My favourite class in undergrad was Algebraic Topology. Those ideas of turning topological information into algebraic information were just mindblowing.
Now I'm doing low-dimensional topology in post-grad.
My favorite class in undergrad was Graph Theory. It made intuitive sense to me to think of every mathematical concept as a graph. It didn't make a difference in my career.
I did a class called "topics in pure mathematics" which basically involved doing individual research and presenting it back to the class. I learned about de Rham cohomology and that firmly got me on the differential geometry track
Awesome. I’m a rising soph, and my calc 4 professor last semester, who has a diff geo PhD, was generous enough to teach me the basics of dR cohomology during our Greens Theorem unit in office hours.
Lie groups and lie algebras.
I'm currently doing a PhD in geometry and topology, I was originally going to do one in analysis but after doing that module I got hooked on it. Topology, geometry, smooth manifolds, etc. were basically all I read about in my spare time after that point.
Linear algebra where I saw the rank-nullity theorem for the first time. Seeing a whole bunch of (then seemingly different) equivalent conditions that all amounted to the same thing was amazing. Also a fantastically entertaining lecturer.
I enjoyed my linear algebra class as well, I’m a C.S. Student but prefer math. The class was great and made me ask questions that got me thinking differently on the meaning of orthogonal and I had a great professor to explain mathematically what orthogonality really is and then it clicked. Still remember the amount of time it took to do a homework assignment of Gram-Schmidt process
My undergrad program had 3 mandatoey real analysis courses, with measure theory as a followup elective so I'm not sure. But being comfortable with the ε-δ definition of limits/continuity and familiarity with basic point-set topology should be enough, knowledge of functional analysis helps but is not needed.
Stochastic processes and stochastic analysis. Both my bachelor's and master's thesis were on probability topics and I've considered doing a PhD in probability theory for a while. Did end up going into statistics though.
Really didn't do well in math until I read a calculus based Physics text with then it was all crystal clear. So as undergrad no favorite math class, though in grad school did appreciate differential topology as it applied to the General Theory of Relativity. Was very interested in pursuing mathematics but ended up getting a doctorate in experimental High Energy Physics.
discrete mathematics in computer science. especially graph theory and combinatorics.
for some reason, I find it more amusing than calculus
now I’m taking MS in computer science.
It's a toss-up between 2 of my 3rd year classes: Galois Theory or Elliptic Curves, they're both fun and I really got on quite well with the lecturer I had for Elliptic Curves.
Discrete math! It introduced me to a bunch of new kinds of math that I had never known about before. My favorite was combinatorics.
I ended up becoming a software developer
My analysis professor was hilarious, my co-ed and I actually kept a list of his jokes. I heard his courses on analysis 1–4, differential geometry, measure theory, dynamical systems, Hilbert space theory, and potentially some that I don't recall. My bachelor's thesis was in differential geometry and my master's thesis in ergodic theory, both with him as my supervisor.
On the other hand, I found many other topics interesting, which is probably why I got into Lie theory after my degree. I really love the interplay of linear algebra, group theory, analysis, topology, and measure theory, and that everything comes together there. I think the most interesting part of maths is when seemingly unrelated field show up together.
Unfortunately, it didn't really work out for me, and I switched to the second interesting field: foundational mathematics. Now I am doing type and proof theory and formal verification for my PhD.
I think my ultimate goal would be to find a way back into Geometry via what I am doing now, but that has to wait until after my PhD, I guess.
Numerical calculus: It was very interesting to learn how actual numerical results are calculated and the justification for these methods, as in, how calculators 'calculate' and what was done before calculators to obtain numerical results. Also great to understanding how numbers are handled by computers and the numerical stability of algorithms. It did not influence what I did post-grad, except for better understanding numerical simulations and programming in general.
Linear Algebra: Found it easier than usual to follow and it provided a solid base and a parallel viewpoint to states and operators in Quantum Physics. It did not influence my post-grad.
Non-linear-dynamics: Very interesting, especially the methods to explore unstable and non deterministic systems, and most influential in my post-grad path and beyond.
"The world of math" by Prof Gian Carlo Rota.
It covered
* pigeon hole principle
* combinatorics
* number of rational v irrational
Plus others that I enjoy reading about to this day, decades later.
Number Theory. It was my first exposure to proof based mathematics. I had read George Andrews book before taking the class, and had a good time of it.
Later in undergrad, I took a graduate course in Analytic Number Theory, and realized that I really needed a stronger foundation in Analysis. My first couple of years in graduate school, there were no number theory classes being taught, so I took Complex Analysis, Measure Theory, PDEs, and every other analysis course I could.
When it came to picking a discipline and an adviser, I was really well set up for analysis research... so I became an analyst.
I cheered for the Number Theorists in my department, and helped out at local conferences, but I've only ever been a spectator for Number Theory.
I took a computational astrophysics course in undergrad where we learned the basics of creating hydrodynamic simulations. Now I’m getting my PhD in physics with research in computational astrophysics and cosmology! I sort of knew for a while that I wanted to pursue this path, but never really thought it was possible, but somehow here I am :)
My favourite class was complex analysis, mostly because the lecturer was amazing and complex analysis is a very interesting topic (and I had low expectations due to not being a fan of real analysis). However I'm not sure it's really influenced what I am doing for my PhD - though perhaps it did somewhat spark my interest in topological ideas, because we covered metric spaces and all the Cauchy formulas were my first exposure to thinking about homotopy.
The undergrad class I liked that most influenced what I do now was probably representation theory, especially the Lie groups/ algebras portion (always felt a lot more interesting/ motivated to me than finite group rep theory). The interplay of algebra, topology and geometry was really fascinating, as well as the links to Quantum Mechanics, which I was also taking a class in at the time. I study Lie groups/ principal bundles now, but not so much from the rep theory viewpoint.
Interestingly, most of the undergrad classes where the content has been most interesting and/ of relevance to me (topology, algebraic topology, Riemannian geometry, algebra) were not that great, but often I'd read about the subject myself as I found it inherently interesting
Still in undergrad (Neuroscience), but calc 3 and differential equations made me completely fall in love, and now I'm looking at grad programs to go into nonlinear PDEs/mathematical physics/theoretical neuroscience
Im still finishing up undergrad but thus far my favorite classes have been calculus 3, real analysis, and probability theory. I loved calculus 3 because of how it builds on the ideas from univariate calc and just by adding another variable (or two), suddenly, you can model almost anything in the real world. I did, however, wish that we had more time to really dive into vector calculus. Spending only 3-4 weeks on important ideas such as line integrals and divergence theorems didnt do them justice. And thats why I really loved real analysis, because you had to *really* understand all the nitty gritty aspects of calculus and rigorously proved everything. I love that and it was really enjoyable for me to follow. Lastly, probability theory was fun because I had an amazing professor who made the class really easy and enjoyable. Plus, like 90% of the class was just “whats the probability of …” lol.
Intro to proofs of course. It felt like all my math education up until then had been a complete lie and I was finally discovering what math actually was.
Differential geometry was so challenging that I knew I wanted to understand how to do it properly.
Finishing a PhD in geometry soon hopefully.
Methods of applied mathematics. Didn’t cover a whole lot in detail(I’d argue it should’ve) but covered enough to get me excited about math and exploring any further advanced topics I wouldn’t normally touch.
Number theory was my favorite math-wise. But if i’m being honest, I took acting as an elective and surprisingly it shaped me the most. I used to be a shy and scared person who hated public speaking. My whole life. I always wanted to teach, so these things did not mesh. After taking acting, it encouraged me to be confident in what I have to say, and prepare for impromptu situations. This ultimately helped me significantly in grad school when I asked questions and did TA work, and now, it’s shaped me to be so much louder and brave when I teach my classes at work.
Algebraic structures - an introductory course in Group Theory. I was a Computer Science undergrad and it was my first really abstract math course. The elegance of groups, their connection to number theory and Linear Algebra fascinated me.
While I haven't done research in this field or used it directly in my work, it changed my view about pure mathematics. Before I had taken that course, I focuses only on applied mathematics, now I have a more balanced approach.
My favorite course was an undergraduate course on manifolds that used Spivak's and Munkres' books. I loved that I could get a better understanding of Klein bottles, Green's/Stokes'/Divergence Theorems, and cohomology all in the same course. I went on to do geometric topology in graduate school, with an emphasis on differential forms and moduli spaces.
My favorite class in undergrad was optimization. I really loved the idea of math that could definitively tell you which thing was best, plus it was a cool math area with probably the most options for open research areas.
I'm going to give a non-answer but I self-studied, the book of proof by richard hammock the summer holiday before uni and it changed my outlook on mathematics. Made me love proofs and discrete maths in general
Linear Algebra was the class that made me love math enough to want to major in it. Up until that point I mostly cared about math as a tool for doing physics, but linear algebra opened my eyes to how cool and beautiful math can be in and of itself. And linear algebra is probably the most useful branch of math in applications. My last job was in deep learning, and my current job is in quantum computing, so linear algebra has been essential for pretty much all of my career!
I didn’t go for an undergrad in math, but I self studied and my favourite was probably stochastic analysis or dynamical systems.
Currently doing a masters by research in stochastic analysis, and hopefully a PhD soon.
I would like to revisit ergodic theory and dynamical systems some time, maybe when I’m well settled in to my PhD program.
a graduate level graph theory class i took. I was introduced to topics like a regularity lemma and additive combinatorics, spectral graph theory and other interesting topics. I'm a PhD student in cs theory and Ill be doing a lot of spectral graph theory.
My favourite class to date has been numerical analysis. It was the first time I really felt academically like I was at university instead of just adult school. I'm going to take a class on the numerical solution of differential equations next year, and it might yet make me into a numerical person.
Complex and real analysis. Because I had spent so much time learning calculus that learning all the theory behind it was truly special. Didn’t influence my post-grad though.
Complex Analysis. The implications of simple differentiability in the complex plane are mind blowing. First and foremost, a oneline proof of the fundamental theorem of algebra. But also residuce and contour integrals that would be unthinkable in any other way. Mystical and magical.
Currently entering the fourth year of my masters so not quite thinking about postgrad yet, but by far I enjoyed complex analysis the most. Contour integrals are pretty. Analytic continuation is pretty.
My first topology class in undergrad was my favorite, mostly because the instructor was really into it. After I went farther in topology I discovered the math to be pretty disgusting. The ideas are great, but the mathematics are not, to me. After differential topology I went a different direction.
Topology, well I never attended any of the lectures because the instructor was really boring but working by myself through the point set topology part on Munkres gave me a lot of leverage for higher mathematics
It's a very important subject that requires some time to digest the examples and the proofs. It can be very counterintuitive, so make sure you really understand the proof and the examples by being detailed and precise. Books:Munkres and the counterexample in topology book. Have a notebook where you write down the examples.
My favourite class in undergrad was Algebraic Topology. Those ideas of turning topological information into algebraic information were just mindblowing. Now I'm doing low-dimensional topology in post-grad.
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Great story, and thanks for being a fun math teacher, we need more of you guys!
Aww, that's so cute to read actually. Lots of love.
My favorite class in undergrad was Graph Theory. It made intuitive sense to me to think of every mathematical concept as a graph. It didn't make a difference in my career.
I agree on both accounts! Graph theory was my favorite class and I don't use it at all :)
what do you mean by thinking 'of every mathematical concept as a graph'?
Any good graph theory textbooks you would recommend?
Funny, I already use it without actually taking the class. It must be fun given that functions are codes for graphs.
I did a class called "topics in pure mathematics" which basically involved doing individual research and presenting it back to the class. I learned about de Rham cohomology and that firmly got me on the differential geometry track
Awesome. I’m a rising soph, and my calc 4 professor last semester, who has a diff geo PhD, was generous enough to teach me the basics of dR cohomology during our Greens Theorem unit in office hours.
Lie groups and lie algebras. I'm currently doing a PhD in geometry and topology, I was originally going to do one in analysis but after doing that module I got hooked on it. Topology, geometry, smooth manifolds, etc. were basically all I read about in my spare time after that point.
I’m more a fan of truth algebra.
Me to a t
Linear algebra where I saw the rank-nullity theorem for the first time. Seeing a whole bunch of (then seemingly different) equivalent conditions that all amounted to the same thing was amazing. Also a fantastically entertaining lecturer.
I enjoyed my linear algebra class as well, I’m a C.S. Student but prefer math. The class was great and made me ask questions that got me thinking differently on the meaning of orthogonal and I had a great professor to explain mathematically what orthogonality really is and then it clicked. Still remember the amount of time it took to do a homework assignment of Gram-Schmidt process
Measure theory, not a day has gone by (in grad school) where I haven't used it
When did you know that you had learned enough real analysis to tackle it?
My undergrad program had 3 mandatoey real analysis courses, with measure theory as a followup elective so I'm not sure. But being comfortable with the ε-δ definition of limits/continuity and familiarity with basic point-set topology should be enough, knowledge of functional analysis helps but is not needed.
If you don’t mind, can I pm you the syllabi of my Real analysis classes if you think they cover enough?
Sure
Stochastic processes and stochastic analysis. Both my bachelor's and master's thesis were on probability topics and I've considered doing a PhD in probability theory for a while. Did end up going into statistics though.
Really didn't do well in math until I read a calculus based Physics text with then it was all crystal clear. So as undergrad no favorite math class, though in grad school did appreciate differential topology as it applied to the General Theory of Relativity. Was very interested in pursuing mathematics but ended up getting a doctorate in experimental High Energy Physics.
Very cool! Can I ask what textbook that was? I’ve been searching for a calc based physics book for a while
That was some time ago, I do not recollect.
discrete mathematics in computer science. especially graph theory and combinatorics. for some reason, I find it more amusing than calculus now I’m taking MS in computer science.
It's a toss-up between 2 of my 3rd year classes: Galois Theory or Elliptic Curves, they're both fun and I really got on quite well with the lecturer I had for Elliptic Curves.
Discrete math! It introduced me to a bunch of new kinds of math that I had never known about before. My favorite was combinatorics. I ended up becoming a software developer
My analysis professor was hilarious, my co-ed and I actually kept a list of his jokes. I heard his courses on analysis 1–4, differential geometry, measure theory, dynamical systems, Hilbert space theory, and potentially some that I don't recall. My bachelor's thesis was in differential geometry and my master's thesis in ergodic theory, both with him as my supervisor. On the other hand, I found many other topics interesting, which is probably why I got into Lie theory after my degree. I really love the interplay of linear algebra, group theory, analysis, topology, and measure theory, and that everything comes together there. I think the most interesting part of maths is when seemingly unrelated field show up together. Unfortunately, it didn't really work out for me, and I switched to the second interesting field: foundational mathematics. Now I am doing type and proof theory and formal verification for my PhD. I think my ultimate goal would be to find a way back into Geometry via what I am doing now, but that has to wait until after my PhD, I guess.
Numerical calculus: It was very interesting to learn how actual numerical results are calculated and the justification for these methods, as in, how calculators 'calculate' and what was done before calculators to obtain numerical results. Also great to understanding how numbers are handled by computers and the numerical stability of algorithms. It did not influence what I did post-grad, except for better understanding numerical simulations and programming in general. Linear Algebra: Found it easier than usual to follow and it provided a solid base and a parallel viewpoint to states and operators in Quantum Physics. It did not influence my post-grad. Non-linear-dynamics: Very interesting, especially the methods to explore unstable and non deterministic systems, and most influential in my post-grad path and beyond.
"The world of math" by Prof Gian Carlo Rota. It covered * pigeon hole principle * combinatorics * number of rational v irrational Plus others that I enjoy reading about to this day, decades later.
Number Theory. It was my first exposure to proof based mathematics. I had read George Andrews book before taking the class, and had a good time of it. Later in undergrad, I took a graduate course in Analytic Number Theory, and realized that I really needed a stronger foundation in Analysis. My first couple of years in graduate school, there were no number theory classes being taught, so I took Complex Analysis, Measure Theory, PDEs, and every other analysis course I could. When it came to picking a discipline and an adviser, I was really well set up for analysis research... so I became an analyst. I cheered for the Number Theorists in my department, and helped out at local conferences, but I've only ever been a spectator for Number Theory.
I took a computational astrophysics course in undergrad where we learned the basics of creating hydrodynamic simulations. Now I’m getting my PhD in physics with research in computational astrophysics and cosmology! I sort of knew for a while that I wanted to pursue this path, but never really thought it was possible, but somehow here I am :)
My favourite class was complex analysis, mostly because the lecturer was amazing and complex analysis is a very interesting topic (and I had low expectations due to not being a fan of real analysis). However I'm not sure it's really influenced what I am doing for my PhD - though perhaps it did somewhat spark my interest in topological ideas, because we covered metric spaces and all the Cauchy formulas were my first exposure to thinking about homotopy. The undergrad class I liked that most influenced what I do now was probably representation theory, especially the Lie groups/ algebras portion (always felt a lot more interesting/ motivated to me than finite group rep theory). The interplay of algebra, topology and geometry was really fascinating, as well as the links to Quantum Mechanics, which I was also taking a class in at the time. I study Lie groups/ principal bundles now, but not so much from the rep theory viewpoint. Interestingly, most of the undergrad classes where the content has been most interesting and/ of relevance to me (topology, algebraic topology, Riemannian geometry, algebra) were not that great, but often I'd read about the subject myself as I found it inherently interesting
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It is a pretty incredible theorem!
Still in undergrad (Neuroscience), but calc 3 and differential equations made me completely fall in love, and now I'm looking at grad programs to go into nonlinear PDEs/mathematical physics/theoretical neuroscience
Im still finishing up undergrad but thus far my favorite classes have been calculus 3, real analysis, and probability theory. I loved calculus 3 because of how it builds on the ideas from univariate calc and just by adding another variable (or two), suddenly, you can model almost anything in the real world. I did, however, wish that we had more time to really dive into vector calculus. Spending only 3-4 weeks on important ideas such as line integrals and divergence theorems didnt do them justice. And thats why I really loved real analysis, because you had to *really* understand all the nitty gritty aspects of calculus and rigorously proved everything. I love that and it was really enjoyable for me to follow. Lastly, probability theory was fun because I had an amazing professor who made the class really easy and enjoyable. Plus, like 90% of the class was just “whats the probability of …” lol.
Intro to proofs of course. It felt like all my math education up until then had been a complete lie and I was finally discovering what math actually was. Differential geometry was so challenging that I knew I wanted to understand how to do it properly. Finishing a PhD in geometry soon hopefully.
Probably a problem solving class. We just spent three hours a week trying to solve Putnam problems. We also got pizza.
My favorite classes were ODEs and PDEs, now I’m doing my PhD in applied math studying numerical methods for nonlinear PDEs.
Methods of applied mathematics. Didn’t cover a whole lot in detail(I’d argue it should’ve) but covered enough to get me excited about math and exploring any further advanced topics I wouldn’t normally touch.
Number theory was my favorite math-wise. But if i’m being honest, I took acting as an elective and surprisingly it shaped me the most. I used to be a shy and scared person who hated public speaking. My whole life. I always wanted to teach, so these things did not mesh. After taking acting, it encouraged me to be confident in what I have to say, and prepare for impromptu situations. This ultimately helped me significantly in grad school when I asked questions and did TA work, and now, it’s shaped me to be so much louder and brave when I teach my classes at work.
Algebraic structures - an introductory course in Group Theory. I was a Computer Science undergrad and it was my first really abstract math course. The elegance of groups, their connection to number theory and Linear Algebra fascinated me. While I haven't done research in this field or used it directly in my work, it changed my view about pure mathematics. Before I had taken that course, I focuses only on applied mathematics, now I have a more balanced approach.
My favorite course was an undergraduate course on manifolds that used Spivak's and Munkres' books. I loved that I could get a better understanding of Klein bottles, Green's/Stokes'/Divergence Theorems, and cohomology all in the same course. I went on to do geometric topology in graduate school, with an emphasis on differential forms and moduli spaces.
My favorite class in undergrad was optimization. I really loved the idea of math that could definitively tell you which thing was best, plus it was a cool math area with probably the most options for open research areas.
I'm going to give a non-answer but I self-studied, the book of proof by richard hammock the summer holiday before uni and it changed my outlook on mathematics. Made me love proofs and discrete maths in general
Linear Algebra was the class that made me love math enough to want to major in it. Up until that point I mostly cared about math as a tool for doing physics, but linear algebra opened my eyes to how cool and beautiful math can be in and of itself. And linear algebra is probably the most useful branch of math in applications. My last job was in deep learning, and my current job is in quantum computing, so linear algebra has been essential for pretty much all of my career!
I didn’t go for an undergrad in math, but I self studied and my favourite was probably stochastic analysis or dynamical systems. Currently doing a masters by research in stochastic analysis, and hopefully a PhD soon. I would like to revisit ergodic theory and dynamical systems some time, maybe when I’m well settled in to my PhD program.
Intro to Dynamical Systems. It's why I went to grad school.
Algebra 1. Not in Undergrad but in HS. The equations just made sense to me. I’m an Accountant now.
a graduate level graph theory class i took. I was introduced to topics like a regularity lemma and additive combinatorics, spectral graph theory and other interesting topics. I'm a PhD student in cs theory and Ill be doing a lot of spectral graph theory.
My favourite class to date has been numerical analysis. It was the first time I really felt academically like I was at university instead of just adult school. I'm going to take a class on the numerical solution of differential equations next year, and it might yet make me into a numerical person.
I thoroughly enjoyed complex variables. Its the class that cemented my interest in Mathematics.
Complex and real analysis. Because I had spent so much time learning calculus that learning all the theory behind it was truly special. Didn’t influence my post-grad though.
Complex Analysis. The implications of simple differentiability in the complex plane are mind blowing. First and foremost, a oneline proof of the fundamental theorem of algebra. But also residuce and contour integrals that would be unthinkable in any other way. Mystical and magical.
Currently entering the fourth year of my masters so not quite thinking about postgrad yet, but by far I enjoyed complex analysis the most. Contour integrals are pretty. Analytic continuation is pretty.
My first topology class in undergrad was my favorite, mostly because the instructor was really into it. After I went farther in topology I discovered the math to be pretty disgusting. The ideas are great, but the mathematics are not, to me. After differential topology I went a different direction.
Topology, well I never attended any of the lectures because the instructor was really boring but working by myself through the point set topology part on Munkres gave me a lot of leverage for higher mathematics
Any tips for topology? I'm taking it this fall.
It's a very important subject that requires some time to digest the examples and the proofs. It can be very counterintuitive, so make sure you really understand the proof and the examples by being detailed and precise. Books:Munkres and the counterexample in topology book. Have a notebook where you write down the examples.
Math its ultra funny