Topology/Differential geometry versus analysis

In summary, the conversation discusses a second year student's interest in physics and mathematical physics, and their dilemma of choosing between majoring in physics and pure mathematics. The student has not taken a second year pure mathematics course and is now considering taking it in their third year along with other advanced mathematics courses such as Metric Spaces, Rings Fields and Galois Theory, Differential Geometry, and Modules & Group Representations. Some participants in the conversation argue that knowledge of analysis is essential for pure mathematics, while others suggest that topology and group representation theory are equally important. It is also mentioned that analysis and topology have various applications in physics, such as in Lagrangian mechanics, electromagnetism, and relativity. In summary, the conversation highlights the interconnectedness
  • #1
jdstokes
523
1
Hi all,

I'm a second year student entering 3rd year with an interest in physics and mathematical physics. Foolishly I decided not to enrol in the second year pure mathematics course ``real and complex analysis''. My current mathematical knowledge comprises the following

First year:
Differential Calculus (Advanced)
Linear Algebra (Advanced)
Integral Calculus and Modelling (Advanced)
Statistics (Advanced)

Second year:
Linear Mathematics & Vector Calculus (Advanced)
Partial Differential Equations Intro (Advanced)
Algebra (Advanced) [a course in group theory]

I am thinking about majoring in physics and pure mathematics, with the following 3rd year maths courses. The course descriptions can be found in the handbook http://www.maths.usyd.edu.au/u/UG/SM/hbk06.html

Metric Spaces (Advanced)
Rings Fields and Galois Theory (Advanced)
Differential Geometry (Advanced)
Modules & Group Representations (Advanced)

Interestingly, none of these courses require knowledge of analysis. So it is possible to major in pure maths without having done any analysis whatsoever. I can't help but feel that my lack of analysis training will come back to haunt me, which is why I'm also considering the following, less interesting combination of courses

Analysis (Normal)
Rings Fields and Galois Theory (Advanced)
Complex Analysis with Applications (Advanced)
Modules & Group Representations (Advanced)

Note that the normal analysis course does not technically satisfy the assumed knowledge for complex analysis, but the lecturers inform me that I ``might be okay'' if I do very well in the normal course and do some extra work in my own time. Of course, this means dropping differential geometry, which I'm not too keen about due to its obvious connections with general relativity. I guess what it boils down to is whether topology or analysis is considered more important in physics. I would appreciate any advice you may be able to give on this question and/or my course selections.

Thanks,

James.
 
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  • #2
jdstokes said:
Hi all,

I'm a second year student entering 3rd year with an interest in physics and mathematical physics. Foolishly I decided not to enrol in the second year pure mathematics course ``real and complex analysis''. My current mathematical knowledge comprises the following

First year:
Differential Calculus (Advanced)
Linear Algebra (Advanced)
Integral Calculus and Modelling (Advanced)
Statistics (Advanced)

Second year:
Linear Mathematics & Vector Calculus (Advanced)
Partial Differential Equations Intro (Advanced)
Algebra (Advanced) [a course in group theory]

I am thinking about majoring in physics and pure mathematics, with the following 3rd year maths courses. The course descriptions can be found in the handbook http://www.maths.usyd.edu.au/u/UG/SM/hbk06.html

Metric Spaces (Advanced)
Rings Fields and Galois Theory (Advanced)
Differential Geometry (Advanced)
Modules & Group Representations (Advanced)

Interestingly, none of these courses require knowledge of analysis. So it is possible to major in pure maths without having done any analysis whatsoever.

The bolded statement is absurd. Metric spaces provide the foundation for analysis. While differential geometry provides the natural link b/w topology, analysis and linear algebra.

As for group representation theory, you got to be kidding me it doesn't use calculus. Unless there's no Lie group there, thing which would be rather absurd.

Daniel.
 
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  • #3
Daniel,

I'm sorry, why is that absurd? You just said yourself that ``differential geometry provides the natural link b/w topology, analysis and linear algebra''?
 
  • #4
Yes, knowledge of multivariable calculus is essential to diff.geom.

You have to know analysis b4 taclkling geometry.

Daniel.
 
  • #5
Personally, I would cut metric spaces and group so I could take the anyalsis courses and diff. geo.

Metric spaces, at least at my school, are part of advanced (riggorous proof based) calculus. From your list I am assuming that you have the background to teach it to yourself.
 
  • #6
I really envy you for your choice of courses.

Differential geometry can be taught without recourse to topology and with only as much analysis as in a good calculus text. However "advanced" suggests a little more familiarity may be expected. I'd suggest checking out the text used.

Among those 3rd year courses, the "Modules and Group Representations" one sounds really cool. Most physics grad students are expected to pick this stuff up by osmosis. I wouldn't miss Differential Geometry myself, it's a beautiful subject.

If you have the time, money, and discipline, I'd definitely take real analysis and topology courses.
 
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  • #7
i think both topology and analysis are absolutely basic. actually point set topology and metric spaces is merely foundations of analysis.
 
  • #8
Does metric spaces as a stand alone subject have applications to anything other than analysis? For example, does topology help with GR/QM/strings independently of analysis?

From my somewhat naive perspective, it seems that applications of analysis (particularly of the real type) to physics are limited compared to topics such as groups and group representations. I'm not sure what the situation is with topology. Does anyone know of a list of applications of analysis to physics and a corresponding list for topology?
 
  • #9
I took topology and analysis simutaneously. I think this they both help me to understand each other. Honestly, I think you will have a basic concept of analysis if you conplete all the course you have listed. Of course, if you really don't want to take a course in analysis, you should still get a book in analysis. I understood my undergrad analysis book before the first time I walk into my class. Knowing analysis makes me to become a more practical person in life :approve:

In the end, everything is just topology, analysis, and algebra. For example, functional analysis is a very applicable in mechanic, i.e energy spaces. Operator Theory is also important in many branch of phys. Calculus of Variations is the base of Lagrangian mechanic; one can find application like least action principle in QM, etc. Tensors Analysis is the language of relativity. Exterior Calculus can be applied to E&M and Thermodynamics. For topology, Morse Theory provides a new insight of conjugate point using differential topology. One can also apply algebraic topology to understand n-dimensional circuit. These are all beautiful applicaions of analysis and topology in practical world.
After all, you, as a physicist, also want to do calculus on a manifold, right? That means you want to know how calculus works on real numbers before you do calculus on a manifold. I don't know anyone can skip calculus in reals before jumping into manifold theory. Then concepts of analysis and topology are nice to have with your algebraic knowledge.

Leon reference:
Bamberg & Stermberg, A course in mathematics for students of physics I & II
Lebedev & Vorovich, Functional analysis in mechaics;
Arnol'd, mathematical methods of classical mechanics;
Hirch, Differential Topology.
 
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1. What is the difference between topology and analysis?

Topology and analysis are both branches of mathematics that deal with the study of shapes, structures, and spaces. However, topology focuses on the properties of objects that remain unchanged under continuous deformations, while analysis deals with the study of functions and their properties using techniques such as calculus and differential equations.

2. How are differential geometry and analysis related?

Differential geometry is a subfield of mathematics that combines the concepts of geometry and calculus to study smooth curved surfaces and spaces. Analysis plays a crucial role in differential geometry as it provides the necessary tools and techniques to study these surfaces and their properties.

3. What are some applications of topology and differential geometry?

Topology has various applications in fields such as physics, engineering, and computer science, where it is used to study the properties of complex systems and networks. Differential geometry has applications in areas such as general relativity, computer graphics, and robotics, where it is used to model and analyze curved objects and spaces.

4. Is topology more abstract than analysis?

Topology is often considered more abstract than analysis because it focuses on the qualitative properties of objects rather than their quantitative properties. However, both fields use abstract concepts and methods to study complex mathematical structures.

5. Can topology and analysis be used together?

Yes, topology and analysis are often used together in various mathematical and scientific fields. For example, in topology, the concept of continuity is essential, which is also a fundamental concept in analysis. Additionally, techniques from analysis, such as the study of convergence and limits, are used in topology to define and study topological spaces.

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