**What is Applied Mathematics? | Satyan Devadoss — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/applied-mathematics-satyan-devadoss/

Exploring the evolving meaning of applied mathematics and its impact on modern data analysis and topology.

## Key Takeaways

- Applied mathematics is no longer limited to traditional subjects but includes all mathematical fields.
- Modern applied math is essential for data encryption, prediction, and understanding complex systems.
- Topological data analysis provides powerful tools to analyze imperfect, noisy data.
- Mathematics is deeply integrated into everyday technologies and scientific advancements.
- The evolving definition of applied mathematics reflects its expanding role in solving real-world problems.

## What the video covers

- Applied mathematics traditionally focused on practical applications like engineering and physics.
- In the 21st century, applied mathematics encompasses all branches of math, including number theory and probability.
- Number theory is crucial for encryption and cybersecurity, employed by agencies like the NSA.
- Probability underpins predictive business analytics, handling vast, imperfect data sets.
- Mathematical patterns explain molecular structures and quantum mechanics foundations.
- Topology studies shapes in a fuzzy, approximate way rather than precise geometry.
- Topological data analysis helps interpret noisy, incomplete real-world data.
- This method can quickly classify medical conditions such as types of diabetes.
- Applied mathematics now integrates diverse mathematical ideas to solve complex problems.
- The speaker expresses a deep passion for the broad and evolving nature of applied mathematics.

## Chapters

1. 00:00 Introduction to Applied Mathematics and its Traditional Meaning
2. 03:00 The 21st Century Shift in Applied Mathematics
3. 06:00 Applications of Number Theory and Probability
4. 09:00 Mathematical Patterns in Molecular and Quantum Mechanics
5. 11:00 Topology and Topological Data Analysis Explained
6. 14:00 Handling Noisy Data with Topological Methods
7. 16:00 Medical Applications of Topological Data Analysis
8. 18:00 Conclusion: The Broad Scope and Beauty of Applied Mathematics

Answers

## Questions about this video

What does 'applied mathematics' mean in the 21st century?

In the 21st century, applied mathematics is not limited to specific subjects like calculus or differential equations; it includes all branches of mathematics applied to real-world problems, from number theory to topology.

How is number theory used in modern applications?

Number theory is essential for data encryption, such as securing transactions when swiping a Visa card, and is employed by organizations like the NSA to protect information.

What is topological data analysis and why is it important?

Topological data analysis studies the overall 'shape' of data in a fuzzy, approximate way, which helps interpret noisy and incomplete data sets, enabling applications like quick medical diagnoses.

## Full Transcript — Download SRT & Markdown

00:00

Speaker A

And one of the things I'm in love with is not just mathematics. My title actually is Professor of Applied Mathematics, and that word "applied" has two meanings. There's the 20th-century meaning, and the 20th-century meaning is applications. It has always been what Boeing would want. You know, let's use mathematics for an engineer to build an airplane, design submarine structures, or build buildings. So mathematics has been applied using the tools of sines and cosines and functions for engineers and scientists and physicists. So Einstein uses the mathematics, and it's applied math. But, you know, this amazing shift has happened in the 21st century. In the 21st century, applied mathematics is no longer about subject. It's not about differential equations or calculus. It turns out every math you could ever imagine is applied math. So if you care about numbers, it turns out that almost everywhere that numbers are multiplied together and factored into pieces is how data is encrypted when you swipe through a Visa card. So the National Security Agency, NSA, they hire mostly math PhDs or number theorists. Number theory turns out to be useful for people anywhere in the world today. If you care about issues of chance and probability, like how often do I get heads or tails in a simple coin toss or rolling dice, it turns out that's the foundation of how you do prediction analysis for business. You want to model these things because you have so much data coming in that you can't get anything 100% right. So now data is almost going to be approximated. You'd be happy to be 95% sure, 90% sure something is going to work because you have terabytes of data coming to you per second. So now, all of a sudden, things that were known for probability turn out to be applied. If you care about patterns of how molecules are organized and the structures and shapes of a Rubik's Cube, for example, it turns out that's the foundation for quantum mechanic issues and foundations for chemistry. So there's that. Almost every mathematical idea that you can think of turns out to be applied, even from number theoretic stuff. And what I care about is shape, and a revolution happened related recently to the kind of shape I care about, which is not geometric shape, which talks about spheres and angles, but topological shape. And topology just means take off your glasses. If you take your glasses off, how does the world look? It looks fuzzy, and then you roughly get the idea of their shape, but you don't get it perfectly. And it turns out there's, and at Stanford, there's a faculty named professor named Gunnar Carlson who I worked with, and he was one of the founders of this field called topological data analysis. So what he does is he looks at data, and he takes up his glasses, and he sees the data approximately in a fuzzy way, and he's able to tell you what the overall shape of the data is in a fuzzy setting, not on a perfect angular measurement setting, but in a fuzzy setting. And that helps a lot because most of the data that we get have noise in it. All right, no data that's coming to us is perfect. A lot of people, when they fill out their forms and you're watching Netflix and you're giving Netflix data, if you're giving data from Amazon or Apple, you don't fill out all the fields, so they don't know everything about you. There's all these holes missing, and that data has noise because you might maybe fill one or two things incorrectly. So the fact you can take off your glasses and get a rough approximation of what's really going on is beautiful. So using topological data analysis, you could immediately tell whether you're normal or have type 1 or type 2 diabetes really quickly. You can quickly see that there are these three branches going on. You're going to fit in one of those things. You don't need to know the exact geometry, but that rough topology is really good. So everything is applied. That's why I love it.

00:16

Speaker A

always been what Boeing would want you know let's let's use mathematics for an engineer to to build airplane design or submarine structures or or build buildings so mathematics has been applied using the tools of sines and cosines and functions for engineers and

00:34

Speaker A

scientists and physicists so Einstein uses the mathematics and it's applied math but you know this amazing shift has happened in the 21st century in 21st century applied mathematics is no longer about subject it's not about differential equations or calculus it

00:48

Speaker A

turns out every math you could ever imagine is applied math so if you care about numbers turns out that almost everywhere that numbers are multiplied together and factored into pieces is how data is encrypted when you swipe through a Visa

01:01

Speaker A

card so the National Security Agency NSA they hire mostly math PhDs or number theorists number theory turns out to be useful for for people anywhere in the world today if you care about issues of chance and probability like how often do

01:17

Speaker A

I get heads or tails in a simple coin toss or rolling dice it turns out that's the foundation of how you do prediction analysis for business you want to model these things because you have so much data coming in that you can't get

01:28

Speaker A

anything 100% right so now data is almost going to be approximated you'd be happy to be 95% sure 90% sure something is gonna work because you have terabytes of data coming to you per second so now all of a sudden things that were known

01:40

Speaker A

for probability turns out to be applied if you care about patterns of how molecules are organized and the structures and shapes of a rubik's cube for example turns out that's the foundation for quantum mechanic issues and foundations for chemistry so there's

01:55

Speaker A

that almost every mathematical idea that you can think of turns out to be applied even from number theoretic 'el stuff and and what I care about is shape and a revolution happened related recently to the kind of shape I care about which is

02:06

Speaker A

not geometric shape which talks about spheres and angles but topological shape and topology just means take off your classes if you take your glasses up how does the world look it looks fuzzy and then you roughly get the idea their

02:19

Speaker A

shape but you don't get it perfectly and and it turns out there's and at Stanford there's a faculty named professor named Gunnar Carlson who I worked with and he was one of the founders of this field called topological data analysis so what

02:32

Speaker A

he does is he looks at data and he takes up his glasses and he sees the data approximately in a fuzzy way and he's able to tell you what the overall shape of the data is in a fuzzy setting not on

02:43

Speaker A

a perfect angular measurement setting but in a fuzzy setting and that helps a lot because most of the data that we get have noise in it all right no data that's coming to us is perfect a lot of

02:53

Speaker A

people when they fill out their forms and you're watching Netflix and you're giving Netflix data if you're giving data from Amazon or Apple you don't fill out all the all the fields so they don't know everything about you there's all

03:03

Speaker A

these holes missing and that that data has noise because you might maybe fill one or two things incorrectly so the fact you can take off your glasses and get a rough approximation of what's really going on is beautiful so using

03:14

Speaker A

topological data analysis you could immediately tell whether you're normal or have type 1 or type 2 diabetes' really quickly you can quickly see that there are these three branches going on you're gonna fit in one of those things

03:25

Speaker A

you don't need to know the exact geometry but that rough topology is really good so everything is applied that's why I love it

Topics: applied mathematics number theory probability topology topological data analysis data encryption predictive analytics quantum mechanics mathematical applications data science

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