She didn't even talk about the fact that the 60's aluminum high-rise that GM built for their SB out flows any Edelbrock I've ever tried. Thought I was going to learn something watching this.
Hi! I have dyscalculia due to autism, which makes mathematical concepts quite challenging for me. While watching your explanation, I tried to understand the ideas through metaphors that resonate with me. I thought of exotic manifolds as humans: seemingly “the same” on the outside but unique and complex internally. I saw corks as neuroplasticity, small changes that completely transform who we are. And finally, I imagined H-cobordisms as language and touch-bridges that connect us to other internal worlds, even when we are so different. I found all of this deeply poetic and fascinating. Thank you for sparking these reflections!
What really inspires me is that so many people clicked, driven by the urge for knowledge...even though most came to the conclusion that they wouldn't understand, it's lovely to also see your comment about how you attempted to understand complex mathmatics through metaphor. This is a very interesting occurance here on youtube. It grows my faith in our future being bright.
This lecture helped me in another area of mathematics, namely statistics. I can now appreciate just how far to the right the intelligence distribution curve extends.
The first four comments I’m like literally LIVING FOR!!! 😆 Thumbs up for each one of you for literally reading my mind! Our minds are intersecting and the ESP has now kicked in!
Hyperfixation bro, this is probably someone who is really invested in her work emotionally as well as intellectually. These are people that read articles on their work for fun. Not being sarcastic at all, these people are out there and they are in love with what they do like obsessively. So their brain just sponges in all the information they can find on it.
She knows she has beautiful back and shoulders, and she loves showing them. Compensates her lack of boobs and proves she is an attractive woman (she is clever either, but we now this already).
The reason why parts of atoms seem to “pop in and out of existence” is that they can’t be perceived in the other dimensions they go to (or movie in). In other words, they’re not disappearing and going into another existence or universe. They’re still here it’s that they’re moving in all dimensions that exist right here, but that can’t be observed. At least not yet. 🤯
@ I think it’s because we didn’t/couldn’t evolve to. But, I won’t tell you that you’re wrong because I can’t prove you wrong. Your idea is a very interesting one. How amazing it would be if our brains could perceive it all, but it’s locked away?!
I think this lecture might go down in history as the only one where topology was illustrated both conceptually and physically, showcasing symmetry, strength, and transformations in reality. Bringing theory to life!
EDIT: Apparently not all of this is 100% right. See comment of @jestingrabbit This talk is about something called "four-dimensional manifolds," which is just a fancy way of looking at shapes that have four dimensions. You’re used to three dimensions (like up-down, left-right, forward-backward), but here we’re adding one more. It's a bit hard to imagine because we can't see four dimensions, but mathematicians can describe and study them with formulas. What’s a Manifold? A "manifold" is basically a shape or surface that can be very simple or super complicated. Think of a line, a circle, or even the surface of a ball-these are all examples of simple manifolds in 1D, 2D, or 3D. Now, imagine that in four dimensions! Classifying Manifolds The speaker talks about how in lower dimensions (like 1, 2, and 3), mathematicians have figured out how to "classify" or organize these shapes into types, kind of like sorting objects into bins based on their characteristics. But when it comes to four-dimensional manifolds, things get trickier. This is because we know much less about them-they’re sort of like mysterious shapes! Smooth vs. Rough Manifolds Another important idea is "smooth" manifolds versus "topological" manifolds. Imagine a smooth manifold as a super-smooth surface, like glass, and a topological one as something rougher, like sandpaper. They’re both kinds of shapes but are different in texture (smoothness). The speaker explains that in four dimensions, these differences get very interesting. Exotic Manifolds Here’s where it gets fun: some four-dimensional manifolds are called "exotic." This means they look the same as other manifolds if you see them from far away but are actually different in their smoothness if you get close. It’s like two identical drawings of a line that feel different when you touch them-one might feel smooth, and the other rough. Gauge Theory and Invariants The last bit is about how mathematicians study these manifolds. They use something called "gauge theory," which is like a set of super-powered tools for telling different manifolds apart. It's complicated, but it involves using equations to find tiny differences between manifolds. If you know how to work with these tools, you can sometimes discover that two shapes are actually exotic versions of each other. So, to sum up, this talk is about exploring strange four-dimensional shapes and finding out if they are exotic by using mathematical tools that can measure differences that aren't always obvious.
This talk is about glutes and deltoids and triceps and pectorals, which are technically covered with clothing, but this lecture would do equally well at Harward and P*rnhub.
@@mbauducco Alright, let’s make it super simple! Imagine you have a ball of yarn, and you make a knot in it. Now, some knots are easy to untangle, but some knots are really tricky. For a long, long time, people tried to figure out if one special knot-the "Conway knot"-could be untangled in a special way. Then, a very smart lady named Lisa came along. She saw this tricky knot and decided to try solving it, just like how you might try a new puzzle. And guess what? She figured out that this knot couldn’t be untangled in that special way! She solved a puzzle that had been too hard for anyone else, and everyone was super impressed. So now, thanks to Lisa, we understand this knot much better! Isn’t that cool?
I thought I learned mathematics. This is the most incomprehensible lecture I have ever listened to. If it was in Chinese I would understand more. I am just glad no one has any questions.
It´s very abstract mathematics. Do not feel bad about it. I am sure, you are pretty good at low abstract math, and honestly, this is going to solve 99% of problems in your life, in the lives of your family, in your stock portfolio, in your job, and in your education by far.
In the 1960s there was a young man that graduated from the University of Michigan. Did some brilliant work in mathematics. Specifically bounded harmonic functions. Then he went on to Berkeley. He was assistant professor. Showed amazing potential. Then he moved to Montana, and blew the competition away.
@WattSounds lol, the concepts at large make sense, the specifics and technical methods are far beyond me. That doesn't mean I can't appreciate the gravity of their study.
i dont even know algebra 1 mostly but watching someone speak about something they are so passionate about is always delightful. you certainly cant accomplish something like this if youre not in love with it ❤
@@regwatson2017lol. you sure do not look like that just because you are skinny. trust me, i know. as others have mentioned, she probably does rock climbing with a back like that. this is pure trained and used muscle
When I was a grad student a classmate of mine used to go to various free meetups around the city for the free food. I almost thought about doing that but there's only so much pizza I can eat.
Thank you Harvard Mathematics Department for putting the CDM lectures on UA-cam. The conference has a long history of very good lectures on recent top research. Piccirillo's talk on 4-manifolds certainly upholds that tradition. I'm not a topologist, but the talk gives me an inkling of what she, her collaborators, and her predecessors have accomplished, and where it fits into our understanding of low-dimensional manifolds.
I'm not a mathematician and I don't remember how to solve a simple equation. Still, the amount of alien knowledge displayed here is perplexing to me, in a good way. If only we could use this level of intelect to actually make the world a better place instead of what seems to be the current regress to the middle ages, that would be awesome.
"Howard", this is "Sheldon", why are you even leaving a comment to this video, never mind clicking this video on? Also, can you or Bernadette drive me to the comic book store tomorrow, because Leonard and Raj are mad at me again and Amy and Penny are going to a spa together.🤣🤣
It's so refreshing to watch a genuine expert discuss her area of specialty and showcase a deep intuition about the subject, with very few notes or supports.
its actually not refreshing at all to see brain power wasted on something that has no practical application in the real world. what value is gained from conjecture about the shapes of 4d objects? cool looking art? thats it?
@@greekthejimmy4107 You wouldn't understand, honestly. But remember, spacetime can be seen as a 4d manifold, just to give you an idea of what you can do with it.
Back in the day, pulling off math magic like that would’ve gotten her a one-way ticket to the stake. Glad we’ve evolved to just being confused and impressed now!
She solved the Conway knot problem, which of course is the problem of how knot to laugh when Tim Conway does his Siamese elephant routine on the Carol Burnett Show.
What about the longitudinal trans vein restrictor on the aft section of the quad-modial switch bearing? Asking for a friend. Also her brain steals all her calories and now shes ripped.
In 2020, Piccirillo published a mathematical proof in the journal Annals of Mathematics determining that the Conway knot is not a smoothly slice knot, answering an unsolved problem in knot theory first proposed over fifty years prior by English mathematician John Horton Conway.
Listening to her makes me miss Boston. In Boston, you ride public transportation and see people reading books... you can also overhear conversations on topics like this, or ai, art, music or robotics... Boston is amazing for being around a society of people who educate themselves and love to share what they learn. Boston also has other humorous social idiosyncrasies... I just don't hear people talk about this positive one very often.
Hmm you’ve sparked my interest in this Boston place you speak of, scholars all around sharing what they’ve learned, love it! Must be an elevated environment!
@@ErnestoHernandez-u2w There are a number of great colleges in close proximity to each other in Boston and neighboring 'hoods. MIT, Harvard, Berklee, Northeastern, Boston College, Tufts, Emerson... just to name a few. It's defiantly a college town. I think this may play a large part as to why.
Boston changed my life. I lived there between the ages of 18-20 and still at 50, I gasp to imagine what would have happened if I had never left Florida. I couldn't do those winters. That's the only reason I left.
I have the utmost respect for science and scientists. I don't feel that respect obligates me to watch a lecture which I haven't the slightest chance of understanding. Do your thing, Dr. Piccirillo!
Lisa Piccirillo is so clear in communicating the ideas, I actually followed (sort of) the construction around 1:08:05 (I'm not a topologist). Really excellent lecture
the algorithm only sent me here, I'm sure, because I like to check out muscular women and female athletes on YT -- i don't even hardly remember how to do long division no more, remember algebra started with a "quadratic equation" but not what that is, and never asked for a mathematics video once in my life. Dr. P., Math Genius, Muscle Goddess....!
I'm flattered that the algorithm thought I would like this.
I am pretty sure that the algorithm determined that this would make me feel stupid, and that was the entire goal in how it delivered suggested content to me. It was correct.
🤣
this comment had me laughing out loud, thank you
Right lol
😂ditto!!!
I think I have the same facial expression watching this as my dog when I'm talking to him.
u r dog looks better than you
I usually have to go over it 20 times just to wrap my mind around it
Especially, when he tilts his head side to side and still looks
I think the chalk tapping is morse code.
I read this comment before watching and 30 seconds into her talking I completely felt this comment 😂😂😂😂
Procrastinated so much that I'm watching a Harvard lecture on 4th dimension
(Wow thanks for the likes and WE MUST STOP PROCRASTINATING 😞🙏 )
wow i just finished my degree 4 Days ago and procrastinating to ask the school administration to send me the photos from our finals.
I was thinking something similar except listening to vocal fry. So difficult to get past “exxxxxotic”
Oh, hello, me.
could be worse dude, you could be on a marathon of Karen videos like I did a couple of years ago, was fun tho.
@@samuelcarstens6152 I couldn't stop focusing on her traps and delts. If she was any skinnier she would collapse into a 4th dimensional manifold
this video explains to me how big of a gap between what I know and what our civilization has learned
And what is really necessary to know
Should I be sad ?
now was not the time for me to read this
@@angelamfranco3583 wdym? Some has to know it for you to be able to buy the final product associated to this 🤷♂
🤣
she built those back muscles from years of chalk hieroglyphics
Can she do the Tommy gun thing like that old math professor though?
lmaoo
@@trentonclark222 🤣
There's a high probability she's an athlete of some sort in an "unrelated" to math activity.
I was gonna say...wow
This is the person that gets recruited to meet aliens
😆 😂
A she looks young as fuck
this is a serious comment, don't take it lightly
Guess I need to watch 'Arrival" again.
Well then, must be why it's been recommended, to a certain populist...take care & engulf this Masterpiece Theater of Mathematica.
After watching im convinced my brain has a Temu sticker on it. 😐
Nah this is a bunch of malarkey
😂
Temu is the new Wish. As such, you have poetically described how your brain comes from a relevant place, even if it's not perfectly made. Our brains are precious, even though they are grown from little genetic material, dependant on what we have consumed and affected by their environment. So many factors that make the outcome vary a bit, just like a Temu product.
TLDR: You're a genius.
😱 😆
I'm pretty sure Temu might convince me to get a YT Premium subscription.
Best part of this is me not knowing why this popped up but 3 seconds in went straight to the comments and got exactly what I wanted lol. Merry Xmas everyone 🤙🏽🤙🏽
Same, the comments are a treasury. I'd like to meet every single one of these commenters personally.
lol im doing same, opene video, put on pause and reading comments, happy new year :)
I know some of these words.
Exactly what I was thinking. Also, I think there is a manifold on my car engine. I don’t think it’s exotic or in the 4th dimension or homeomorphic.
and after watching this i know some more of them. is scary
@@Rob_132 differentiable engine manifold - it probably means that it is marked so that it can be distinguished from others of the same kind.
😁
this comment....im dying
I suddendly realize that I've been watching the video because she made me feel that i was understanding everything when i did not understand a word.
I now know exactly how Penny felt every time Sheldon tried to explain what he does for a living.
🙃
@@velocitymg hey thats why we are here. we both like big bang theory and our algorithm is basically calling us penny
She got no bra on cuz 💫
Poor harvard students paid for admission 😭😭
She's ripped and never skips brain day. My inspiration
I blinked for a second..
Love that 😂
Anorexic*
being soo skinny that all your muscles are visible isn't being ripped.
Ripped? She's a skeleton? You can't be ripped when literally have no muscle. I could practically see through her.
I am mostly shocked that after all these years no one has designed a better working eraser.
🤣🤣🤣
no doubt, same as video cameras, somehow the key moments are recorded on 1973 video technology when we can already support 8k resolution
I didn't understand a single word of this, but I made it through 10 minutes. Cheers.
Me either 😂😂
Hint: it is all made up.
Me too! 10 minutes!
3 minutes hell yea
9.25
just learned that this lady gained significant recognition for solving a longstanding problem concerning the Conway knot, a complex structure in knot theory. In 2018, as a graduate student, she demonstrated that the Conway knot is not "slice," resolving a question that had puzzled mathematicians for over 50 years. Congratulations
Right! I remember reading about her back ago, about the knot theory, when she was “just” a student. Pure inspiration, beautiful mind. Glad to watch a lecture of her
@AdamFontenet-g3f the tech youre commenting on happens to be a development aided by the imaginary solutions you are talking about. Whatever works man
Einstein famously stated that, "Imagination is more important than knowledge."
In a world of fluid dynamics, flux capacitance is king...✨️🤙💫
@AdamFontenet-g3fevery problem is imaginary until it isn’t, the real
problem you refer to wouldn’t have been a problem without war. You lost here. Take the L.
@@DWS205lloosseerrrr
Dumb down Summary:
1. In the flat, 2D world of a piece of paper, we can easily understand different shapes, like circles and squares. But in the 3D world we live in, shapes get much more complicated.
2. Mathematicians are really interested in understanding 4-dimensional shapes, which are even harder to picture. They want to know if there are different types of 4D shapes that look the same on the outside, but are actually different on the inside.
3. Mathematicians have come up with a few different ways to study 4D shapes. They can try to build different 4D shapes and then figure out how to tell them apart. They also use special math tricks called "invariants" to help identify differences between shapes.
4. Over the years, mathematicians have gone through a few different periods of studying 4D shapes. In each period, they've gotten better at both making new 4D shapes and finding new ways to tell them apart.
5. Recently, mathematicians have started using some new, clever tricks to study 4D shapes. They're finding new ways to construct 4D shapes, and they're also finding new "invariants" that can help them figure out if two 4D shapes are really the same or different.
6. One of these new tricks is called the "slic approach." It involves finding a special loop or knot in one 4D shape that doesn't exist in another 4D shape. This can show that the two shapes are different, even if they look the same on the outside.
7. Mathematicians are also using computers to help them find new 4D shapes that might be different. They're making lots of different 4D shapes and then using machine learning to try to figure out if any of them are really different on the inside.
8. One really cool idea is that the differences between 4D shapes can be hidden in a tiny, simple part of the shape. Mathematicians call this part a "cork," and they've shown that this cork is the key to understanding how 4D shapes can be different.
9. Using this idea of the "cork," mathematicians have been able to make some of the simplest possible examples of 4D shapes that are actually different on the inside, even though they look the same on the outside.
10. By understanding these simple, "corky" 4D shapes, mathematicians are hoping to get a better idea of where all the different types of 4D shapes come from, and how they're related to each other. It's like solving a big puzzle, one piece at a time!
Example - The amplituhedron.
It's a mind-blowing multi dimensional geometric shape that simplifies particle physics. Instead of using Feynman diagrams to calculate particle interactions, you can just study this shape, which encodes all the info you need. What's crazy is it works without space or time! It suggests that space-time might not be fundamental, but instead something that emerges from deeper geometric rules. Think of it as a cheat code for the universe: a single shape that predicts particle behavior.
Thanks for the high level breakdown! My friend did a PhD in manifolds so I know nothing about them.
Dimensions are multi and can be manifested by the individual who has the correct resonance.
So if I understand your summary, that makes me dumb? Looks like I’m smart after all.
Thanks for the breakdown! This is some wicked shit to fall asleep to. Hopefully my brain absorbs it
Tl;dr?
Its great to see a million views on a math lecture in just a couple of months. Truly a math moment.
what math?
I subscribed because I find it fascinating that I'm listening to my native language and a foreign language simultaneously.
Amazing comment. Yes. This.
Excellent reason for subscribing my dear fellow !!!
No you subscribed because it's a woman teaching advanced physics and you're simping
Epic comment! 😂
@@niket527physics?
thank you harvard math department for uploading stuff like this it helps me fall asleep every night
lol
This is another amazing lecture which also has a lulling quality - you ever see this great physicist, Dr. Witten? ua-cam.com/video/IE_8596AYsk/v-deo.html
I looked up her dissertation, which begins: "Knot traces are elementary 4-manifolds built by attaching a single 2-handle to the 4-ball; these are the canonical examples 4-manifolds with nontrivial middle dimensional homology. In this thesis, we give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, we show that there are knot traces where the minimal genus smooth surface generating homology is not the canonical surface, resolving a question on the 1978 Kirby problem list. We also use knot traces to give a new technique for showing a knot does not bound a smooth disk in the 4-ball, and we show that the Conway knot does not bound a smooth disk in the 4-ball. This resolves a question from the 1960s, completes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot." I stopped there, went outside, and, like an early hominid, gazed up in wonder at the luminous moon.
If I would have known how clever and funny so many of the commenters would be on this channel I would have been here
sooner. I don't know if I'm any smarter but
I feel like I am. If ever I want to blow up my brain, I'll just watch the video longer.
Fascinating and frightening that there are
people this smart. But good for her. She
found her lane and she's cruising and crushing.
Congratulations for being on social media yet still managing to retain a sense of wonder.
I will join you 😅
@@thomasknott7432well it’s almost scary to a certain extent. Take this level of complexity. Look at Geometric langlands. People understand this stuff somehow. But we still die of cancer. We haven’t colonized other planets. We haven’t even gone back to the moon.
We are either screwed. Or this is all theoretical and bases itself ON itself. And therefore means nothing in the end.
@@Bubbychungs Yes, some humans still die
of cancer; and yes humans haven't colonized other planets; and yes humans haven't even returned to the moon. Humans will probably
continue to die from cancer or something
else. But nuclear physics is founded and advanced utilizing theoretical mathematics isn't
it? And isn't it true that from this basis that
radiation treatments, medical imaging( X-ray, CT
scan, MRI) were developed. I'm quite sure that the astrophysics, computer science, mechanical
engineering and electronics utilized to put humans on the moon the first time and subsequently; had origins In theoretical mathematics and other forms of advanced
mathematics. If cancers are ever eradicated,
or humans ever return to the moon, or humans
ever colonize other planets; all of these accomplishments will have been achieved
because some mathematician posed the
question; "What if ?". All philosophies, from ancient classical to modern classical cannot
be proven objectively. They have their origins
in one person's opinion. They are by definition;
theoretical; including Atheism and Nihilism.
Let's all please be respectful of the validity of other people's lived experience. I hope you
well.
chat gdp's explanation of this:
Imagine you're playing with a Lego set. You build two castles with the same exact pieces, and when you look at them, they look identical. But then you try to take them apart or change how the pieces fit together, and suddenly you realize… they’re not built the same way at all. You can't just rearrange the pieces the same way in both castles. Weird, right?
That's what Lisa Piccirillo is talking about in her lecture. She's studying 4-dimensional shapes (manifolds) that look the same from the outside, but if you try to smooth them out or manipulate them, they’re fundamentally different inside.
Why is that important?
Because this ONLY happens in 4 dimensions, and no one really knows why. It’s one of the biggest mysteries in math, and solving it could change how we understand space, time, and even physics.
Key takeaway?
Mathematicians thought they had a system to classify all shapes, but 4D breaks the rules. There are “exotic” 4D shapes that mess with their theories. Lisa is trying to figure out what makes these shapes so weird, why they exist, and how to tell them apart.
Why care?
Because the universe might actually have 4 or more dimensions (hello, sci-fi vibes). Solving this puzzle could help us understand how the world really works, from black holes to quantum physics.
Also, Lisa solved a math problem that stumped people for decades (the Conway Knot problem), so she's kind of a badass. 😎
thank you I needed this!!!
Thank you for sharing. I would not have cared otherwise one bit except to checkout Lisa's impressive back muscles.
cool
Must include giving lectures on differential topology in my back and shoulders workout routine
Mine is diffeomorphic to hers, so I shouldn't work that hard... Anyway great lecture Lisa!!!
🤣🤣 she's ripped fr
xD
Bro 😂😂😂
The STATE of Harvard Math: Commentary on Lisa Piccirillo’s (fabulous!) presentation and the reaction of the sexist, egoist judgmental society offered by her would-be peers, as depicted by the Harvard Math dept.
This channel has roughly 18,000 subscribers - a niche group. This presentation was posted 2 days prior and ALREADY saw more than 17,900 views, and almost 500 likes. An amazing response given the following TRUTHS Miss Piccirillo faces:
Since the field is DOMINATED by egotistical, semi-intelligent males who compete with each other (and go VERY hard against ALL women in the field), we can SAFELY estimate MOST of these views were men. Men! Being so incredibly intelligent, each view represents up to FOUR men, all secretively salivating, huddled around a single screen.
Why? So as to NOT allow their desires to be known to Miss Piccirillo nor how badly they wished to view her (excellent, informed, intreprative, state-of-the-art) summations, techniques and conclusions. Gentlemen? The MATH DOESN’T LIE!
THE TRUTH BEHIND THE STATISTICS: 17,900 views by Harvard Males = 64,000 (approx) MALE viewers who were too cowardly to allow their ‘view count’ to be added to the total views for this pres-o.
A very niche mostly-male group DID manage to vote, ‘awarding’ her almost 500 likes (probably 10 percent female while the remaining are extremely generous male peers or out-right horndogs, as the comments imply).
500 LIKES from 17,900 views? A rare and highly positive response given the aforementioned egotists. Based upon the comments, the ONLY thing that would have given her sexist colleagues MORE reason to LIKE would be if she disrobed and conducted her presentation topless - The comments about her physical appearance dominate the ‘discussions’ and provide further evidence as to our own conclusions regarding the sexism faced by women in academia. 500 LIKES across a field of 18,300 subscribers exceeds and is comparative in equivalence to internet porn.
COMMENTARY: The Pee-nut gallery has Only a few comments which are to be taken seriously. In her chosen field, even ONE comment NOT from her team of friends, family and advisors (i.e., even a SINGLE serious questions or comment NOT from her closest collaborators) is an amazing result.
Thank you, Miss Piccirillo! From an all-male, non-academic who appreciates intelligence, talent and the ability to communicate at the highest level!
Lifting weights and doing maths as ours big ancestors.
Aristotle would be proud
edit: this was just an humoristic comment, there's no need of overthink about it.
She's a beautiful hard working woman that sures makes exercise and share with us about topology and knot theory, appreciate it and cheers for her, no need to be disrespectful...
Iguess, she´s climping a lot
And Plato, who liked to flex after an argument.
I think it comes from extensive chalkboard writing
Aristotle would have no idea about anything she wrote
just skinny ...
Randomly got recommended this video and decided to watch. After ten minutes I HAD to come see the comments and you all did NOT disappoint. I had SO many questions and I have so many answers 😅
Lucky you, I came here to the comments with so many questions and I am now leaving with so many questions 😂
@@its_clean LOL 😆🤣
I don't understand a thing on this, but I am so extremely fascinated by Mrs. Piccirillo's passion towards her field and her passion for teaching that it gives me inspiration to pursue the field that I want and continue to share knowledge to others as best as I know how to.
This got recommended to me and I randomly watched 28 minutes of it for no reason
Oh, there was a reason! Just admit it.😂
@@timprescott4634 you caught me I tried to play it off like it didn’t matter but the algorithm knew I couldn’t run from it any longer
me too
I'm here to learn about manifolds
Watch the real 4D muscle orchestration. 😊
Bold of UA-cam to recommend this to me. They must have seen the gold star on my recent math test.
I can't even do school maths yet here I am 😂😂
I have AI do my math for me, easy button stupid
😂😂
It's because we watched the Korean cheerleaders and Ukrainian teen dancers.
The comment section is something I CAN understand, and enjoyed very much
They're hilarious!
I'm loving these comments! 😆
LMAOO😭
Interest in math = 0
Interest in sexy mathematician = 4th dimension
guilty as charged 😂
lol
me
Real 😂 😂
Now I know how my goldfish feels if I leave the house without turning off the t.v.
Lol
😂
I WILL HAVE TO ROLL ANOTHER ONE FOR THIS.....
lol me toooo !
Same
lol best coment xD I'm in the same page dude, about to min 40 xDD
I'm just going have to turn it off, .. go back to drinking my beer and sniffing my farts.
High five
Primary takeaway from this lecture: My back workout is reeeeeeally lacking.
A search suggested she has bouldering as a hobby. (powerful rock climbing moves)
@@PB-sk9jn I could tell it was a climber's back. Bodyweight exercises just look a certain way.
Women lecturers should be forced to dress professionally. It's meant to be a catalyst for the greatest young minds, but this will obviously be hindered when they are inevitably distracted.
Baby got back 😭🤦♂️
She knew what she was doing with that shirt
The guy leaving at 6:08 is my spirit animal
What really amazes me is that some people (her parents and teachers) must have (I hope) recognized her talent at a young age, nurtured it, encouraged her to be where she is today. When you think about it a little more, you will realize that many, many brilliant people are either born in poverty or die before they can achieve anything significant. But not her. She is unmistakably one of the most brilliant minds in math in the country right now (do check her wikipedia page).
This makes me feel an infinite amount of awe and joy, even as I watch (and understand nothing) in this video. The human mind is an amazing thing, but without the right environment, it can't achieve anything of significance.
Right. That were my thoughts too.
When I started to learn math at university and had problems with not having money for living expenses mid throw, and I couldn’t find any solution while trying to approach people. They were telling me that not everyone learns math, and go to such university; “relatives” told me to go to work on factory. Not even proposing something specific, but as a metaphor wish for the baddest work. Though I worked lots hard of works in life. Everyone made sure to put me down.
Though there are loans for students and there must be some solutions. They didn’t give me it. And no one just was able to talk constructively with good willing, lacking jealousy or bad attitude. And I was very young to be able to deal with such amount of hostility. And even today I couldn’t be able to.
Plotline to White Tiger
in a way, the fact that this video is public and accesible to everyone was only a dream 30 years ago...still other many human factors need to align (i.e. the Anna Karenina effect they called) but the access to such sources like this can contribute to democratize the knowledge for the ones who are eager to learn more and improve ...
Not to be a jerk, but this theory isn't as brilliant as one might think. Just another language which has been established for decades. Kudos to all who seek knowledge!
The majority of women are in poverty and many are forced to dress and behave certain ways. Unfortunately, being brilliant in math doesn't solve the issue of the majority of women being in poverty and practically enslaved.
When your math professor comes in with slippers, golden pants and being absolutely ripped :D
It's weird, very very weird, not a good look. She looks like she has a tinder date after the lecture.
Soo fine ❤
Two important things she forgot though
@sungazer454 Most modern women don't know how to dress anymore. They will get mad when you notice.
And a thong😮
I feel like I just left a party half drunk and walked in here while looking for the bathroom.
LOL 😂
I understand this comment more than I understood what this entire video is about 😅
The only thing i understand is that the door on the very far left is not an exit.
legit.
And it was around 4:00pm according to the clock to the left of the door. ;-)
Is it allowed to draw on door thou?
I can honestly say I didn't understand a single sentence of this wow
I did NOT expect YOU here
I'm not even sure there was a sentence 😂
What about the introduction? I don't mean to brag, but I totally understood everything in the first 60 seconds!
I'm a retired mechanic - I heard "manifold" like a dog hears "Squirrel"
Oh well if that's the case I'm gonna stop watching it right now😂
The fact that there's a person out there with the job title "low dimensional topologist" makes the world a better place!
Right?? Restores your faith in human intelligence at least 😂
I was just sitting on the toilet wondering about this. Glad YT recommended this at such a crucial and convenient time!
😂😂😂😂
😂😂
yes i totally searched for this unlike yall other plebs
After 40 minutes of watching this video, all I know is my name. 🤯
We can conclude that to solve 4-dimensional manifold problems, you really need to put your back into it.
That’s great! 😂
or remove calories from your diet...
@@senexa01exactly she has no fat that is why her muscles show she does not have a gym body but a starvation body too many trips to those other dimensions no time to eat or maybe no food in those dimensions 😂😂😂😂
I am not a genius. This lecture has really driven home that point.
Well if you've not come across them before you're not like;ly to know what they mean. If you picked up a novel and started reading it half way through then you wouldn't be surprised if you didn't know who a lot of the characters were.
If it helps, I'm a mathematician who knows enough of the field of Algebraic Topology to pick up a broad idea of what she's on about - but if a mechanic started talking about the manifold in my car then I'd be confused and go glassy eyed very quickly. We're all good at different things - and that's good.
@@steviebudden3397 I think it's funny that people think they are never going to be capable of something because they have never done it before. That's the point of learning, you don't know something, you do some work to understand it, and then you have learned it.
@@steviebudden3397 very valid point
Are there practical applications to what she’s discussing? Not to say that there “should” be, but I’m curious if there are, and what some of them might be?
@@Rob_132I was curious about that as well. Hopefully someone answers…
That rug really ties the room together.
😅😂😅😂😅😂😅
You're too silly 😅😂😅
Would be a shame if someone pissed on it
This guy peed on it.
@ at least I’m house broken.
Smokey, my friend, you're entering a world of pain. A world of pain.
I feel like a worm. I'm going to cry.
😂😂😂😂 Like most of us …….
worms do not cry! stay worm, bro
Right off the bat, I had to Google search "what are manifolds in mathematics", and now watching "Elf" for the approximately the 500th time seems even more appealing.
SAME LOL and the definition made me have to google “Euclidean,” and THAT definition had the word “axioms” so I gave up 😂😂
@ 😂
@@devonkelly44thank you for saving me time googling only to find more I don’t understand
Basically An area that looks similar to another area? But that area doesnt exist within the object
@@rachinvocat9587 You can think of it as a shape, that when you zoom into any area it 'looks' flat. It's like being a flat earther -- they see the world as flat from their point of view, but if you were to zoom out, you would see that the earth is actually a sphere (ish) :) This is effeciecly the definition for a manifold, without the extra mathematical things that make it well-posed in a rigorous context :)
i took discrete math with Dr. Piccirillo last year and took an interest in abstract math shortly afterward. one of the best educators and individuals i've been able to meet
That's really cool. I love that these lectures are available online for anyone to access who has a desire to investigate these subjects. Surely nothing beats actually taking the class, but to someone who is interested and might not otherwise have access, this is wonderful.
Did you take her class at UT Austin? I live in Austin.
I had a romance with her. She is a great lover.
@@spawnterrorsure thing buddy
She was able to prove that the Conway knot is not a smoothly slice knot, a problem that had stumped mathematicians for half a century!
When she said “The one you’re thinking of… a nice topological space…” made me laugh out loud as the manifold I was thinking of is attached to the exhaust system of my car.
Exactly the same here 😅
As a math guy I had no idea cars had manifolds. Apparently they have holes in them so they're not simply connected. This means they're more complicated than the objects discussed in this lecture.
LOL same. The one currently plaguing me has a slow leak grrr
Same ….. 😂
HAHAHAHAHAHAHA
I am really high, but i am enjoying this class. It is clear that the professor is a person who has achieved excellence in her field of study and that is something really interesting. We are used to seeing the body at its maximum potential but not the minds, this is also something admirable to see.
excellence at understanding fake man made ideas.....
Forreal
Hear hear 💀
@@pqlfn Get over yourself.
00:02 Lisa Piccirillo speaks on exotic phenomena in dimension four
02:28 Dimension four manifolds and their classification
07:40 Smooth 4-dimensional manifolds are still not well-understood and lack classification theorems.
10:39 Classical process of building Exotica in dimension 4
16:23 Manifolds are built from simple surfaces and basic building blocks called handles.
19:02 Challenges in computing gauge theory explicitly
23:50 Development of Exotica in Different Eras
26:16 Recent work in 2021 has resulted in the first example of a pair of exotic manifolds distinguished by the Slic approach.
31:45 Exotic manifolds in dimension 4 with definite forms and their recent progress
34:00 Ske lasagna module introduced for compact exotic manifolds
40:44 The argument may disprove the ponre conjecture using exotic phenomena.
43:39 Research on the P conjecture and candidates in B4
48:49 Explanations on Exotica origins and co-bound products
51:07 Understanding H cobordism and its relation to exotic pair manifolds
56:00 Existence of exotic contractable manifolds
58:35 Handles are building blocks for creating manifolds.
1:03:37 Building exotic manifolds using two handles and carving
1:06:36 Building different manifolds with the same boundary
1:12:12 Constructing pair of manifolds using disc attachments
1:14:53 Exotic phenomena quantification through cork twisting and construction
1:21:58 Existence of contractable pairs with surprising complexity levels
1:24:34 Exotic four manifolds exist with unique properties
1:30:13 Alpha invariant for a four manifold with a B3
1:32:32 Constructing manifolds with desired invariants
Crafted by My college degree from GMU.
who cares? totally useless to the real world
Great work 😅
1.32.39 my brain ignites
Thank you, I can expedite my confusion.
I have a feeling that this is probably completely amazing for those who understand
It is. Just the fact that there are topological manifolds that are not smooth manifolds is pretty wild and she starts with that.
@@abebuckingham8198 What does that mean and why is it wild?
@@DelFlo A manifold is pretty easy, it's a space that locally looks like Euclidean space. A classic example is a sphere which locally looks like a two dimensional plane. You can see this because the Earth is spherical but near us on a small scale it's flat. So you can think of it as a deformation of some flat thing like a line, plane, or 3d space.
Smoothness here means the opposite of jagged. It's related to the existence of derivatives if you have some calculus. At pinched points these won't exist so you get some jagged.
What's wild about this is that you don't typically think of something jagged being flat. So these topological spaces that are not smooth but are still locally flat defies intuition.
@@abebuckingham8198 So would a wormhole be an example of a non-smooth manifold? Because it essentially teleports you across the manifold of space-time while locally it still feels like you’re moving through regular 3D space. Thanks for your answer by the way.
@@abebuckingham8198 😉
Wicked !
I was so entertained by your explanation. Even as a graduate engineer (having studied a 💩-load of math, calculus, matrices and vectors…) I have no idea whether you are texting truth or just replying in a manner that your peers will find entertaining !!
Brilliant. I thank you !!! 🫡
She didn't even talk about the fact that the 60's aluminum high-rise that GM built for their SB out flows any Edelbrock I've ever tried. Thought I was going to learn something watching this.
bahhaahaha
Hi! I have dyscalculia due to autism, which makes mathematical concepts quite challenging for me. While watching your explanation, I tried to understand the ideas through metaphors that resonate with me. I thought of exotic manifolds as humans: seemingly “the same” on the outside but unique and complex internally. I saw corks as neuroplasticity, small changes that completely transform who we are. And finally, I imagined H-cobordisms as language and touch-bridges that connect us to other internal worlds, even when we are so different. I found all of this deeply poetic and fascinating. Thank you for sparking these reflections!
Wow you are amazing. Look into how Kabbalah (jewish Mysticism), theory of information, and quantum mechanics intersect. You are spot on
It really is crazy that the jewish sages of old really already had this advanced knowledge of the way the world works.
This is beautiful ❤
but the real question is, "whoislucia1"?
What really inspires me is that so many people clicked, driven by the urge for knowledge...even though most came to the conclusion that they wouldn't understand, it's lovely to also see your comment about how you attempted to understand complex mathmatics through metaphor. This is a very interesting occurance here on youtube. It grows my faith in our future being bright.
i watch shorts of cats and family guy, why would youtube do this to me
😂😂😂😂
You have potential
They observe you, not just online
which pill did you take, red or blue? (in case you are too young, this was a movie reference 😊)
I got recommended this after watching a guy doing pullups lmao
After having so much fun reading the comments it's probably best to conclude that's what the yt algorithm had in store for me here.
This lecture helped me in another area of mathematics, namely statistics. I can now appreciate just how far to the right the intelligence distribution curve extends.
its so impressive how someone has this amount of knowledge at the top of her head
I’m just impressed at her chalkboard writing speed. Now back to reading my 45th romance novel for the year.
At least you're reading!
Nice job on the table cloth, people. Really nice work.
They are the kind of people who don't notice such things
Well, it doesn’t distract from the chalkboard at least… 😂
@@neuroticnation144 there was a chalkboard in the video, haha.
The first four comments I’m like literally LIVING FOR!!! 😆 Thumbs up for each one of you for literally reading my mind! Our minds are intersecting and the ESP has now kicked in!
I did not learn anything from what see said but I got an excellent anatomy lesson on shoulder and back muscles from what she wrote.
Man...she is something else!!No notes at all and she recalls everything at lightning speed!What a mind!!
thats how it works when you understand what you're talking about :D
I think she’s AI 😂
Hyperfixation bro, this is probably someone who is really invested in her work emotionally as well as intellectually. These are people that read articles on their work for fun. Not being sarcastic at all, these people are out there and they are in love with what they do like obsessively. So their brain just sponges in all the information they can find on it.
for the record, if a man were giving this presentation while showing a back this ripped, I would definitely mention it.
Professor Leonard, thank me later
ripped? you don't go to the gym do you.
She knows she has beautiful back and shoulders, and she loves showing them. Compensates her lack of boobs and proves she is an attractive woman (she is clever either, but we now this already).
@@ebog4841 some of us do BOATH 😅
@@ebog4841 i know both and ur mom's back math
The reason why parts of atoms seem to “pop in and out of existence” is that they can’t be perceived in the other dimensions they go to (or movie in). In other words, they’re not disappearing and going into another existence or universe. They’re still here it’s that they’re moving in all dimensions that exist right here, but that can’t be observed. At least not yet. 🤯
Only because that part of our brain has become dormant..
@
I think it’s because we didn’t/couldn’t evolve to. But, I won’t tell you that you’re wrong because I can’t prove you wrong. Your idea is a very interesting one. How amazing it would be if our brains could perceive it all, but it’s locked away?!
I think this lecture might go down in history as the only one where topology was illustrated both conceptually and physically, showcasing symmetry, strength, and transformations in reality. Bringing theory to life!
XD
i think she looks a little bit muscular because her body fat is so low but she also has so little body muscle and body mass.
I concur.
EDIT: Apparently not all of this is 100% right. See comment of @jestingrabbit
This talk is about something called "four-dimensional manifolds," which is just a fancy way of looking at shapes that have four dimensions. You’re used to three dimensions (like up-down, left-right, forward-backward), but here we’re adding one more. It's a bit hard to imagine because we can't see four dimensions, but mathematicians can describe and study them with formulas.
What’s a Manifold?
A "manifold" is basically a shape or surface that can be very simple or super complicated. Think of a line, a circle, or even the surface of a ball-these are all examples of simple manifolds in 1D, 2D, or 3D. Now, imagine that in four dimensions!
Classifying Manifolds
The speaker talks about how in lower dimensions (like 1, 2, and 3), mathematicians have figured out how to "classify" or organize these shapes into types, kind of like sorting objects into bins based on their characteristics. But when it comes to four-dimensional manifolds, things get trickier. This is because we know much less about them-they’re sort of like mysterious shapes!
Smooth vs. Rough Manifolds
Another important idea is "smooth" manifolds versus "topological" manifolds. Imagine a smooth manifold as a super-smooth surface, like glass, and a topological one as something rougher, like sandpaper. They’re both kinds of shapes but are different in texture (smoothness). The speaker explains that in four dimensions, these differences get very interesting.
Exotic Manifolds
Here’s where it gets fun: some four-dimensional manifolds are called "exotic." This means they look the same as other manifolds if you see them from far away but are actually different in their smoothness if you get close. It’s like two identical drawings of a line that feel different when you touch them-one might feel smooth, and the other rough.
Gauge Theory and Invariants
The last bit is about how mathematicians study these manifolds. They use something called "gauge theory," which is like a set of super-powered tools for telling different manifolds apart. It's complicated, but it involves using equations to find tiny differences between manifolds. If you know how to work with these tools, you can sometimes discover that two shapes are actually exotic versions of each other.
So, to sum up, this talk is about exploring strange four-dimensional shapes and finding out if they are exotic by using mathematical tools that can measure differences that aren't always obvious.
ChatGPT, pasted the transcript of the first 30 minutes with the prompt "explain to 12 year old kid"
knots
Can you prompt it for a three year old. I'm still confused😮
This talk is about glutes and deltoids and triceps and pectorals, which are technically covered with clothing, but this lecture would do equally well at Harward and P*rnhub.
@@mbauducco Alright, let’s make it super simple!
Imagine you have a ball of yarn, and you make a knot in it. Now, some knots are easy to untangle, but some knots are really tricky. For a long, long time, people tried to figure out if one special knot-the "Conway knot"-could be untangled in a special way.
Then, a very smart lady named Lisa came along. She saw this tricky knot and decided to try solving it, just like how you might try a new puzzle. And guess what? She figured out that this knot couldn’t be untangled in that special way! She solved a puzzle that had been too hard for anyone else, and everyone was super impressed.
So now, thanks to Lisa, we understand this knot much better! Isn’t that cool?
I thought I learned mathematics. This is the most incomprehensible lecture I have ever listened to. If it was in Chinese I would understand more. I am just glad no one has any questions.
It´s very abstract mathematics. Do not feel bad about it. I am sure, you are pretty good at low abstract math, and honestly, this is going to solve 99% of problems in your life, in the lives of your family, in your stock portfolio, in your job, and in your education by far.
For sure. In Chinese you talk about the same things as in English. This talk is about something you don’t even know it exists.
@@PandaPanda-ud4neIt solves ALL problems unless you are a mathematician.
@@entropica what I mean is I do not speak Chinese
Joke has flown over some heads there.
In the 1960s there was a young man that graduated from the University of Michigan. Did some brilliant work in mathematics. Specifically bounded harmonic functions. Then he went on to Berkeley. He was assistant professor. Showed amazing potential. Then he moved to Montana, and blew the competition away.
This is so cool, truly shows you how complex the universe is. We are barely scratching the surface of understanding its basal language.
You're just trying to sound clever aren't you... admit it.. the only sentence she said that made sense was "Any questions"
@WattSounds lol, the concepts at large make sense, the specifics and technical methods are far beyond me. That doesn't mean I can't appreciate the gravity of their study.
Exactly
I resemble a 4-manifold invariant, myself. The complexity instantly morphs processionally.
Iterations
I'm glad there are people who love studying this type of stuff. If it was up to me, we would be at stone age forever.
Well we wouldnt be building nukes or ai to outlive our own use so maybe we would be better off.
i dont even know algebra 1 mostly but watching someone speak about something they are so passionate about is always delightful. you certainly cant accomplish something like this if youre not in love with it ❤
I watched a good portion of this. I have no idea of what any of it means but I was enthralled.
21,000 people like her back muscles. 513 people were disappointed the 4th dimension was explained on a chalkboard.
😂
Exotic Phenomena😂
How you see the dislikes?
@@rjai4890 Return UA-cam Dislike extension :)
@@rjai4890I am questioning the Same and would love to know.
The fact that no one had questions and she didn’t go Socratic healed something in me.
Ah, you didn’t hang in there long enough for the questions.
@ I am referring to her first request for questions. Just standing there while everyone is thinking “Please don’t call on me.”
“it’s not my problem anymore” 😂
WOW, what a mind she has and you can also feel her energy and passion for her domain of expertise and I absolutely love that.
Came for the curiosity, stayed for the door marked not an exit.
Underrated, meta, comment.
Right like what is it then
Lmao thought I was the only one
Not a manifold
She needs to make a tutorial on back workouts as well.
Ngl the muscle tone is amazing.
No she is just a skinny frame so all her bones and muscles are visible.
Rock climbing my friend
@@varunrathi36I think you nailed it by suggesting she does rock climbing. It’s very popular in Austin.
@@regwatson2017lol. you sure do not look like that just because you are skinny. trust me, i know. as others have mentioned, she probably does rock climbing with a back like that. this is pure trained and used muscle
@namesashhousewares8337 If that's the case then she is narcissistically showing off. Like a guy with big biceps wearing a muscle shirt.
When she asked "questions?"...and no one asked anything, i realized i wasn't the only one who is clueless on this topic.
Luring unsuspecting pedestrians with free food and filming their reactions to lectures like this would be UA-cam gold.
yesss!
When I was a grad student a classmate of mine used to go to various free meetups around the city for the free food. I almost thought about doing that but there's only so much pizza I can eat.
Thank you Harvard Mathematics Department for putting the CDM lectures on UA-cam. The conference has a long history of very good lectures on recent top research. Piccirillo's talk on 4-manifolds certainly upholds that tradition. I'm not a topologist, but the talk gives me an inkling of what she, her collaborators, and her predecessors have accomplished, and where it fits into our understanding of low-dimensional manifolds.
"I'm not a topologist". That's a good one.
@@SimonMilesresearch but I know where Paris is on a map...
I'm not a mathematician and I don't remember how to solve a simple equation. Still, the amount of alien knowledge displayed here is perplexing to me, in a good way. If only we could use this level of intelect to actually make the world a better place instead of what seems to be the current regress to the middle ages, that would be awesome.
She has the back of a ballerina. . . And the brain in the 8th dimension…respect.
You have never seen real ballet dancers. We look different and have much better posture.
@@karadiberlinoshow us
@@karadiberlino You're literally just sh!tting on this woman with every comment you make.
Fk your feelings.
On a sidenote, this passion is inspiring.
Saying this as an engineering student that has no idea what is going on here.
"Howard", this is "Sheldon", why are you even leaving a comment to this video, never mind clicking this video on? Also, can you or Bernadette drive me to the comic book store tomorrow, because Leonard and Raj are mad at me again and Amy and Penny are going to a spa together.🤣🤣
Yeah I'm a forklift driver...
It's so refreshing to watch a genuine expert discuss her area of specialty and showcase a deep intuition about the subject, with very few notes or supports.
It's fascinating
its actually not refreshing at all to see brain power wasted on something that has no practical application in the real world. what value is gained from conjecture about the shapes of 4d objects? cool looking art? thats it?
@@greekthejimmy4107I see you are on a much lower intellectual level
@@greekthejimmy4107 You wouldn't understand, honestly. But remember, spacetime can be seen as a 4d manifold, just to give you an idea of what you can do with it.
@@Rafaeu777 so cool looking art like i said?
i read she solved a long standing math problem that stumped everyone for over 50 years. In one week casually in her spare time.
Back in the day, pulling off math magic like that would’ve gotten her a one-way ticket to the stake. Glad we’ve evolved to just being confused and impressed now!
It’s amazing how most of her shirt is in the fifth dimension
Most of her pants too 😮
Lmao
Criminally Underated
👍
You understood most of the lecture.
What blackboard? @@John-wd5cb
This has got to be the funniest comments section ever !!! Also @ 6:21 she said "doo-dads" which I fully understand.
Yeah, it IS pretty damn good!🤣🤣
GTA 6 brought me here. 😎
She does a phenomenal job of laying out and explaining every step. I like that. Very well explained.
Lisa has a great presentation prowess. 👏
Exotica is an alien car with an awesome manifold?
Serious Mathlete right there. Respect. 💪🏼🙏🏼
My Mom used to tell me stories like this at bedtime...
Then you are Jesus
I’m sure you fell asleep relaxed and full of wonder which manifested your sweetest dreams.
@@apocalypse_frauExactly
She solved the Conway knot problem, which of course is the problem of how knot to laugh when Tim Conway does his Siamese elephant routine on the Carol Burnett Show.
Highbrow humor. Nice.
She didn’t solve the Conway knot… she proved it’s not a smoothly slice knot…
What about the longitudinal trans vein restrictor on the aft section of the quad-modial switch bearing? Asking for a friend.
Also her brain steals all her calories and now shes ripped.
In 2020, Piccirillo published a mathematical proof in the journal Annals of Mathematics determining that the Conway knot is not a smoothly slice knot, answering an unsolved problem in knot theory first proposed over fifty years prior by English mathematician John Horton Conway.
What an accomplishment
I mean, knots are hard to slice smoothly
Best ever math lecture if you're an art student.
finally, a visual artist... now we can impress people in a art gallery, ey did you know there are 4D shapes? tootally worth it man.
You should watch lectures by Maggie Miller (or look at her papers), she has a lot of beautiful illustrations lol.
Listening to her makes me miss Boston. In Boston, you ride public transportation and see people reading books... you can also overhear conversations on topics like this, or ai, art, music or robotics... Boston is amazing for being around a society of people who educate themselves and love to share what they learn. Boston also has other humorous social idiosyncrasies... I just don't hear people talk about this positive one very often.
Hmm you’ve sparked my interest in this Boston place you speak of, scholars all around sharing what they’ve learned, love it! Must be an elevated environment!
@@ErnestoHernandez-u2w There are a number of great colleges in close proximity to each other in Boston and neighboring 'hoods. MIT, Harvard, Berklee, Northeastern, Boston College, Tufts, Emerson... just to name a few. It's defiantly a college town. I think this may play a large part as to why.
Boston changed my life. I lived there between the ages of 18-20 and still at 50, I gasp to imagine what would have happened if I had never left Florida.
I couldn't do those winters. That's the only reason I left.
I can’t wait for a series about this to come out on Netflix!
I have the utmost respect for science and scientists. I don't feel that respect obligates me to watch a lecture which I haven't the slightest chance of understanding. Do your thing, Dr. Piccirillo!
Lisa Piccirillo is so clear in communicating the ideas, I actually followed (sort of) the construction around 1:08:05 (I'm not a topologist). Really excellent lecture
Build half way between a ballet dancer and rock climber. Good for her!
the algorithm only sent me here, I'm sure, because I like to check out muscular women and female athletes on YT -- i don't even hardly remember how to do long division no more, remember algebra started with a "quadratic equation" but not what that is, and never asked for a mathematics video once in my life. Dr. P., Math Genius, Muscle Goddess....!