What are Lie Groups and Algebra?

[mnb96]

Hello,
I am not a physician but I am a new Ph.D. student in computer vision.
Despite this, I often get the feeling that the mathematics involved in computer vision or computer graphics is somehow similar to the one used by physicians, though this is not always true.

Few examples are vector calculus, curvilinear coordinates, tensor calculus, quaternion algebra, projective geometry, geometric algebra concepts...and so on...

In the future I would also like to learn something more about spinors and twistors. I would just feel pleased to understand what they are and learn how to make some calculations with them: just for pure interest.


I guess that in order to to face successfully these topics, some strong background knowledge is required. What do you suggest me to study first?
I would like to build a personal background knowledge that would enable me to grasp the aforementioned concepts less painfully and faster.

Sometimes I feel that studying new maths material is a slow and hard process because one is missing some strength in the background, which would make the learning process much more fluent.

Thanks in advance!

 

[cristo]

What's your current background in mathematics?

By the way, the word you're looking for is physicist: a physician is a medical doctor.

 

[Vanadium 50]

What's your current background in mathematics?

By the way, the word you're looking for is physicist: a physician is a medical doctor.

"Physicien" is French for "physicist" which may be the source of confusion.

 

[mnb96]

Ops,
sorry for the mistake: I always get confused because of this:

mathematics => mathematiciAN
physics => physici...STS!

Concerning my background: I graduated from computer science, so I am familiar with topics like Calculus, Combinatorics, Probability Theory, Linear Algebra, Vector Calculus.

As a new Ph.D. student I am supposed to choose and learn new topics and take exams; that's why I was writing here. I wanted a clear idea of what is the mathematical background I am looking for.

Thanks

 

[mnb96]

...would learining Geometric (Clifford) Algebra be useful for the purposes I described?

 

[timur]

Start with basic differential geometry, matrix Lie groups etc.

 

[mnb96]

I am already diving into differential geometry, but honestly I am totally ignorant about matrix Lie groups. Could someone point out what that theory is about and why it is useful and perhaps some resources for study?

 

[Troponin]

I am already diving into differential geometry, but honestly I am totally ignorant about matrix Lie groups. Could someone point out what that theory is about and why it is useful and perhaps some resources for study?

http://www.phys.uu.nl/~thooft/theorist.html

He should have links to courses, lecture notes, etc., for everything you need. I believe there is a specific link to Lie groups, etc.

 

[mnb96]

Thanks!
Those links are extremely useful!
I also skimmed the notes on Lie Groups which appeared to be accessible.

Perhaps it could be helpful to know before starting to study, what Lie groups/algebra are all about.