"Curvature is a local phenomenon. If you want to talk about global phenomena... you're talking about topology, which is a whole other discipline of mathematics."
"The fundamental group in general is the set of topologically equivalent maps from S1 into your space X."
"The fundamental group of the plane minus two points is not just two copies of the integers; in fact, it is a non-abelian group."
"Homotopy is mapping spheres into spaces and asking how many topologically different ways are there to map a sphere into a space."
"The set of topologically equivalent maps from the circle into some space is known as the fundamental group of X."
"The fundamental group...is not a number; it's a whole space. The topological invariant is Z, the integers."
"Now we have perfectly workable topology, the topology here is actually really solid and the textures which comes along with as well is also really solid which means that you can use it as a base."
"The Chern-Simons invariants is really a number, that's a numerical invariant of the three-manifold."
"Creating that detailed 3D model will require proper topology."
"Proper topology makes you a lot more employable."
"This is not going to be in terms of the deep principles of topology, it's purely going to be what tools to use."
"Getting a nice topology is the foundation of good modeling."
"So by connecting that displacement node, it's going to actually start pushing topology around."
"Topological techniques that would be in pure mathematics, but now with very real applications."
"Good edge flow and topology make separating and unfolding easier."
"If your topology is too low in resolution, you will have a greater amount of distortion."
"The real dynamical laws are going to take the form of some claims about the topology of the distribution of matter and fields."
"Now we have the topology created, the rules assigned, all based on subtypes."
"Good topology in the beginning really pays off."
"But, uh, yeah, remeshing is basically just, um, it recreates the mesh based on that, so it kind of gives you something with nice topology."
"It's impossible to tell where or if we just look at the heads which one has better topology."
"This kind of topology can definitely work for some things."
"An argument for having good topology is the fact that it becomes a lot easier to even map something like this."
"A combination of algebraic geometry and algebraic topology methods."
"Topology won't make a good character, but without it, it can break a character."
"Better topology can really improve your character."
"The Möbius strip is a geometric figure with the unusual quality that it only has one side."
"If I want to just completely rework the topology, I will just use quadra. I'll use another technique where sometimes if I just want to just completely rework the topology, I will just use quadra."
"The lower the topology, the faster that you go, right?"
"...this is why it's so good to have good topology, consistent design, and a well-thought-out mesh..."
"There is no perfect topology, but there are lots of really bad topologies out there."
"Topology is the study of those properties of a space that are not changed by homeomorphisms."
"Topology is the study of those invariants that do not change under continuous transformation."
"If you want to tell spaces apart and they're homotopic, then you need to use stronger invariants."
"Deformation retraction is a homotopy from the identity to a retraction."
"The ultimate object of topology should be to classify spaces up to homeomorphism."
"I have two open sets which cover the top half of Twister space, one is the left-hand part which includes the dotted part in the middle, one is the right-hand part which includes the dotted part in the middle."
"OSPF topology table is also known as the Link State Database (LSDB), containing Link State Advertisements (LSAs)."
"Now we can find the Euler Characteristic for any dimensional shape, and it still does exactly what we want it to do: count the number of holes."
"In the absolute simplest terms possible, the Poincaré Conjecture claims that if you have any four-dimensional solid and every loop that you draw on that solid can be shrunk to a point, then you can morph that solid into a 3-sphere, or as I said before, a glome."
"The outside of a black hole is the same as the inside, resembling a Klein bottle with only one surface."
"A Möbius strip or Loop is a one-sided object known as a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns."
"what do you think about balanced topology is it something that you'd like to try or something that you've tried and got experience of."
"Being simply connected implies that the first cohomology group of u vanishes."
"So now I've gone through and basically just instantly re-topologized the mesh and ZRemesher was smart enough since those creases were really harsh where it was cutting the model and now I have, you know, nice clean geometry for this page here."
"The live boolean system is fairly robust, but every once in a while, if you have some weird shapes or geometry in your mesh, you could end up getting some topology errors."
"Every three-manifold has a decomposition into canonical geometric pieces."
"A star topology is an advancement over the bus. Each host is connected to a central connecting device like a hub or a switch."
"We changed the manifold, so we have to worry about the topological effect of these changes."
"For any reachy flow with surgery, after some finite time, all the manifolds have trivial pi2."
"These collapsed pieces also satisfy the geometrization conjecture by a topological classification."
"Witten observed that Chern-Simons theory is a topological theory and it's three dimensions."
"These sorts of patterns exist throughout formal topology and throughout life."
"Understanding topology is the most important thing you can do."
"One of the best things any 3D artist can do is understand topology."
"I've read this entire book and I think it's excellent. I highly recommend this to anyone wanting to learn topology."
"Simplicity is really the key to good topology because you can always take simple topology and then just subdivide it later."
"If two spaces are topologically isomorphic, then of course they also set theoretically isomorphic; this is a stronger notion, I look at the sets more closely."
"The beauty of topology is that you can change them, you can smoothly distort them and they stay the same."
"Topology says that a donut is different than a muffin, and a sphere is different from a coffee cup."
"Whole numbers cannot change continuously, and so this means this whole topological stuff is much more stable against default, dirt, defects, or anything."
"Topology and geometry were related by luck in the beginning of the 19th century by this amazing discovery by Gauss."
"It's topology that's saved the day."
"Topological insulators... a class of electrical insulators in condensed matter physics whose mathematical underpinning turns out to have a lot to do with the field of topology."
"You can think of, for example, the circle and the square... there exists a continuous deformation between them, so that they're said to be topologically equivalent."
"As long as you keep this baseline in mind, even if you go a bit too high or a bit too low in topology, it doesn't quite matter; you can still get a lot out of the retopology as long as the most important features are captured."
"It's nice and even topology, nice and quadded."
"Topo Shaper from Fredo 6 gives you the ability to create meshes using quads."
"It's all about experimentation, all about seeing what happens when you add weird topology in different places."
"One way to show that a set is closed is to show that the complement is open."
"The Mobius band is a very, very interesting object in mathematics."
"Given any point P on some n-dimensional manifold, then local to that point, the manifold is homeomorphic to n-dimensional Euclidean space."
"A topological invariant is something which is not changed under small deformations."
"The Hall conductivity is also serving as a topological invariant."
"One of the defining attributes of a topological phase should be a bulk edge correspondence."
"The basic idea is that if you just look at one electron moving in the solid and think about its wavefunction carefully using some topological concepts, you can explain a number of amazing experiments."
"Topological order is a totally different kind of order that we were forced to think about because of experiments."
"One example of this are the topological edge states... where the Berry curvature ended up playing the role of a topological invariant."
"Hybrid topology consists of more than one topology in the same network."
"A simplicial complex is more or less a bunch of vertices and edges and triangles and tetrahedra and higher dimensional analogs."
"A topological space talks about how points are connected up without saying where specifically they are in space."
"Exterior calculus and discrete exterior calculus make a very nice clean separation between topology and geometry."
"There's more than one way of putting a smooth structure on four dimensions."
"Any surface which is compact and bounded, which you can picture in a finite part of the space and which has no edges, is exactly one of these: sphere, donut, and so on up."
"The mathematical name for a donut is called a torus."
"Continuity is defined for topological spaces and there's nothing set about a topology on V or W."
"You can draw K33 on a coffee cup with a handle without any edge crossings, and we can't do that in the plane."
"The Gauss-Bonnet theorem says the total Gaussian curvature, if we integrate K over the whole surface M with respect to area, we get 2pi times something called the Euler characteristic Chi of the surface."
"Total curvature is a topological invariant of a surface."
"You can really see there's a topological difference between the two types of periodic motion."
"The Gauss-Bonnet theorem says for a smooth surface that the total Gaussian curvature of the interior plus the total geodesic curvature along the boundary is always equal to two pi times the Euler characteristic."
"By summing up all the curvature over a surface we get to see something about its global topology."
"If you were to calculate the Gaussian curvature... on the outer rim of the torus here, you would find it's positive."
"The integral of the curvature is called the Euler number and it's related to the number of holes and handles."
"A sphere less one point becomes the infinite plane."
"Gauss curvature is a topological invariant; it doesn't matter if we change the geometry of the surface."
"A simplicial k complex is manifold if the link of every vertex looks like a k minus one dimensional sphere."
"Topology is basically the process of simplifying a complex 3D model into a low resolution simplified mesh of polygons."
"Boolean plus a topo is the most common technique I use."
"We want to have some kind of mathematical way of showing that this trefoil is actually different than the unknot."
"If I take any curve and I look at the disc that's spanned by this curve, notice that this segment is in that region, and there's no way of drawing this kind of curve without the segment being in that region."
"The curve that goes all the way around the hole shows it's in fact not simply connected."
"Homology is a way to make precise the notion of a hole."
"A mathematical object has a hole if you can prevent it from shrinking to a point."
"Homology based on the idea that we want to measure holes."
"The ground states will be associated to the cohomology classes of the manifold."
"At least in this kind of toy model, we were able to completely capture the quantum field theory on this general topology."
"As you change the boundary of the structure, you also be able to change the topology."
"The solid torus is an invariant object, surface plus interior, as well as that the surface is invariant also."
"A contour is a piecewise smooth arc."
"A domain is said to be simply connected if any closed curve within it encloses only points of the domain itself."
"Every topological spherical polyhedron can be metrized in such a way that its edges all touch a sphere."
"We are really looking for very, very fundamental properties of a space, namely just the connectivity and just the number of holes that we can find."
"By compactness, there exists a finite subcover."
"Anything homeomorphic to the sphere has the property that faces plus vertices is two more than edges."
"These are all fundamentally the same thing and they're all different from anything homeomorphic to the sphere."
"Space filling curves will actually fill the entire space within which it is contained."
"I think you could get much wider generalization of the system if it understood the topological aspects rather than just the geometry."
"Point set is sort of your foundational concept of topology."
"A lot of algebraic topology is about a way of using group theory to describe the fact that certain topological spaces have holes."
"The property of being homotopic is an equivalence relation."
"The presence of that hole has a huge effect on the number of homotopy classes."
"A manifold is a topological space where if you look at any point on it, that point has some little neighborhood that looks like Euclidean space."
"In a compact metric space, every sequence has a subsequence converging to a point of X."
"A space X is compact... it's equivalent to saying that any collection of closed sets satisfies the finite intersection property."
"A base... is a collection that has the following property: you give me any open U and any point X."
"A set E is open if every point of E is an interior point of E."
"The goal here will be to learn embeddings for subgraphs that capture well the topology of interactions between nodes."
"The persistence is going to look very different because the cycle that's born here is going to fill in; it's going to take until it gets across the whole circle."
"The sort of topology of this is just staggering to me."
"We're shooting for topological spaces that can be used to model the real world."
"Every spot is locally homeomorphic to a subset of Euclidean space, and that's what a topological manifold is."
"One way we can characterize and group shapes together is by counting holes."
"The tangent bundle plays a fundamental role in studying the topology of smooth manifolds and also in studying their differential geometry."
"These are both fiber bundles over M where the fiber is a Euclidean space."
"The manifold can be given a differentiable structure stably in the sense that after you multiply it by some high dimensional Euclidean space, then it can be smoothed."
"This construction gives us three important examples of topological spaces whose homotopic theory is essential for the problem of understanding what topological manifolds are."
"All differentiable manifolds are topological manifolds, but not all topological manifolds are differentiable manifolds."
"The gammas are homeomorphic maps that take us to Euclidean space Rd."
"Topology is almost capturing something that is more fundamental than geometry."
"When you stretch the two-dimensional plane, the topological structure, the notion of connectedness, isn't going to change."
"A topological space can be built from any arbitrary set that you have."
"The empty set and the entire set capital X are always in the topology."
"We're going to define the open sets of points from this plane as these sets which do not contain their boundaries."
"The topology is a way of understanding the notion of nearness."
"A topological homeomorphism is a structure-preserving map in topology."
"That is the formal definition of a topological homeomorphism."
"Vector spaces are an environment in which you can talk about linear concepts and linear maps; topological spaces are an environment in which you can talk about continuity and continuous concepts."
"The collection of open sets is closed under arbitrary unions and finite intersections."
"We believe that this standard topology is capturing the topological structure of the two-dimensional plane."
"A topological manifold... is going to be a patchwork quilt basically of subspaces which are topologically homeomorphic to open sets in R2."
"The most intuitive examples of general topological transformations are the deformations."
"Topology comes from the Greek 'topos' and 'logos', meaning place and study."
"A doughnut is topologically the same as a coffee cup."
"A donut equals a coffee cup; they both have one hole. Well, they're homeomorphic."
"A homeomorphism gives us a way to say when two spaces are the same from the perspective of topology."
"A homeomorphism is an isomorphism in the category of topological spaces."
"Size is not a topological property."
"Shape is of course a topological property."
"Topology really describes a more general notion of shape."
"This strongly suggests that the space we obtained by identifying the endpoints of this closed interval will be homeomorphic to a circle."
"We're pushing the topology that we have on X forward to the space Y along this surjective function Q."
"The quotient topology is going to be the largest topology we can put on Y such that the map Q is going to be continuous."
"Topology can help us to elucidate these things."
"The key to topology is keeping mostly circular loops that are following the muscle structure of the face."
"Mastering topology is absolutely crucial."
"In order to get this structure and to have some sense of control over the structure, we will need to have good clean topology."
"The topology is simply connected but compact and finite."