I am a professor at the Indian Institute of Technology Palakkad working in Complex Analysis and I am going to STRONGLY RECOMMEND this series to the students here. It would be great if you complete this amazing series all the way till the last chapter on homology.

@mariusfurter

2 жыл бұрын

That is great to hear! I have personally profited a lot from the videos made within the Indian NPTEL program. I think it is wonderful that technology allows us to share teaching in this way. I am planning to continue the series at least until the chapter on compact surfaces. I will start a new series for algebraic topology once I get there, but this will take some time.

@iamppk

Ай бұрын

@@mariusfurter Jaikrishnan Sir's NPTEL lecture series on Real analysis is great, it's made up of cute, small videos accompanying by the exercises... If Anyone wants to learn real analysis well, I will suggest www.youtube.com/@kumarhcu channel AND kzread.info/head/PLyqSpQzTE6M_UG841pwaUaAYKI1iQFXp9&si=d5HVSbbrcVQ4nhns

@keziahNjiraini-nh2rh

Ай бұрын

Just mix this video and Sadhguru spiritual spaces 😅😅

Thank you for all of your hard work Marius. The series is totally brilliant.

@mariusfurter

Жыл бұрын

You are very welcome! I'm always happy when I can help people learn.

Beautiful introduction to Topology, It was so interesting that these 40 minutes seemed like 5 minutes to me.Thank you.

I loved your analogy which is related to politics, as a bachelor math student who is interesting with pol-sci. It is inspired me

Thank you very much for making this series! Very informative and helpful!

Correct energy level for presentations. Readable handwriting. Good use of colours. 9/10

@mariusfurter

2 жыл бұрын

Thanks for the positive feedback!

@cade2770

24 күн бұрын

You gave an imperfect score. Etiquette is to name reason(s) why.

I came here thanks to your ad. Amazing playlists! Subscribed.

@mariusfurter

2 жыл бұрын

Great to have you here!

Wow, great delivery! _Crystal_ clear. I'll definitely finish the series. AND you have category theory (yay), which I was also looking for. Thank you, sir!

@mariusfurter

3 жыл бұрын

Thanks, good to hear! Currently, I intend to talk about categories and topology in a follow-up series to this one.

@tacoexpressSEEDEEholeeveryones

2 жыл бұрын

Sutherd

@evionlast

2 жыл бұрын

Wait... The man is on the surface so he is a 2d man, a true flat earther 🤣

Lee's Introduction To Topological Manifolds is a really outstanding reference.

Hi, I am a masters student wanting to get my grips on topology, and i'm forming a study group with some people , to self study along with you. Just wanted to say thank you for this, you're calm and easy to follow explanations are amazing, hopefully my aim to go into riemann geometry and algebraic geometry.

@mariusfurter

Жыл бұрын

Thanks for the positive feedback. I'm happy to hear you are benefiting from the videos.

thank you for this.

Conservation of Spatial Curvature: Both Matter and Energy described as "Quanta" of Spatial Curvature. (A string is revealed to be a twisted cord when viewed up close.) Is there an alternative interpretation of "Asymptotic Freedom"? What if Quarks are actually made up of twisted tubes which become physically entangled with two other twisted tubes to produce a proton? Instead of the Strong Force being mediated by the constant exchange of gluons, it would be mediated by the physical entanglement of these twisted tubes. When only two twisted tubules are entangled, a meson is produced which is unstable and rapidly unwinds (decays) into something else. A proton would be analogous to three twisted rubber bands becoming entangled and the "Quarks" would be the places where the tubes are tangled together. The behavior would be the same as rubber balls (representing the Quarks) connected with twisted rubber bands being separated from each other or placed closer together producing the exact same phenomenon as "Asymptotic Freedom" in protons and neutrons. The force would become greater as the balls are separated, but the force would become less if the balls were placed closer together. Therefore, the gluon is a synthetic particle (zero mass, zero charge) invented to explain the Strong Force. An artificial Christmas tree can hold the ornaments in place, but it is not a real tree. String Theory was not a waste of time, because Geometry is the key to Math and Physics. However, can we describe Standard Model interactions using only one extra spatial dimension? What did some of the old clockmakers use to store the energy to power the clock? Was it a string or was it a spring? What if we describe subatomic particles as spatial curvature, instead of trying to describe General Relativity as being mediated by particles? Fixing the Standard Model with more particles is like trying to mend a torn fishing net with small rubber balls, instead of a piece of twisted twine. Quantum Entangled Twisted Tubules: “We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct.” Neils Bohr (lecture on a theory of elementary particles given by Wolfgang Pauli in New York, c. 1957-8, in Scientific American vol. 199, no. 3, 1958) The following is meant to be a generalized framework for an extension of Kaluza-Klein Theory. Does it agree with some aspects of the “Twistor Theory” of Roger Penrose, and the work of Eric Weinstein on “Geometric Unity”, and the work of Dr. Lisa Randall on the possibility of one extra spatial dimension? During the early history of mankind, the twisting of fibers was used to produce thread, and this thread was used to produce fabrics. The twist of the thread is locked up within these fabrics. Is matter made up of twisted 3D-4D structures which store spatial curvature that we describe as “particles"? Are the twist cycles the "quanta" of Quantum Mechanics? When we draw a sine wave on a blackboard, we are representing spatial curvature. Does a photon transfer spatial curvature from one location to another? Wrap a piece of wire around a pencil and it can produce a 3D coil of wire, much like a spring. When viewed from the side it can look like a two-dimensional sine wave. You could coil the wire with either a right-hand twist, or with a left-hand twist. Could Planck's Constant be proportional to the twist cycles. A photon with a higher frequency has more energy. ( E=hf, More spatial curvature as the frequency increases = more Energy ). What if Quark/Gluons are actually made up of these twisted tubes which become entangled with other tubes to produce quarks where the tubes are entangled? (In the same way twisted electrical extension cords can become entangled.) Therefore, the gluons are a part of the quarks. Quarks cannot exist without gluons, and vice-versa. Mesons are made up of two entangled tubes (Quarks/Gluons), while protons and neutrons would be made up of three entangled tubes. (Quarks/Gluons) The "Color Charge" would be related to the XYZ coordinates (orientation) of entanglement. "Asymptotic Freedom", and "flux tubes" are logically based on this concept. The Dirac “belt trick” also reveals the concept of twist in the ½ spin of subatomic particles. If each twist cycle is proportional to h, we have identified the source of Quantum Mechanics as a consequence twist cycle geometry. Modern physicists say the Strong Force is mediated by a constant exchange of Gluons. The diagrams produced by some modern physicists actually represent the Strong Force like a spring connecting the two quarks. Asymptotic Freedom acts like real springs. Their drawing is actually more correct than their theory and matches perfectly to what I am saying in this model. You cannot separate the Gluons from the Quarks because they are a part of the same thing. The Quarks are the places where the Gluons are entangled with each other. Neutrinos would be made up of a twisted torus (like a twisted donut) within this model. The twist in the torus can either be Right-Hand or Left-Hand. Some twisted donuts can be larger than others, which can produce three different types of neutrinos. If a twisted tube winds up on one end and unwinds on the other end as it moves through space, this would help explain the “spin” of normal particles, and perhaps also the “Higgs Field”. However, if the end of the twisted tube joins to the other end of the twisted tube forming a twisted torus (neutrino), would this help explain “Parity Symmetry” violation in Beta Decay? Could the conversion of twist cycles to writhe cycles through the process of supercoiling help explain “neutrino oscillations”? Spatial curvature (mass) would be conserved, but the structure could change. ===================== Gravity is a result of a very small curvature imbalance within atoms. (This is why the force of gravity is so small.) Instead of attempting to explain matter as "particles", this concept attempts to explain matter more in the manner of our current understanding of the space-time curvature of gravity. If an electron has qualities of both a particle and a wave, it cannot be either one. It must be something else. Therefore, a "particle" is actually a structure which stores spatial curvature. Can an electron-positron pair (which are made up of opposite directions of twist) annihilate each other by unwinding into each other producing Gamma Ray photons? Does an electron travel through space like a threaded nut traveling down a threaded rod, with each twist cycle proportional to Planck’s Constant? Does it wind up on one end, while unwinding on the other end? Is this related to the Higgs field? Does this help explain the strange ½ spin of many subatomic particles? Does the 720 degree rotation of a 1/2 spin particle require at least one extra dimension? Alpha decay occurs when the two protons and two neutrons (which are bound together by entangled tubes), become un-entangled from the rest of the nucleons . Beta decay occurs when the tube of a down quark/gluon in a neutron becomes overtwisted and breaks producing a twisted torus (neutrino) and an up quark, and the ejected electron. The production of the torus may help explain the “Symmetry Violation” in Beta Decay, because one end of the broken tube section is connected to the other end of the tube produced, like a snake eating its tail. The phenomenon of Supercoiling involving twist and writhe cycles may reveal how overtwisted quarks can produce these new particles. The conversion of twists into writhes, and vice-versa, is an interesting process, which is also found in DNA molecules. Could the production of multiple writhe cycles help explain the three generations of quarks and neutrinos? If the twist cycles increase, the writhe cycles would also have a tendency to increase. Gamma photons are produced when a tube unwinds producing electromagnetic waves. ( Mass=1/Length ) The “Electric Charge” of electrons or positrons would be the result of one twist cycle being displayed at the 3D-4D surface interface of the particle. The physical entanglement of twisted tubes in quarks within protons and neutrons and mesons displays an overall external surface charge of an integer number. Because the neutrinos do not have open tube ends, (They are a twisted torus.) they have no overall electric charge. Within this model a black hole could represent a quantum of gravity, because it is one cycle of spatial gravitational curvature. Therefore, instead of a graviton being a subatomic particle it could be considered to be a black hole. The overall gravitational attraction would be caused by a very tiny curvature imbalance within atoms. In this model Alpha equals the compactification ratio within the twistor cone, which is approximately 1/137. 1= Hypertubule diameter at 4D interface 137= Cone’s larger end diameter at 3D interface where the photons are absorbed or emitted. The 4D twisted Hypertubule gets longer or shorter as twisting or untwisting occurs. (720 degrees per twist cycle.) How many neutrinos are left over from the Big Bang? They have a small mass, but they could be very large in number. Could this help explain Dark Matter? Why did Paul Dirac use the twist in a belt to help explain particle spin? Is Dirac’s belt trick related to this model? Is the “Quantum” unit based on twist cycles? I started out imagining a subatomic Einstein-Rosen Bridge whose internal surface is twisted with either a Right-Hand twist, or a Left-Hand twist producing a twisted 3D/4D membrane. This topological Soliton model grew out of that simple idea. I was also trying to imagine a way to stuff the curvature of a 3 D sine wave into subatomic particles. .---------------

Perfect presentation! What software are you using?

@mariusfurter

2 жыл бұрын

Thanks! I'm using GoodNotes on iPad. For the recording I previously used Reflector, but I've since switched to recording directly on iPad. I then edit out all pauses and mistakes, either in Premier Pro on PC or Lumafusion on iPad.

Great video. May I ask what are you using to illustrate?

@mariusfurter

2 жыл бұрын

What you're seeing in the videos is screen capture of goodnotes on ipad.

3:57 ... That's really hinting at the fundamental idea, isn't it? ... that topology generalizes the idea of "nearness" without reference to a distance or a metric.

@tacoexpressSEEDEEholeeveryones

2 жыл бұрын

Sutherd

Man, great video. I praise youtube algorithms for recommending your video. However can i ask a question in case you check your comments. Look, i really grasped idea of how topology helps us to define the most general idea of continuity. However, i understand it and especially understand open sets when they are defined only on real numbers. You might think that this is obsession, but im struggling to understand topology on finite set like your {1,2,3}. When you defined your topology on this set, what did we obtained conceptually? how can i even understand topology on finite set? is it about nearness, closeness? but they are discrete, i cant even create some epsilon ball on such set

@mariusfurter

3 жыл бұрын

I'm happy youtube brought you here too :). I think there are two issues here. The first is "what do open sets of a topology mean?". The second is "what happens when the topological space is finite (or in some sense discrete)?". Regarding the first, I'm not sure I have a satisfying answer. In some cases, thinking of points in a neighborhood of y (open set around y) as being "close" or "near" y seems reasonable. For example, in the case of convergence of a sequence, "being arbitrarily close" means eventually being contained within every neighborhood. But here the quantifier "every" is actually doing a lot of the work. In other cases, sharing neighborhoods seems to have nothing to do with "nearness" (any two points in a space share the entire space as a neighborhood), especially if the topology is not based on some metric. One should probably think about topological spaces as tools for being able to say what continuous maps are. This is analogous to defining vector spaces in order to say what linear transformations between them are. In linear algebra, one really cares most about the linear transformations, not the vector spaces. Similarly, one could argue that continuous maps are actually the important thing in topology (later on, many of the objects one works with are just special types of continuous maps). From this viewpoint, the topology on as space determines which functions into / out of that space are continuous. For example, any function out of a space with the discrete topology is continuous, but any function from an interval to the discrete space must be constant (because the interval is connected, and any subset of the discrete space with more than a single point is disconnected). Thus, the topology on two spaces constrains which function between them are continuous. Regarding finite sets: You are right that many of the ideas change considerably when one considers finite sets. The idea of "closeness" we have for euclidean space no longer seems to work, because in a finite space, each point has a smallest neighborhood (one that is contained in all others) which one gets by intersecting all the finitely many neighborhoods of the point. Therefore, "arbitrarily close" means being within this smallest neighborhood. Moreover, if one can separate any two points in a finite space by open sets, then it must carry the discrete topology. One can still put a topology based on a metric on a finite set. For example, if you think of {1,2,..,n} as a subset of R with its usual topology induced by the euclidean metric, then you can still talk about open balls. However, if you choose epsilon ≤ 1, then the open epsilon ball around a point, say 2, will only contain 2 (since integers have distance 1 from each other). Hence, the topology you will induce is again the discrete topology. In fact, if your points are separated by some minimal distance (which will always be the case in a finite set with a metric), then the metric topology will be discrete. So we've seen that it's easy to just get the discrete topology, when one applies the usual thinking to finite sets. One way to think about the finite case is given by the fact that for a finite set A, there is a bijective correspondence between preorders on A (reflexive, transitive binary relations) and topologies on A (see Wikipedia en.wikipedia.org/wiki/Finite_topological_space#Specialization_preorder for more details). Thus one could think about topologies on a finite set as an ordering of the points, and continuous functions as order-preserving maps. For example, the topology {{},{3},{2,3},{1,2,3}} on the set {1,2,3} corresponds to the preorder where 1≤1, 2≤2, 3≤3, 1≤2≤3. If you draw this as a directed graph it is a linear chain. I hope this is helpful. I think in the beginning one just has to get used to general spaces by seeing how people work with them. The topology on the real numbers is also a bit weird the first time one encounters it (at least it was for me). Then one gets used to it. Feel free to ask more questions.

@vittoriobeghelli3561

2 жыл бұрын

@@mariusfurter Thank you! I had the same questions

So a unit circle is also an S^1 sphere.

@mariusfurter

Жыл бұрын

Yes, exactly.

I'm here because someone put a good word for your channel in Reddit

@mariusfurter

2 жыл бұрын

Okay, great! Happy you're here.

@binamra5521

2 жыл бұрын

@@mariusfurter would definitely be staying here mate. Gonna start watching your graph theory playlist tommorow.

please can you give me research topological space

I thought this was a 3ds max tutorial :))

straws have 1 hole, right?

@mariusfurter

2 жыл бұрын

Yes! Both homotopy and homology (the two most common invariants for detecting 'holes') would classify a straw (seen as the surface of a bounded cylinder) as having a single hole. Homotopy uses the idea of contracting a loop which I discussed in the video. The only loops that can't be contracted are the ones that go around the hole some number of times.

10:40 If topology T is mentioned as a 'subset of X' then it implies that X is an element of T. I don't see any reason why 'X to be an element of T' should be mentioned as a condition to satisfy.

@mariusfurter

Жыл бұрын

A topology T is a set of subsets of X, not a subset of X. The elements of T are subsets of X. Hence it makes sense to require that X is an element of T.

@nehalkalita

Жыл бұрын

​I still did not understand. Instead of subset you have used plural form 'subsets'. Even if T is a set of subsets but all the values / data in those subsets are of the superset X.

@mariusfurter

Жыл бұрын

It is confusing because there are two levels of sets happening. Maybe an example will help: Define X := {1,2,3}. Then X has six subsets, namely { }, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}. A topology T on X consist of a set of these subsets. So we need to chose some of the previous six subsets of X, for example, T := { { }, {1}, {1,2}, {1,2,3} }. Each of the elements of T is a subset of X. For T to be a topology it also has to satisfy the conditions in the definition. It must contain both { } and X, and must be closed under unions and finite intersections.

@nehalkalita

Жыл бұрын

Many websites are using the same definition as yours but in Wolfram Mathworld, the first condition is written as: 'The (trivial) subsets X and the empty set are in T' which I think is clearer. What is your opinion?

Increase the volume lmao

@quantumgaming9180

3 ай бұрын

Increase your volume 😂

Very useful examples and clear intro :)