Oh, man, where do we even start with Grigori Perelman? This guy is the rock star of the math world, and he doesn’t even care! I mean, talk about beating to your own drum, right? This guy solved one of the seven Clay Millennium Problems, the Poincaré Conjecture, and what did he do? Said no thanks to a million-dollar prize. If that’s not rock ‘n’ roll, I don’t know what is.
So let’s delve deep into Perelman’s genius. Now, for those who don’t speak ‘math,’ the Poincaré Conjecture is a big deal in the world of topology. It’s the sort of problem that makes normal equations look like 2+2. This issue was hanging around for about a century, and people started to think maybe it’s one of those impossible tasks. But Grigori walked up to this giant problem and showed it who’s boss.
Forget the complicated jargon; the man took Ricci flow, a concept as confusing as assembling furniture without an instruction manual, and made it work for him. Think of Ricci flow like the universe’s way of smoothing things out, like how a river smooths out a stone. Grigori used this to simplify complex shapes down to their most basic forms, allowing him to crack the conjecture.
When you look at Perelman’s methods, you realize the guy sees the world differently. Most of us look at a ball and think, “Hey, it’s a ball.” Grigori looks at a ball and sees a universe of possibilities and equations. I mean, the guy even applied geometric analysis and Riemannian geometry to break this thing down. It’s like watching Beethoven compose; you can’t quite grasp the process, but you know something amazing is happening.
But get this: the man didn’t even patent his methods. Nope! He put it out there for everyone. He uploaded his work for peer review instead of going for instant recognition. Collaboration over competition; that was his motto. You can imagine the academic world was buzzing like a beehive. Everyone wanted to understand, to learn, to be part of this monumental achievement.
Let’s not forget that Perelman is an enigma outside of the math world too. Born in Leningrad, now known as St. Petersburg, this guy was a math prodigy from the get-go. Imagine being so good at math that by your teen years, you’re already outperforming adults with Ph.Ds! Then, he goes off to the Steklov Institute of Mathematics. But does he boast about it? Nah, he’s as humble as they come.
There’s something magnetic about his reclusiveness. He’s like the J.D. Salinger of mathematics. Interviews? Nah. Public appearances? Forget about it. Perelman has this air of mystery that just adds to his allure.
And yet, despite his aversion to the limelight, Grigori’s work has made a lasting impact on mathematics, cosmology, and even quantum mechanics. You know you’ve made it when even those who don’t understand your work are talking about it. That’s the Perelman effect for you.
So, what’s the legacy of a man who is a riddle, wrapped in a mystery, inside an enigma? It’s unquantifiable, just like his contributions to mathematics and our understanding of the universe. You can’t really put a number on genius like that, and maybe that’s just the way Grigori likes it.
How Grigori Perelman Solved the Century-Old Poincaré Conjecture
So, where should we start? Ah, yes, the Poincaré Conjecture itself. This problem in topology is like the Everest of math problems. In simple terms, it asks whether every shape that doesn’t have any holes can be transformed into a simple sphere. Sounds deceptively simple, right? But this baby had mathematicians scratching their heads for nearly 100 years.
Enter Grigori Perelman. He tackled the problem using a method called Ricci Flow, introduced by mathematician Richard S. Hamilton. Ricci Flow is a process that smooths out the curvature of a surface. Imagine a lumpy, misshapen ball of clay turning into a perfect sphere as you roll it between your hands.
The math behind Ricci Flow is complex, involving all sorts of tensor calculus and differential geometry. Now, Ricci Flow had already been used in many different ways. Still, it was Perelman who pushed the boundaries of its application.
One of the game-changing moments in solving the Poincaré Conjecture was Perelman’s introduction of ‘surgery’. No, not the medical kind! In this context, surgery means cutting and gluing parts of a shape to alter its topology without changing its essential characteristics. This was groundbreaking because it helped deal with singularities, points where the Ricci Flow went haywire.
The big deal here is that Perelman didn’t just use these methods as they were. He modified them, optimized them, and even created new mathematical methods. His entropy formula for the Ricci Flow is another marvel. It measures the complexity of the evolving shape, allowing mathematicians to predict how it will change over time.
While we’re geeking out on the math, let’s not forget the Yamabe Flow, another tool that Perelman utilized. Yamabe Flow works like Ricci Flow but considers the whole shape rather than just its curvature. It was like bringing in the heavy artillery.
Perelman’s genius doesn’t end with solving the problem. He posted his work online on ArXiv for the whole world to see, without bothering to get it formally published in a peer-reviewed journal. The man’s confidence level? Infinity.
You’d think winning a Millennium Prize and a cash award of one million dollars would be the cherry on top. But in a move that shocked everyone, he declined the prize money. Said he wasn’t in it for the money. How’s that for academic integrity?
So, to recap, Grigori Perelman didn’t just solve a century-old problem. He created new methods, applied existing ones innovatively, and opened up new possibilities in topology, geometry, and even quantum mechanics and cosmology. And he did it all while basically saying, “Eh, money’s not the point.”
How Grigori Perelman’s Papers on ArXiv Revolutionized Math
Oh boy, let me tell you, when Grigori Perelman decided to post his groundbreaking papers on ArXiv, it wasn’t just a drop in the mathematical ocean. It was a full-on cannonball that sent ripples throughout the academic world.
But wait, what’s ArXiv? For those who might not know, ArXiv is an open-access repository where researchers can upload preprints of their research papers. The idea is to get scientific discoveries out there ASAP, without waiting for lengthy peer reviews. Talk about a game-changer!
Now, on to Perelman. You see, this guy wasn’t your everyday mathematician. He was a bit of an enigma, both intensely private and mind-blowingly brilliant. His focus? Topology, the study of properties that stay the same through deformations, twistings, and stretchings. Think of it as advanced geometry, but way, way cooler.
Grigori Perelman came to solve one of the greatest problems in topology, the Poincaré Conjecture. Now, I know what you’re thinking. Why upload something that major on ArXiv? Well, that’s Perelman for you—always unpredictable and truly revolutionary. He believed that his work should be publicly accessible, a noble thought if you ask me.
But let’s dive into the papers themselves. What did they contain? Perelman employed Ricci Flow, a concept from the world of differential geometry. This is basically a process that smoothens out the kinks in a geometric shape, like ironing a wrinkled shirt but way, way more complex.
Ricci Flow wasn’t new. What was new was how Perelman applied it to tackle the Poincaré Conjecture. He introduced something he called entropy formulas to measure this flow. This is some hardcore math, but in essence, these formulas provided a framework to understand how geometric shapes evolve over time.
The ArXiv papers also dove deep into surgery procedures. No, not the kind you’re thinking of—these are methods to simplify complex shapes by cutting them and then sewing them back in simpler ways. Think of it as shape “plastic surgery.” Perelman was a virtuoso in this field, and his ArXiv papers showcased his skill in all its glory.
It’s important to note that Grigori Perelman didn’t just upload a single paper; he uploaded a series of papers that complemented each other. Together, they provided a comprehensive look at his solution to the Poincaré Conjecture, adding layer upon layer of complexity and insight.
This wasn’t just a dump-and-run. Perelman’s papers were carefully plotted, each one building upon the last. Every upload was like a new episode in the most riveting Netflix series you’ve ever binged, but for math. And the math community was here for it.
Criticism and reviews poured in, but guess what? Perelman took them like a champ. You see, posting on ArXiv means you’re opening yourself up to real-time scrutiny from your peers. It’s like live-streaming your discoveries. And just like that, the math world went into a frenzy, attempting to understand and verify his works.
In the end, Perelman’s decision to publish on ArXiv changed how we think about mathematical research and open-access journals. His move bypassed traditional gatekeepers, making way for immediate, global scrutiny and discussion. It was an approach as unique and groundbreaking as the man himself.
And hey, did I mention that he declined a million-dollar prize for solving the Poincaré Conjecture? Yeah, that happened. But that’s a story for another time.
So, if you’ve ever questioned the power of open-access, just remember Grigori Perelman and his wild ride through ArXiv. The man, the myth, the legend, and his digital treasure trove of breakthroughs. A tale that’s not just about numbers, but also about breaking barriers and setting a new status quo.
Grigori Perelman’s Application of Ricci Flow
First off, you need to get the hang of Ricci Flow. Picture it as a smoothing tool for irregular shapes. Imagine you’ve got a crumpled paper ball. What Ricci Flow does is gradually “uncrumple” it, without tearing the paper or gluing any part of it. Sounds neat, right?
Grigori Perelman, this enigmatic Russian mathematician, took Ricci Flow and used it to crack open the century-old problem, the Poincaré Conjecture. Now, Perelman didn’t come out of the blue; he stood on the shoulders of giants like Richard S. Hamilton, who initially proposed Ricci Flow as a method to understand the geometry of three-dimensional spaces.
So, the guy introduced what he called entropy formulas to measure the flow. This isn’t about disorder or chaos, as you might think from the word ‘entropy.’ Instead, it’s a meticulous way to monitor how the geometry of a shape changes as you apply Ricci Flow. It’s like the speedometer in your car, but for shapes morphing into simpler forms.
Hang on tight, because this is where it gets technical. Perelman exploited partial differential equations in his work, which are equations involving unknown functions and their partial derivatives. To simplify, these are the mathematical tools you’d use to describe things like heat distribution over time, but in this case, they detail how shapes evolve through Ricci Flow.
Ah, surgery procedures! Don’t freak out; it’s not what you’re thinking. Perelman had to deal with something called “singularities,” essentially the ‘trouble spots’ in shapes where Ricci Flow didn’t work smoothly. Using a series of complex procedures, he would ‘cut out’ these singularities and replace them with simpler geometrical bits. Picture taking a scalpel to that crumpled paper ball to remove any stubborn creases.
Hold on to your hats; we’re not done yet! Perelman also dabbled with eigenvalues—the factors by which a linear transformation produces a change in a vector’s magnitude or direction. Alongside this, he used isoperimetric inequalities to further delve into the shapes’ intrinsic properties. In layman’s terms, these inequalities help us understand how to enclose the greatest volume using the least surface area. Perelman used these in conjunction with Ricci Flow, solidifying his solution.
Last but not least, Perelman applied iterations. You see, all of these steps weren’t just a one-time thing. The process of applying Ricci Flow, conducting surgery, and checking the properties through entropy formulas and inequalities had to be repeated many times to arrive at a more straightforward topological object. Imagine looping your favorite playlist until you get every lyric; that’s what Perelman did but in a math-geek way.
Look, Grigori Perelman and his application of Ricci Flow were no small feat. The man took a sledgehammer to one of the biggest walls in mathematics and left us in awe. His detailed methods and groundbreaking conclusions are the kind of stuff that legends are made of. But remember, Perelman’s work isn’t just about solving a mathematical conjecture. It’s about redefining the way we understand the universe, one shape at a time.
Grigori Perelman and the Geometrization Conjecture
Now, most of us know that Perelman made waves with his resolution of the Poincaré Conjecture. But how about the Geometrization Conjecture? Well, you can think of it as the bigger, badder, and even more comprehensive version of Poincaré. It’s the Godfather sequel to the original, so to speak. And let me tell ya, it is a sight to behold.
Geometrization isn’t just a mouthful; it’s a conceptual feast. In essence, it’s the practice of breaking down complex shapes into simpler, understandable parts. Think of your most complicated IKEA furniture. Now imagine breaking it down into its most basic shapes. That’s geometrization for you.
Folks, Perelman wasn’t playing around when he delved into the realms of hyperbolic and elliptic geometry. Picture a saddle for the hyperbolic, and for elliptic, think of a sphere. Perelman’s work made sense of how these geometries can be used as building blocks for other, more complex shapes. And believe me, if you can understand a saddle and a soccer ball, you’re well on your way to getting the gist of these concepts.
Just when you think it couldn’t get more interesting, here comes Ricci Flow with Surgery. What a term, right? This is where Perelman really shook things up. Imagine a plastic surgeon for shapes, smoothing out the wrinkles and making them look like a million bucks. Or rather, making them easier to understand. This is the nuts and bolts of how Perelman approached geometrization.
Don’t nod off yet; we’re getting to the fun stuff. Moduli spaces are sets that represent different but equivalent geometric structures. Think of them like the settings on your phone’s camera, tweaking the appearance but not the essence. Perelman used moduli spaces as a framework to investigate these geometric structures.
A-ha, Laplace Operators! These little nuggets are what you might call the calculators of change within a function. Named after the French mathematician Pierre-Simon Laplace, these operators play a key role in analyzing geometric shapes. They’re like the background apps running to ensure everything is copacetic.
Okay, hold your applause, because here comes canonical metrics. These are a set of ‘standard measurements’ that help us understand shapes in a uniform way. Just like you’d measure ingredients for a cake, canonical metrics give you the recipe for understanding the essence of a shape.
Last but far from least, let’s chat Uniformization Theorem. This theorem helps to take a complex shape and make it uniform using a particular kind of geometry: spherical, flat, or hyperbolic. It’s like taking a wild, out-of-control garden and turning it into a neatly trimmed hedge maze.
There you have it, the grand tapestry of Grigori Perelman‘s exploration into the Geometrization Conjecture. These concepts, tools, and methods are what Perelman and other mathematical geniuses use to understand the crazy complex world of shapes and spaces. And let me tell you, the math world hasn’t been the same since.
The Enigmatic Saga of Grigori Perelman’s Unpublished Works
Wowza, hold onto your hats, folks! If you’ve ever found yourself curious, even just a teeny bit, about Grigori Perelman, then you’re in for a treat! The guy is a math whiz, a recluse, and a genius. The kind of person who makes headlines not for what he says, but for the groundbreaking work he does…or doesn’t publish.
Okay, picture this: ArXiv, the preprint repository where scientists and mathematicians upload their latest discoveries. Now imagine that amidst all these complex equations and theorems, there are these unpublished works by Perelman that only a select few have laid eyes on. Intrigued? You should be.
Remember that time Perelman threw the math world into a tizzy with his exploration into Riemann Surfaces? You could say this unpublished gem was like a teaser trailer for what would become his full-length movie on the Poincaré Conjecture and Geometrization Conjecture.
Now, if we’re getting down to the nitty-gritty, you’ll want to know about his Complex Manifold Hypothesis. Look, we’re talking higher-dimensional spaces, we’re talking properties, and we’re talking geometries that’ll make your head spin faster than a top.
Oh boy, let’s not forget his work on hyperbolic spaces. I mean, this stuff is so out there, you’d need a telescope to see it. I’m not kidding. Perelman dives deep into the curvature and geometry of these spaces. We’re talking about mind-bending, eye-popping math here, people.
Did you know Perelman used to correspond with other mathematicians? Yes! In those letters, he would often hint at work he was doing but had yet to publish. Like, really, really complex stuff involving topological invariants and differential equations. Trust me, this isn’t chit-chat; it’s like leaving breadcrumbs for the next generation of math wizzes.
Now, I don’t know if you’ve heard of p-adic numbers, but let’s just say they’re not your run-of-the-mill integers. These are mathematical constructs that function in a whole different number system, and you guessed it, Perelman had something to say about them, albeit unpublished.
If you’re in the mood for something spicy, how about Lie Algebras? Perelman’s unpublished works suggest he had quite the interest in them. And if you’re unfamiliar, these bad boys help describe the symmetries of differential equations.
Hold your horses, because gauge theory is up next. Perelman apparently had a whole set of ideas around this bad boy, but alas, we’ve yet to see them in a published format. This is the stuff that deals with fields, forces, and particles in mathematical language. Yup, real-deal cutting-edge stuff.
So, there you have it, an inside scoop into the uncharted territory of Grigori Perelman’s unpublished works. Each one is like a treasure chest of mathematical insights waiting to be discovered. And until that day comes, we’ll just have to sit tight and wonder what mind-blowing revelations are sitting in a drawer somewhere, etched on stacks of paper.
Grigori Perelman: The Math Olympiad Gold Medalist
First, a bit of background: We’re looking at the late ’70s, when bell-bottoms were the rage, and mathletes were the unsung heroes of academia. Perelman wasn’t just any competitor; he was the real deal. This is a guy who saw numbers, shapes, and equations not as schoolwork, but as a language all its own.
Let’s zoom in on 1975. Perelman was just 8, and his parents discovered he had a knack for numbers. Fast forward a couple of years, and bam! He’s at the Mathematical Olympiad, the pinnacle of youth math competitions. No biggie, right? Except it is! We’re talking about a contest where future Fields Medalists, Abel Prize winners, and genius-level folks cut their teeth.
Okay, let’s get geeky. In 1982, Perelman, not even 17, snags gold. And not just any gold; he was perfect. Scored a full 42 out of 42 points. For those not in the know, that’s like pitching a no-hitter in the World Series but for math.
And it’s not just about getting the answers right; it’s about how you get ’em. Perelman tackled problems related to number theory, combinatorics, and geometry, showcasing a knack for divergent thinking. That’s where his work on Diophantine equations really shined. These equations are named after the ancient mathematician Diophantus and are basically algebraic equations where only integer solutions are allowed.
In the same vein, Perelman waded into graph theory, turning heads with his solutions on vertex coloring and edge coloring. These are basically ways to color a graph’s vertices and edges so adjacent ones have different colors. The issues are as complex as they sound, but Perelman was a natural.
Let’s not forget his problem-solving technique, often featuring an elegant use of inductive reasoning, a logic tool used to determine mathematical proofs. His approach was less about cranking out calculations and more about connecting dots in unexpected ways.
So, you’re wondering, how did all this Olympiad jazz shape Perelman’s later life? Oh, it did, my friends. It set the stage for his eventual tackling of the Poincaré Conjecture, a problem so tough and infamous it had remained unsolved for nearly a century. But that’s another tale for another day.
Grigori Perelman: Turning Down the Millennium Prize Like a Boss
Alright, sit tight because we’re diving deep into one of the most intriguing, jaw-dropping episodes in the realm of mathematics. Yessir, we’re talkin’ Grigori Perelman and that bombshell move of his when he said, “No thanks” to the Millennium Prize. How’s that for shaking the establishment?
So, imagine this: You’ve cracked a problem that’s stumped the brightest minds for over a century. We’re talking the Poincaré Conjecture, one of the seven Millennium Prize Problems. Named after the French mathematician Henri Poincaré, this bad boy is all about topology. In simple terms, topology is the mathematics of shapes that can be stretched or squeezed but not torn or glued.
Specifically, the conjecture posits that in three dimensions, every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. The theorem has its roots in Riemannian geometry and Ricci flow, concepts that deal with the curvature and dimensions of space. Perelman published papers outlining a proof for this theorem using Ricci flow with surgery, a term that sounds like a hospital procedure but is actually a mind-bending mathematical concept.
Now, let’s talk proofs. Mathematicians get all giddy when they find proofs. It’s their bread and butter. The rub? They’re also sticklers for peer review. So when Perelman dropped his Ricci flow with surgery proof on arXiv, an open-access site, it was like setting off fireworks. Mathematicians worldwide dissected his work, and, spoiler alert: they confirmed it was legit.
Enter the Clay Mathematics Institute. They’re the ones who set up the Millennium Prize, a cool $1 million purse for anyone who could solve any of the seven biggest puzzles in math. So, they’re looking at Perelman’s proof, and they’re thinking, “Bingo, we’ve got our guy.” Only one problem: Perelman was having none of it.
Yup, you heard that right. The guy solved one of the biggest riddles in math and then told the Clay Institute to keep their million bucks. In an unprecedented move, he declined the prize money. When asked why, Perelman said the recognition of his peers was his reward. That’s like being offered the keys to the city and saying, “Nah, I’d rather have a good chat with the mayor.”
So, you might ask, “What’s the big deal?” Well, this isn’t just about turning down a wad of cash. It’s a statement about the very nature of scientific inquiry. Perelman shattered the notion that external validation is the be-all and end-all. He emphasized the purity of academic pursuit over financial gain.
Perelman’s refusal to accept the Millennium Prize wasn’t a snub; it was a philosophical stand. And let’s face it, in the hallowed halls of academia where publication and prizes often define one’s worth, turning down a million bucks is as iconoclastic as it gets. The man essentially dropped the mic on the traditional academic paradigm.
So, if you’re still tracking, Grigori Perelman’s actions set a new precedent. A line in the sand, if you will. It’s a narrative that goes beyond the confines of algebraic topology and differential geometry to touch the very essence of intellectual integrity. The dude didn’t just solve a problem; he questioned the underlying motives that drive people to solve problems in the first place.
And there you have it, the nitty-gritty on why Perelman’s Millennium Prize declined is more than a footnote. It’s a manifesto, a challenge, and perhaps, an enigma wrapped in a riddle. Let’s just say it’s got all of us still scratching our heads and maybe, just maybe, rethinking what really counts.
The Legend of Grigori Perelman: Rewards, Recognition, and a Legacy
Now, let’s start with what makes this guy tick, or rather, the reason he became a household name in the world of advanced mathematics. It all began when he solved the Poincaré Conjecture, a problem related to topology that had been boggling minds since 1904. Yeah, that’s over a century of befuddlement!
Perelman’s proof turned up on arXiv, a preprint archive for papers in math and physics. That’s kind of like dropping a hot new single on Spotify. No big label, no gatekeepers, just raw talent. He applied Ricci Flow in his proof, which is all about how shapes morph in geometry. Mind you, we’re talking 3-manifolds, which is a fancy term for shapes you can’t just visualize easily. The point is, his paper sparked an academic frenzy, leading to thorough peer reviews and ultimately, validation. Yessiree, the guy was legit!
So, the big question: What rewards and recognitions did this mathematical wizard rack up? You’d think there’d be a treasure trove, right? Not exactly. Sure, there was the Fields Medal, the “Nobel Prize” of mathematics. But here’s the kicker: he declined it. Yup, no flashy ceremony, no trophy on his mantle. When asked about it, he shrugged it off, saying that he wasn’t in it for the recognition but rather for the pure joy of discovery.
And then comes the Millennium Prize. With a bounty of $1 million, this was the big kahuna of academic accolades. Drum roll, please… he declined that too! This isn’t just being humble; this is like turning down a lottery ticket that’s a guaranteed win.
But hey, let’s not forget that prizes and medals aren’t the only form of recognition. The guy’s name is pretty much synonymous with intellectual integrity. In academic circles, you say “Perelman,” and people know you’re talking about a level of dedication and purity that’s as rare as a perfect circle.
Okay, let’s talk legacy. What does a guy who declines every major accolade leave behind? First up, a renewed sense of what it means to be a scholar. He turned the spotlight away from external validation and redirected it to the essence of academic pursuit. He made us all question: What drives us? Is it fame, money, or the simple, unadulterated thrill of cracking a problem no one else can?
He also stirred the pot in how academics share their discoveries. By putting his proof on arXiv, Perelman broke from the traditional mold of peer-reviewed journals and academic gatekeeping. He essentially democratized knowledge, a move that’s still rippling through the academic world.
And of course, the Poincaré Conjecture itself. His solution didn’t just solve a problem; it opened a new avenue in mathematical research. Scholars now use his approach in Ricci flow to study other 3-manifold problems, delve deeper into Riemannian geometry, and even explore the realm of quantum physics.
And so we’re left pondering the paradox that is Grigori Perelman. He’s both an enigma and an open book, a recluse who shared his groundbreaking work freely with the world. His rewards and recognition may be unconventional, but his legacy? That’s set in stone—or maybe in equations, the universal language of the cosmos.
Conclusion
Who needs an Oscars-style acceptance speech when you can let your work talk, right? That’s Perelman for you. The man didn’t just decline the Fields Medal and the Millennium Prize; he pretty much said, “Thanks, but no thanks” to the entire system of academic rewards. It’s a move that makes us all question our motives, doesn’t it? Are we in it for the bling and the clout, or do we yearn for the eureka moments, the pure unfiltered joy of cracking the cosmic code?
I mean, talk about setting the bar high! His monumental proof of the Poincaré Conjecture isn’t just a solved equation gathering digital dust in a server. No siree! It’s a blueprint, a map for future scholars to explore new galaxies within geometry, topology, and even quantum mechanics.
While he might be a recluse in the eyes of the media, in the world of mathematics and science, the man is a bonafide rockstar. No gold records, no platinum albums, but a legacy that will echo through the hallowed halls of academia for generations to come. And that, my friends, is the epitome of real recognition.
So, what are we left with? A legacy that defies norms, breaks barriers, and recalibrates our understanding of academic glory. It’s not about the laurels or the medals; it’s about the indelible mark you leave on human knowledge. And if that’s the yardstick, then Grigori Perelman is off the charts, setting a course that’ll inspire mathematicians and laypeople alike for ages. How’s that for a lasting impression?
References
- The Puzzling Case of Grigori Perelman
- Solving the Unsolvable: The Poincaré Conjecture
- Grigori Perelman: A Reclusive Genius
- The Man Who Declined a Million: Perelman’s Story
- Intellectual Integrity: The Grigori Perelman Way
- The Geometry and Topology of Three-Manifolds: Perelman’s Legacy
