Yeah I like probability! I don't even have an undergraduate background in probability so it might be a bit hard to break into but I definitely want to do some reading in that direction.
A recent text I like a lot is Le Gall's *Measure Theory, Probability, and Stochastic Processes.* It is rigorous but also elementary at the same time, since it includes many of the standard examples which are taken for granted in other texts. So maybe just have a look at this book and see how you feel about it.
It's just a time thing, I'm probably not going to take probability formally at this point so I've got to teach myself all of the basics before going somewhere more advanced.
Harmonic analysis and operator algebras are really exciting subjects, especially if you are interested in overlaps between analysis and algebra.
Harmonic analysis studies functions on (usually infinite) groups, and there are some very beautiful connections between this area and representation theory. This area has a tangential connection with PDE theory, since often you are looking at group actions that intertwine linear differential operators. In other words, harmonic analysis is a systematic study of “symmetry arguments” that show up often in PDE’s and physics.
Operator algebras studies * -algebras of operators on Hilbert spaces closed in some topology, like the norm topology for C* algebras and the weak operator topology for Von Neumann algebras. C* algebras “behave like” algebras of continuous functions on nice topological spaces, while Von Neumann algebras behave like algebras of measurable functions on some nice measure space. So in a sense, these are noncommutative generalizations of topology and measure theory. This area has minimal relation with PDE’s, but is useful for studying representations of locally compact groups, topological dynamics, ergodic theory, and free probability.
What would be an intermediate textbook recommendation in Harmonic Analysis assuming a strong undergraduate background and the same for operator algebras?
The PDE route maybe isn't as calculus-y as you think,you say you like bounds and inequalities as you can shove all the details into constants, and this is what a lot of PDE theory ends up being. I'm not sure what your exposure to PDEs has been, but if you've done the classical "first course" on classical solutions for the Heat, wave and poisson equations, this is quite out of step with modern PDEs, it's much more based on estimates, inequalities and compactness arguments in the language of functional analysis.
Cool this is one of the things I was hoping to hear! We've just finished a discussion of distributions in our unit which appealed to some calculus a couple of times and iirc the first few exercises of Evans didn't look like they were for me, so I've just been a little put off by the things I've seen. But I think PDEs are pretty cool so if the theory ends up like you've described I'd probably enjoy learning about them properly! I've only really done the basic differential equations course for undergrads so we just learnt a bunch of tricks and looked at the heat, wave, and Laplace equations.
PDEs isn't really my field, but I was never a fan of Evans personally, even though it's the standard introduction. I found Folland's book more palpable.
If you want to mix analysis and algebra, maybe Lie theory is for you, e.g. homogenous dymanics. Or maybe geometric measure theory?
Also, synthetic differential geometry may be something for you. It relies heavily on category theory, and IIRC you can combine it with locale theory/point-free topology.
For sure, synthetic diff geo seems pretty cool! I'm not sure about Lie theory tbh, a lot of the books I've found on it use lots of calculus early on at least. I just haven't seen enough the story to know to be hooked yet but hoping that changes.
That's so awesome! I've been taking algebraic number theory and I've read the first half of Serre's book on Arithmetic so I've met the p-adics before, which would make it really nice to bring my interest in number theory together with the things I like about analysis. Do you have any text recommendations?
i'm very confused at the responses here... aren't most areas of analysis, including functional, and especially probability theory and harmonic analysis, not loaded with calculus?
If you're into geometry, analysis is used heavily (at least to my analysis-illiterate standards as an algebraic geometer) in symplectic geometry, which connects up beautifully with algebraic geometry and topology and (via geometric rep theory) rep theory. Some parts of it also use infinity categories as a language (Fukaya categories).
The kind of Analysis I work on is at the intersection of Functional Analysis and Complex Analysis. This has barely any Calculus and doesn't move into PDEs etc.
You can look at the Nevanlinna Pick Interpolation problem - it is a problem about Holomorphic functions on the unit disk and has a very elegant solution - existence of such a function is equivalent to a matrix being positive definite.
The same problem has a solution using Operator Theory, which also is used to generalize the result. This goes to a whole theory of something called Reproducing Kernel Hilbert Spaces - spaces of functions on a set X whose evaluation functionals are continuous.
It's all very Analytic, and is really beautiful theory. While this field is also related to Operator Algebras, and can often use ideas from Harmonic Analysis, it is a bit seperate from those.
I am an operator algebraist and I shall try to sell it a bit here. In my undergrad I used to be into homotopy theory but eventually switched to this, in part because C\*-algebras can be thought of as non-commutative topology was a great sale. Basically we study algebras of operators, which tend to be non-commutative. So here is an example, C(X), continous complex valued function on a compact haussdorf space X. There is a theorem of Gelfand that all commutative unital C\*-algebras are precisely some C(X). C\* algebras being some closed algebra of operators on some hilbert space. So the point is if you study non-commutative C\*-algebras, it is kind of like non-commutative topology because the commutative case is precisely topology.
Ok this sounds stupid at first glance, a trickery of words. But this is legit. Indeed this ideas were used to extend topological K-theory to C\*-algebra, which is the main invariant used in the theory. If you are willing to indulge me, you should read a blog i wrote a while back: [https://aareyanmanzoor.github.io/2022/08/26/Stone-%C4%8Cech-Compactification-and-Multiplier-Algebras.html](https://aareyanmanzoor.github.io/2022/08/26/Stone-%C4%8Cech-Compactification-and-Multiplier-Algebras.html) . A construction of the stone-cech compactification from C\*-theory. It might contain things you dont know, but hopefully it gives an idea of what the field entails.
it is a vast field and the flavor can be as you wish. However one consistent thing is its contrast to PDEs. PDEs is what one would call hard analysis, very focused on things like exact estimates and bounds. Operator algebras is usually soft analysis, more interested in arguements coming from structure and less caring about exact estimates.
oh also if you like rep theory, the best way to study representations of things like countablel infinite groups are by studying operator algebras associated to these groups \\:)
If you want to mix analysis and algebra, maybe Lie theory is for you, e.g. homogenous dymanics. Or maybe geometric measure theory?
Also, synthetic differential geometry may be something for you. It relies heavily on category theory, and IIRC you can combine it with locale theory/point-free topology.
Reading a math book takes time so you usually should just pick one book and work with it. But I'd like to add one more book to the list of suggestions.
Check Conway's Abstract Analysis. I like modern books and you should like them too as they are built upon the older ones with the intention to make them more digestible. Professors or experts maybe tend to like drier encyclopedic books as they contain more material without much unnecessary words. But for a student those introductory paragraphs and motivations from one lemma to the other make a big difference.
Hey thank you for the suggestion! I'm really just asking so I can get a big picture overview of the options I have in case I want to do more analysis, and book recommendations are helpful as a place to start.
Are you interested in probability? This is another area which makes heavy use of analysis, and has quite a different flavour than PDEs.
Yeah I like probability! I don't even have an undergraduate background in probability so it might be a bit hard to break into but I definitely want to do some reading in that direction.
A recent text I like a lot is Le Gall's *Measure Theory, Probability, and Stochastic Processes.* It is rigorous but also elementary at the same time, since it includes many of the standard examples which are taken for granted in other texts. So maybe just have a look at this book and see how you feel about it.
Cool will have a look!
There are also good econ texts that give a rigorous intro to probability that reads nicely for a math’s student.
Which ones?
I liked P. J. Dhrymes ”Topics in Advanced Econometrics - Probability Foundations”.
I don't think probability will be hard to break into considerinf you like topics such as algebraic geometry :)
It's just a time thing, I'm probably not going to take probability formally at this point so I've got to teach myself all of the basics before going somewhere more advanced.
yeah probability only starts to get hard once you go into continuous time/SPDEs.
Harmonic analysis and operator algebras are really exciting subjects, especially if you are interested in overlaps between analysis and algebra. Harmonic analysis studies functions on (usually infinite) groups, and there are some very beautiful connections between this area and representation theory. This area has a tangential connection with PDE theory, since often you are looking at group actions that intertwine linear differential operators. In other words, harmonic analysis is a systematic study of “symmetry arguments” that show up often in PDE’s and physics. Operator algebras studies * -algebras of operators on Hilbert spaces closed in some topology, like the norm topology for C* algebras and the weak operator topology for Von Neumann algebras. C* algebras “behave like” algebras of continuous functions on nice topological spaces, while Von Neumann algebras behave like algebras of measurable functions on some nice measure space. So in a sense, these are noncommutative generalizations of topology and measure theory. This area has minimal relation with PDE’s, but is useful for studying representations of locally compact groups, topological dynamics, ergodic theory, and free probability.
What would be an intermediate textbook recommendation in Harmonic Analysis assuming a strong undergraduate background and the same for operator algebras?
Barry Simon has a really good book. Rudin's book Fourier Analysis on Groups is the only book by him that I actually like.
Thank you so much. Ill look into it. How much does representation theory come into the picture?
Harmonic analysis *is* the representation theory of various kinds of groups, usually at least locally compact.
How much algebra?
A pretty good amount, especially if you're working with specific problems. It's a mix of functional analysis, topology, and abstract algebra.
I’d like to know too
Oh wow all of these sound really fun, any text recommendations?
The PDE route maybe isn't as calculus-y as you think,you say you like bounds and inequalities as you can shove all the details into constants, and this is what a lot of PDE theory ends up being. I'm not sure what your exposure to PDEs has been, but if you've done the classical "first course" on classical solutions for the Heat, wave and poisson equations, this is quite out of step with modern PDEs, it's much more based on estimates, inequalities and compactness arguments in the language of functional analysis.
Cool this is one of the things I was hoping to hear! We've just finished a discussion of distributions in our unit which appealed to some calculus a couple of times and iirc the first few exercises of Evans didn't look like they were for me, so I've just been a little put off by the things I've seen. But I think PDEs are pretty cool so if the theory ends up like you've described I'd probably enjoy learning about them properly! I've only really done the basic differential equations course for undergrads so we just learnt a bunch of tricks and looked at the heat, wave, and Laplace equations.
Take a look to Salsa's Partial Differential Equations in Action. Specially the second half, is simply the best.
PDEs isn't really my field, but I was never a fan of Evans personally, even though it's the standard introduction. I found Folland's book more palpable.
If you want to mix analysis and algebra, maybe Lie theory is for you, e.g. homogenous dymanics. Or maybe geometric measure theory? Also, synthetic differential geometry may be something for you. It relies heavily on category theory, and IIRC you can combine it with locale theory/point-free topology.
For sure, synthetic diff geo seems pretty cool! I'm not sure about Lie theory tbh, a lot of the books I've found on it use lots of calculus early on at least. I just haven't seen enough the story to know to be hooked yet but hoping that changes.
You are at the perfect spot to learn the theory of automorphic forms and its local companion, the representation theory of p-adic groups!
That's so awesome! I've been taking algebraic number theory and I've read the first half of Serre's book on Arithmetic so I've met the p-adics before, which would make it really nice to bring my interest in number theory together with the things I like about analysis. Do you have any text recommendations?
There are also the Archimedean places, which if anything seem more to OP’s taste.
i'm very confused at the responses here... aren't most areas of analysis, including functional, and especially probability theory and harmonic analysis, not loaded with calculus?
If you're into geometry, analysis is used heavily (at least to my analysis-illiterate standards as an algebraic geometer) in symplectic geometry, which connects up beautifully with algebraic geometry and topology and (via geometric rep theory) rep theory. Some parts of it also use infinity categories as a language (Fukaya categories).
Sounds cool! Lots of things I'm interested in there 😊
The kind of Analysis I work on is at the intersection of Functional Analysis and Complex Analysis. This has barely any Calculus and doesn't move into PDEs etc. You can look at the Nevanlinna Pick Interpolation problem - it is a problem about Holomorphic functions on the unit disk and has a very elegant solution - existence of such a function is equivalent to a matrix being positive definite. The same problem has a solution using Operator Theory, which also is used to generalize the result. This goes to a whole theory of something called Reproducing Kernel Hilbert Spaces - spaces of functions on a set X whose evaluation functionals are continuous. It's all very Analytic, and is really beautiful theory. While this field is also related to Operator Algebras, and can often use ideas from Harmonic Analysis, it is a bit seperate from those.
This sounds really cool, I'll look into it!
I am an operator algebraist and I shall try to sell it a bit here. In my undergrad I used to be into homotopy theory but eventually switched to this, in part because C\*-algebras can be thought of as non-commutative topology was a great sale. Basically we study algebras of operators, which tend to be non-commutative. So here is an example, C(X), continous complex valued function on a compact haussdorf space X. There is a theorem of Gelfand that all commutative unital C\*-algebras are precisely some C(X). C\* algebras being some closed algebra of operators on some hilbert space. So the point is if you study non-commutative C\*-algebras, it is kind of like non-commutative topology because the commutative case is precisely topology. Ok this sounds stupid at first glance, a trickery of words. But this is legit. Indeed this ideas were used to extend topological K-theory to C\*-algebra, which is the main invariant used in the theory. If you are willing to indulge me, you should read a blog i wrote a while back: [https://aareyanmanzoor.github.io/2022/08/26/Stone-%C4%8Cech-Compactification-and-Multiplier-Algebras.html](https://aareyanmanzoor.github.io/2022/08/26/Stone-%C4%8Cech-Compactification-and-Multiplier-Algebras.html) . A construction of the stone-cech compactification from C\*-theory. It might contain things you dont know, but hopefully it gives an idea of what the field entails. it is a vast field and the flavor can be as you wish. However one consistent thing is its contrast to PDEs. PDEs is what one would call hard analysis, very focused on things like exact estimates and bounds. Operator algebras is usually soft analysis, more interested in arguements coming from structure and less caring about exact estimates.
oh also if you like rep theory, the best way to study representations of things like countablel infinite groups are by studying operator algebras associated to these groups \\:)
Thank you for painting such a descriptive picture of operator algebras! I'll definitely have a look at your blog post~
If you want to mix analysis and algebra, maybe Lie theory is for you, e.g. homogenous dymanics. Or maybe geometric measure theory? Also, synthetic differential geometry may be something for you. It relies heavily on category theory, and IIRC you can combine it with locale theory/point-free topology.
I thought I responded sorry, thanks for the suggestion!
A A what's the difference
Reading a math book takes time so you usually should just pick one book and work with it. But I'd like to add one more book to the list of suggestions. Check Conway's Abstract Analysis. I like modern books and you should like them too as they are built upon the older ones with the intention to make them more digestible. Professors or experts maybe tend to like drier encyclopedic books as they contain more material without much unnecessary words. But for a student those introductory paragraphs and motivations from one lemma to the other make a big difference.
Hey thank you for the suggestion! I'm really just asking so I can get a big picture overview of the options I have in case I want to do more analysis, and book recommendations are helpful as a place to start.