**Topological Intelligence: AI Planning w/ Persistent Homology — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/topological-intelligence-ai-planning-persistent-homology/

Explore topological intelligence and AI planning using persistent homology to tackle complex global monetary and economic systems.

## Key Takeaways

- Topological intelligence leverages advanced mathematics to enhance AI planning in complex environments.
- AI can simulate and analyze multi-agent global monetary systems with unprecedented complexity.
- Persistent homology and topological methods provide a new language for AI to discover strategic patterns.
- This approach moves beyond traditional reinforcement learning by focusing on abstract mathematical structures.
- The research has significant implications for economic policy, financial markets, and geopolitical forecasting.

## What the video covers

- Introduction to a new AI technology called topological intelligence based on recent mathematical research.
- Application of AI to simulate and analyze complex global monetary policies and economic systems.
- Discussion of multi-player, multi-dimensional numerical simulations involving monetary, trade, and policy dynamics.
- Use of advanced mathematical abstractions from topology and persistent homology to discover new AI planning methods.
- Explanation of how AI can identify known and unknown patterns in high-dimensional spaces for strategic decision-making.
- Presentation of a novel AI mechanism that transcends traditional reinforcement learning approaches.
- Insight into a 60-page mathematical paper from the Army Engineering University of PLA introducing invariant strategic subgoal control.
- Comparison of semantic/syntactic complexity in language with numerical and mathematical complexity in AI systems.
- Demonstration of how topological concepts like homotopy classes and zero homology groups help AI understand task trajectories.
- Emphasis on the real-world impact of this research on financial systems, pension funds, and global economic foresight.

## Chapters

1. 00:00 Introduction to Topological Intelligence and New AI Technology
2. 02:21 Real-World Implications: Monetary Costs and Global Economic Impact
3. 04:36 Mathematical Abstractions and AI Mechanisms for Complexity
4. 07:01 Overview of the Key Mathematical Paper and Its Significance
5. 09:20 Reinforcement Learning and AI Task Learning Approaches
6. 13:27 Topological Concepts: Homotopy, Homeomorphism, and AI Planning
7. 16:29 AI Learning Structural Logic from Successful Trajectories
8. 19:39 Advanced Mathematical Tools for AI Strategy and Planning
9. 24:08 Summary and Deep Dive into the Research Paper

Answers

## Questions about this video

What is topological intelligence in the context of AI?

Topological intelligence refers to using advanced mathematical concepts from topology, such as persistent homology, to enable AI systems to understand and plan within complex, high-dimensional environments.

How does this AI approach differ from traditional reinforcement learning?

Unlike traditional reinforcement learning that relies on recorded experiences or interactions, this approach uses mathematical abstractions to discover strategic patterns and subgoals, enabling AI to handle more complex and abstract tasks.

What are the real-world applications of this research?

This research has significant implications for simulating and forecasting global monetary policies, economic systems, and geopolitical scenarios, impacting pension funds, social security, and trillions of dollars in financial markets.

## Full Transcript — Download SRT & Markdown

00:00

Speaker A

Hello, community. Yes, we have a new paper, and it is just amazing. Let's talk about topological intelligence. We have a new technology. Now, you know, in preparing this video, I have quite a lot of papers. Every day, I scan through the

00:13

Speaker A

papers, and you have this list here provided by me here if you are a member of the channel. And today, we're going to look at this particular paper. Now, let's play a game. Let's play a game of a global simulation of a complexity that

00:26

Speaker A

a human mind, a human brain cannot process. Let's go with monetary policies of nations. So we have known political actions from a certain pattern. We have now action patterns that have consequences. Those consequences form another pattern. And now we have the

00:43

Speaker A

task for an AI system. Hey, play a more complex game. Increase the complexity of the gameplay itself. Find new strategic dimensions.

00:53

Speaker A

Now I received some paper here for some kind of evaluation. I just want to tell you there are currently quite a lot of economic think tanks that use artificial intelligence to have a simulation of scenarios of what can happen.

01:07

Speaker A

Let's go for a monetary system here, and let's go for the complexity of the global monetary system on our planet. On our planet, so we have the bond market, we have the yield structure, we have the 10-year treasuries, we have the complex

01:19

Speaker A

monetary policy from different nations, we have different trade policies implemented by different nations, and we have maybe even some innovation policy, some technology policy, and now this forms a dense network here of interlinked problems, and we are looking here for

01:35

Speaker A

known and unknown patterns to reach our goal. Now you might say, okay, we have the abstraction agent here. So we have an LLM that builds new strategic dimensions, and along those new strategic dimensions that emerged now by the artificial

01:50

Speaker A

intelligence, we can have now a much deeper understanding of the dynamics here of the global monetary system, and yeah, you are right, and we can integrate economic complexity, and we can integrate the trade complexity with a certain numerical representation, and then we

02:04

Speaker A

just let the AI work, and I think this is the main course that we have artificial intelligence let discover this AI machine new patterns. And you might say, is this anymore just a mathematical problem? No, this now becomes a real-world problem because

02:21

Speaker A

remember this is about the costs of money. This is about pension funds. This is about social security funds. This is about trillions of US dollars globally.

02:30

Speaker A

So this has real-world implications. Now let's start with a simple dunking experiment. Let's say we have a player A here in North America, and then we have a player B, let's say here the one location of China. Now you understand that a

02:46

Speaker A

numerical simulation has multiple players and multiple dimensions in multiple monetary streams across multiple financial dimensions itself, across multiple policy dimensions, across multiple trade dimensions. So we have a numerical simulation with a complexity that is really hard to understand. And now

03:04

Speaker A

we have that not only you have two players, player A and player B, but player A is now looking here for strategic partnerships, other nations, other companies, other groups of people, other cities, other macrodynamics to influence here the global political

03:21

Speaker A

system here to have a particular effect, a goal that it wants to implement here.

03:26

Speaker A

Let's say some impact on player B. So you see it's not anymore a dual player system. This is now a real complex system with n players.

03:36

Speaker A

Now there are currently some simulations interesting how to imply AI, let's say in the implementation of a combined view of interest rates here of Federal Reserve, central banks here, short-term interest rates, the development of the bond market

03:51

Speaker A

in general, what factors influence here the bond market, how can we price in the gold dynamics, let's call it this way. So those papers are not published yet, but I just want to tell you AI is really

04:04

Speaker A

becoming kind of a foresight instrument here for geopolitical, monetary, and economic scenarios. Now you understand that the dependencies that we're dealing here are if you abstract it just patterns. We have known patterns, we have unknown patterns, and from the unknown patterns, we can further

04:23

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subdivide it into known unknowns that we know what we do not know. But we also have unknown unknowns where we have no idea that this pattern exists at all. If we go here in, I don't know, a high

04:36

Speaker A

dimensional mathematical space and we map the relationship between the players. So let's go here, and this is not about a monetary policy. This is here a video about a new AI mechanism that we apply for a complexity that has nothing to do

04:52

Speaker A

with some social media post or do a little coding. No, because think about the way we represent a complexity can be manifold. No, we can have human words.

05:03

Speaker A

So we have a semantic and we have a syntactic complexity in the English language. Beautiful. But you only can express here a certain complexity with human words. If you really go here for numerical number crunching and you have

05:17

Speaker A

to have a code base and, yeah, software engineering and beautiful everything in it. So what we really going for and please have a look at my last video although it was a member video. We looked at the numerical simulation of

05:28

Speaker A

specific scenario dynamics where we have an integrated approach where we have an LLM or an agent structure that is steering now the huge numerical simulation here, the deterministic code execution on supercomputers, but now we have another language and today I want

05:45

Speaker A

to show you we have here the language of mathematics, of a pure mathematical abstraction. We will go in a different world where we use a different mathematical language you, if you're not familiar with mathematics, you will never heard before. We will use objects that

06:00

Speaker A

you are never heard before that are completely unknown to you. But I just can tell you about the last 250 years in pure mathematics research there were some amazing developments, and we use now those mathematical developments to get a

06:15

Speaker A

better understanding how to implement new AI technologies. Let's have a closer look. This is the study of today, and yeah, a lot of people ask me, hey, do you go only with the famous institution? No, not at all. I go

06:29

Speaker A

here for the technical abstracts, and then I read here the first pages here to understand what a paper is all about. You see this here, this is an institution I've never heard in my life before, Army Engineering University of PLA, and I'm

06:42

Speaker A

here as [clears throat] a European not able to pronounce this city name, I suppose here in any correct form. So yeah, a beautiful team here in China here. They published here September 10, 2026, topological necessities mechanism invariant strategic subgoal

07:01

Speaker A

for a cross embodiment of goal condition control, and you might say, hmm, sounds like an insignificant paper, but you might be completely wrong. Those are 60 pages of pure mathematical abstract logic applied to possible AI system, and this provides us with a new

07:19

Speaker A

perspective on AI. This provides us with a new technology that we can use within AI. So let's make it simple. If you're new to AI, you know, you have different framings that you can use. I want to show you two different frames. If you

07:34

Speaker A

start learning AI, you know, you have something what we call reinforcement learning. And in this reinforcement learning, you have multiple, let's go with an offline reinforcement learning.

07:42

Speaker A

This simply means you have an offline data set, a record of experiences of other executors at scale. You have 10,000 reasoning traces from a fable 5.

07:52

Speaker A

You have 100,000 reasoning traces from an open UI system. Whatever, you have a robotic system, and then the job is acquiring now a task from that experience without that you as an AI machine have any interaction, any new

08:06

Speaker A

interaction with the real environment. So we are totally offline, and you just have to learn as an AI machine from recorded experiences. But those recorded experiences can be, I don't know, 100 million human using AI machines, and you

08:22

Speaker A

can analyze now this particular complexity or as I showed you at the beginning of this video you analyze the economic interdependence here of a global monetary system we have a goal conditioned term here. So this means we have we have a starting

08:39

Speaker A

position and a goal that we want to achieve and this is now here the reinforcement learning of course on our path to our goal we have sub goals so step one step two step three you got it and those are produced in reinforcement

08:52

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learning think about the value function we have latent actions and we have a policy output a particular strategy that the I machine applies now the sub goals defined through the quantities tied to one executor let's say this is a

09:04

Speaker A

humanoid robot or you have some other abstract robotic machine can absorb now some execute a specific variation. Now such variation is noise with respect to the task and it can deier the transfer of knowledge of experience when the

09:20

Speaker A

executor or the embodiment changes. Think about you want to explore a new planet. Let's make it really science fiction. Yeah. And you have different probes. You can send in a probe that explores here the water. You can send in

09:33

Speaker A

a probe that is a little Mars rover that travels here on the solid surface of Mars or you can have things that fly in the atmosphere.

09:43

Speaker A

Depending what robotic system you choose, you have different control mechanisms for this particular AI machine. What you want to have is an abstraction of a mathematical logic.

09:54

Speaker A

what are the option that if I send this atmos this probe into the atmosphere of Mars whatever then I have strategic option and goals I want to achieve and this can be now modeled here in a new way using mathematics. So given now a

10:10

Speaker A

theoretical complexity of thousands of possible actions. So if think about the monetary system you have more then we have all the observations we have all the known patterns from economic theory of the last 150 years a known unknowns

10:24

Speaker A

in this system. We know that sometimes we do not understand a certain dynamic evolution of a subsystem either monetary policy, economic systems, whatever. And the job is now find a unifying but a complete system description of all theoretical possible solution for this

10:40

Speaker A

particular complexity. And yeah, an example is monetary policy or you explore a new planet or whatever. So this is a highly complex task. This is not that you write an email and you post something on social media.

10:54

Speaker A

Now analyze the mathematical space of all possible system state is of course your first idea you want to implement.

11:00

Speaker A

Yeah. So given now here a certain domain complexity or multiple domain complexity. You want to analyze what is the complete mathematical space that I have and then I can have a search algorithm that searches here on the search base for my particular solution

11:14

Speaker A

given my condition here my start condition end conditions and you got it and then you want to describe the temporal and the path dependence dynamics. Think about famous puff integral quantum field theory that will lead you to a defined solution. Either

11:27

Speaker A

you have a goal or a sub goal or whatever. This is your kind of a fractal system.

11:32

Speaker A

Now frame two is when the task knowledge is tied to what robotic machine you send out to explore the planet or what task knowledge and monetary policy you want to use as a tool to have an influence on

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Speaker A

the US central banks. execute transfer the task means learning it again. So task knowledge separated from the particular executor from the tool that we use to execute something we want to have an abstraction and an abstraction that is in a mathematical way that we

12:05

Speaker A

can share it with other executor and reuse directly. So what we're looking for a mathematical representation of an executor independent task structure but remember this task structure we have now to extract here from observation from a particular executor specific robotic

12:24

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system. So we do have to have the first level of abstraction and whether it can be recovered from the experience of others. Can we have a general abstraction? This determines here the cross executed learning. Is it possible at all for reinforcement learning? and

12:39

Speaker A

then we go for non-distributed. So we do not start here just to make it clear with a complete world model where we have a mathematical descriptor that provides it with a complete analytical description of the dynamic of this

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Speaker A

simulated world model. No, we go now and we collect observation observation from the internet for whatever you have your robotic AI system your AI machine operating now for thousands of hours here in a particular domain and then you

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Speaker A

just collect all the information because you are looking for patterns for unknown undiscovered patterns. So we are taking in the raw experience here from the data streams and now we do not have a wrong model because now we have a mathematical

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Speaker A

knowledge a mathematical theory that will provide us with some extreme powerful tools and this kind of abstraction as I showed you we already had here this video of the abstraction agent here where we create a new state space geometry. So we can create here

13:43

Speaker A

our mathematical space already but now of course we go another step. So we are going to embed we have to build now an topological space. Now a topological space if you're new to metamatic is a set endowed with a structure called a

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Speaker A

topology which allow defined continuous deformation of subspaces and more generally all kinds of continue operation and the deformations are of course considered in the topology what we call a homeomorphism or homotopes. If you study mathematics, you know this is

14:12

Speaker A

exactly semester five when you finally start to learn topology. Now removing now this particular executor, this particular robotic machine entirely leaves behind something what you can call an order of unavoidable stages that have been observed by all different machinery by

14:32

Speaker A

all the probes who sent out to explore Mars. You know there was something that was always there in the data that was a pattern that was independent. If you go for water, if you go for atmosphere dynamics, for soil chemistry, you'll

14:45

Speaker A

found patterns that were coherent. Now, if you define a goal on this domain, so to reach a goal, every successful behavior must commit to one side of each loop in a free space and then traverse the corresponding region in a fixed

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Speaker A

order. If we see this through the topological glosses to the topological perspective. So my job is now to tell you I cannot present you in 20 minutes on YouTube 20 pages of pure mathematics and this is not a classical mathematic. This is here

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what I would call already advanced mathematics. So my job now and I thought about this is to show you an image.

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Speaker A

So I choose now another frame. I choose now a visualization where I show you the mathematical tools of topology homotropy how to apply this what they achieve for EI and I will then come back and tell you what mathematics can do for this new

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Speaker A

EI technology. So here we go. So here you see different kind of robotic system. No humanoid then something like a mechanic dog or something from Star Wars here. Perfect. Yeah. Different sensor regions whatever. And now you have a goal. And here on the right hand

15:58

Speaker A

side you have here this beautiful shining goal. And now you have strategic options how to reach your goal. Remember this is not the uklidian space. This is not what you have in the normal world.

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Speaker A

This is here an absolute abstract mathematically syntactic space that we construct not on ukidian or any other simple mathematics. This is here what we call yeah homot classes and carrier structures. So let's talk about topology.

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Speaker A

This image depicts the EI system that learns the structural logic of a particular task from the observed trajectories from successful trajectories rather than memorizing here the exact movements of one robotic system. So yeah, of course you can have here the memory. You read

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Speaker A

out the memory files and you know exactly at which time period what kind of robot moved in what particular sequence of motion touched what moved to what position and you got it. But you do not want this because you want to

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Speaker A

abstract it away and you want to have the pure logical abstraction from a task. a task now in a space that you cannot imagine that does not exist but we will build out of pure logic and we will find a mathematical representation

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Speaker A

of this abstract logic with the help of some mathematical tools. Great. So the 3D surface that you see here as this wobble in space is what we call here a data supported world model. So we do not have an analytical solution. We do not

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Speaker A

have Maxwell equation to explain the dynamic of this system. We just have millions millions of observations of data points what we observed particular action takes place and then we have in the environment here an observation. So this surface bubble whatever you like to

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Speaker A

call it here is what we call a carrier. What is a carrier? A car is a geometric model that is constructed from previous successful trajectories of other robotic system or of reasoning traces of large language model or vision language model.

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So a geometric model constructed from successful trajectories. Now in AI terminology if we move from pure mathematics back to artificial intelligence those are just some region of state space for which the system dynamics has evidence that a successful

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behavior was possible and on the path on the reasoning trace path to the solution we crossed around here in this region of the space that is now building here our complete geometric model here. So you see this is a very particular way to

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Speaker A

mathematically build the world. Now each location that you see here it's each point on this carrier is not a real XY Z coordinate of the real world. This is here absolute mathematical abstraction and you have much more information than the physical position

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XY Z of a robot. You have configuration files, you have the object state descriptors, you have the context state descriptors, you have general abstraction of the task process of all the features that are relevant in the control cycle. So this is a pure

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mathematical abstraction. You cannot imagine this. This is abstract. So the carrier but in our logic in our complete mathematical coherent girdle proofed logic is this carrier is not a literal map of a room or something. It is a

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topological task state manifold that is assembled now in our world building from successful demonstration where a machine reached its goal and now we take millions and millions of those demonstration and the eye has not the task to find common patterns. So if you

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want in my simple words this is kind of an abstraction of all the possible path to the goal. If you have in monetary policy a goal those are you want to know all possible path you don't want to know

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only what happened in history what happened in 1958 here in the United States or what happened in the 1980s in Japan no you want to have a complete system dynamics that integrates your known patterns and unknown patterns and

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you want to explore the unknown patterns the unknown [clears throat] political options for example or trade policies that you can apply to reach your goal.

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Speaker A

Now you see here that we have here because there is something in the middle and I will have a look at this what this is. We have now homotopic classes in our mathematical description. You cannot imagine this. This is nothing you will

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encounter in the real world. This is a mathematical logic that was developed to solve certain mathematical problems.

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Yeah, this is yeah university semester 5. The two colored pathways are not intended to show only two individual rollouts. Remember this is here a state of states. So each ribbon here that we have the orange one and the blue one

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represents families of similar successful trajectories. So inside one ribbon the light individual lines show different execution paths but all the trajectory in that particular bundle and this is a mathematical expression implement the same highle route strategy.

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So you want to know your strategic option and you want to analyze all strategic option in an unknown complexity space. So you have here and I tried to generate this picture here as a homotropy class gamma minus and gamma

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plus. And you say okay I identify now two different path homotopy classes. If you're not familiar with mathematics, a pathomotopic class is mathematically speaking a group or more precisely an equivalence class of trajectory that implement the same topological route.

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Speaker A

Okay. So this means but a path from one colored bundle cannot be transformed continuously with any mathematical mapping operation into a p from the other bundle. So you cannot map here the blue one to the orange one onto the

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orange one without crossing here this particular obstacle and this will be a topological hole structure leaving here or leaving here the demonstrated space or returning and going around here the obstacle in a different way. So this means for the eye this means that these

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are not merely variations of noise like in diffusion operation these are really generally different planning option and here we go now because you want to reach your goal. So you are planning now a strategy to reach your goal and you want

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to identify okay I don't want to have tiny variation from one path to reach my goal I want to have here a complete understanding of all my option that are mathematical possible so you want to have the genuine different playing

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option and you see this is where you build exactly this mathematical structures now let's talk about this this thing in the middle this this obstacle the dominance 3D object and center is a region that successful trajectories avoid. You will find in your observation

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in this particular synatic mathematical space there is no data point in this region of space. This is absolute empty space. So the surrounding path blue and orange here produce now in this mathematical representation a stable hole in the trajectory coverage and this

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is what the image identify and if you are a little bit familiar with mathematic you know this is H1 this is our first homology group. So this means we have mathematical tools to build this to identify this and really see oh we do

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have here a particular topological structure. Now homology itself is an algebraic tool that counts and classifies here in a very simplified version dimensional holes in a particular space by looking here at the boundary condition and H1 specifically counts here on one-dimensional holes.

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Now you know if you have a PhD in mathematics this is really a simplification but I hope to just give you a feeling an image why we are doing this because not all option that are in this space available are really

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available because we will discover that we have regions where there's no solution possible which brings up a new mathematical problem but more about this later. So and then we have winding numbers. You see how w uh equal + one here for the um

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blue one and w minus one for the orange. What are those? Now those are mathematical ideas that come also here from kind of the winding numbers. So those encode how each path winds around the obstacle. Now in the eye how we use

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this the winding value works like a stable route identifier. You went to the north or to the south and you know exactly what it means.

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And once the agent commits to one side here, so either blue or orange, the winding signature remembers this decision becomes here our topological memory structure in EI terms. No, but operationally this prevents now a planning step of our AI machine from

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repeatedly switching between the routes merely because the other route seems temporarily a little bit closer and we have not defined what closer means in this particular mathematical space.

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So the winding signature tells here the planner if you want hey I'm already executing here a gamma plus route here.

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So the blue one already continue using here the checkpoint. So we do have particular points on this mathematical space that now belong exactly to the implementation of this particular reasoning strategy or of this particular strategy if you have a robotic system to

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reach here a certain point. So therefore the winding structure if you want is for the eye implementation now a compact topological memory of a long-term planning commitment in the decision process of the EI machine. Great. Now there is now no distance. Now we have no

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distance defined. You cannot tell here like in a vector space or in other spaces here. So what we do now guess what there were really years and years of mathematical research in the past. So they found here a pure abstract

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mathematical construct they call iconal shells and this is kind of a global progress representation and you yeah I tried to find here this image or ask machine to generate here this image how I have a feeling how I see this. So

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maybe it helps you here this visual to understand what we are talking about because we have no distance in this mathematical space. So we have to introduce distance. Now distance is simply here a distance to our goal because all is here goal centered. So

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the repeated translucent surfaces that are here now uh built here by this EI image processor are equal distance to the goal shells. And of course we have here shell structure in this topological space. So every state on one shell has

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approximately the same remaining legal path distance to the goal. But it's not a ukidian distance. This distance is measured through here a valid carrier and it has a mathematical complexity you cannot even imagine if you have not studied mathemat mathematics at the

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university. So this shells give here the a common global progress coordinate the outer shell here if you think about this percentia states system state dynamic states that are farther away from our goal that we want to reach. The inner

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shells represent states that are real close to the goal. And moving across the successive shells here in the topological space means making progress towards our goal in the task structure.

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Beautiful. Now you remember the classical um learning we had back propagation. We had to have a direction of learning here for the neural network. Now here in the topological space we have something similar. We need a direction that our

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reasoning trace or here the path in the decision process of a robotic system has to go to reach the goal. Of course, this is an mathematical abstraction. So what is here a direction? So we have here also a direction. This is our knob R

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represents the local direction in which the remaining goal distance decreases most rapidly. So perfect. We have exactly here on a manifold now a preferred direction to our goal. So this acts now if you want in simple terms a

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global navigation signal pointing towards smaller distance to goal values in a complete abstract mathematical space. Now there's a whole mathematical theory how those shells can be created.

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I have a simple explanation for you can be understood as a backward wave emitted here from the goal. This is like a pulsar that emits here in astrophysics constant time signals. It emits your wave front and you know exactly where is

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the goal. So the wave expands through every data supported routing and bends around certain obstacles and yeah there will be effects. I cannot explain to you with plain mathematics.

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Now there are something about the degree of freedom and the degree of choices I have for my process. how the I machine will decide on what options I have and what amount of options I have because as you can see we have here some

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bottlenecks. Now what is a bottleneck in an abstract mathematical space? At first it is a shell compression methodology.

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No. So this means it indicates here a loss of behavioral freedom for reaching here my particular goal. But this is great because remember in the abstraction I told you I want to find abstract patterns and this is kind of a

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pattern. This is a point where all the observation have to go through. I mean [laughter] if you watch TV I don't know if you know the TV series Doctor Who there are something like a fixed point in time. No, this is something similar.

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Of course it's just a Yeah, but you got it. There are some points where all the decision processes have to go through.

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This is here something that is valid for all the different option all the different robotic systems that will explore this planet. Now we have to find those particular bottlenecks and because they will become gates and they will become absolute important decision

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points here for to reach our goal. So near a narrow passage now and we talked about shell compression the same shell becomes compressed almost all successful trajectory must pass through a smaller state region in this mathematical space.

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So therefore the system measures here the effective size of each shell and ask now how many successful option remain available at this particular stage.

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Now we take now the mathematical toolbox and we discover another beautiful advanced mathematical tool and this is the zero homology group. Now an H0 operation we use now as a kind of filter for false bottlenecks because guess what

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like in the classical AI yeah sometimes this is a real bottleneck and sometimes this is a not real bottleneck. My goodness if my mathematical teacher would hear me. So therefore we have the zero homology group and it simplifi

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simply counts at a number of path connected components in a topological space. Great. Now a persistent H0 is used here as a filter as a robustness filter. What it means it simply determines which dips here in the freedom signal complexity remain

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Speaker A

distinct across a wide range of detection thresholds and then those stages will become real gates. And this is what we are interested in. We want to find those gates for our process. So a shared gate is if you want an

32:04

Speaker A

unavoidable strategic point in time in space in a topological space and time this is an unavoidable checkpoint that we have to go through with our reasoning process with our whatever robotic action with our mathematical simulation whatever. So at the gate itself the route freedom

32:27

Speaker A

collapses. Everything has to pass through this particular gate. And this is great because these are kind of fixed points in our complexity. So we know exactly where we have to go through at some point in time. And after passing

32:41

Speaker A

through the two strategy classes here, our blue and our yellow ribbon, those two strategy classes merge now into a common continuation towards the goal. So you see exactly here back to topology homology and whatever that we have now a

32:59

Speaker A

new mathematical understanding our strategic option now converge to a final option. This is what we have to do if we want to preserve here uh particular interest rates in the US. So I think we're coming almost to an end because

33:16

Speaker A

every valid solution as I told you in these route classes must pass through this particular gate produce this particular region and it simply turns thousands and thousands of continuous trajectory states and all those observation into a very small number of

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Speaker A

meaningful task milestone. So a successful task becomes now really abstracted in a topological space and we found here really the gates that define here the process of reasoning to define here all my option all my actions I can

33:49

Speaker A

take. So this means this topological representation can therefore transfer if you want the task skeleton the most important backbone of a task while allowing each agent each different robotic system to supply its own controller.

34:06

Speaker A

Great. So this is another kind of representation I played around here with AI machines to show you. So the paper on topological necessity is simple this you have here h1 route classes remember route class is a homotopic class of

34:22

Speaker A

paths itself this is a non-trivial mathematical space we have our holes where we have no observed data in no of our successful uh trajectory ever a data point was recorded in this particular region of the topological space and we

34:38

Speaker A

will talk about this later why this happens but we have now we've had conal shells and Here we have a H0 persistence here and our gates were that we understand now what is this gate? We have now kind of a pathway to our goal.

34:52

Speaker A

So this means I cannot find theoretically, mathematically hopefully all possible solution in a complex system dynamics and this is what we set out to do.

35:05

Speaker A

What is now looking back what have we achieved? We have some cross embodied transfer between different robotic system. We have a strategy that can survive any changes in the robotic body itself or what sensor the robot carries.

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Speaker A

No temperature, pressure, whatever. We have some long horizon planning. This means we have really compressed thousands of different system states into a short sequence here in a new representation mathematical representation to get meaningful commitments. We have a route stability

35:38

Speaker A

with our winding memory that provides now the memory aspect here for the continuous root switching. We have a persistent that separates here the stable structure from the sampling artifacts. We have found our global gates that point us to the global

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Speaker A

waypoint execution. We have kind of found an interpretability of the system and we have kind of a data efficiency.

36:01

Speaker A

Let's be here positive here because the AI plan extracts here this shared abstract structure from ex existing successful demonstration paths.

36:11

Speaker A

So this is it. You see this is now an image that explained hopefully to my best knowledge I can do this here. This produce this video. This shows here an AI transforming raw successful rollout paths into a reusable highly abstract

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Speaker A

planning document. So this means if whatever system it was a Fable 5, a robotic system or a little three billion model here on your local PC, if there was a successful trajectory from a beginning to a goal independent

36:49

Speaker A

what domain and complexity it was, if I can abstract it into a topological space, if I can find a solution. You know in the good old times we had skill markdown files and then we had procedural uh files that we described

37:03

Speaker A

and now we are in a much higher mathematical space and we are much more abstract. We are not model dependent. We are not harness dependent. We are not even domain dependent. We are really going in the abstract mathematical space

37:17

Speaker A

where anything that a human intuition is is not there if you do not have a PhD in mathematics.

37:26

Speaker A

So if you ask me now and you know whenever you see this green this beautiful green bright bar here at the end here. So what do we build? We first build a data supported strategy supported topological task space a

37:39

Speaker A

manifold and then we measure here some global progress toward our goal. We discover where the successful options split around certain obstacles certain holes in our topological representation.

37:53

Speaker A

We will separate this option into distinct root classes ways to find a solution. We remember the selected class through a winding signature. Then we identify stable collapse of behavioral freedom when we pass through our gates and converts these stable collapses here

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Speaker A

into the unavoidable gates that are our waypoint to our goal. And now you have it. Now you have here my feeling, my kind of try to explain you 60 pages of dense mathematical argumentation in a single image. And

38:35

Speaker A

this is why I called this image on this video here topological intelligence because it portrays here a machine intelligent not as a pure memorization of successful trajectories but as a discovering here the invariant mathematical structures shared by many

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Speaker A

possible executions. I hope you enjoyed it. I hope I provided you now the main idea the main mathematical understanding. We went through the complete process and now you can really take a weekend enjoy 60 pages of pure mathematics. Now you know

39:11

Speaker A

exactly the outline. You know exactly the reasoning. You know exactly how we use each mathematical tool. If you have now a deep dive, I think hopefully you can really now enjoy this particular paper into the orus. Absolutely amazing.

39:26

Speaker A

I love this idea. I love this new framing. I love this new perspective that you bring into the next development of AI technologies.

39:35

Speaker A

I hope to see you in my next video.

Topics: topological intelligence AI planning persistent homology global monetary system economic simulation reinforcement learning mathematical AI complexity strategic AI geopolitical forecasting


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