Really cool! I'm doing a PhD in Theoretical CS (in particular I'm interested in aspects of the mathematical foundations of machine learning, and more broadly optimization/ probability). One of the things I've been thinking about is designing loss functions for machine learning problems. Turns out there's a fair amount of convex geometry involved in the underlying proofs. I was wondering if you'd be open to a chat sometime? Feel like we're looking at similar problems from different lenses!
Hey! What did your undergrad look like? I'm currently working on a CS degree with a minor in mathematics because I'd like to work with machine learning. I'm curious if it might be better to go for a math major with a minor in CS or something else. What sort of classes were you taking in your undergrad?
My undergrad was in math and stats. Lots of emphasis of probability, stats and some pure math. Ranged from more applied stat modelling courses to stuff like real analysis and measure theoretic probability. I enjoyed the stuff on the proof-heavy end of the spectrum considerably more than the applied modelling courses. Also worked in industry afterwards, and just really missed actual proof-based math, so returned to academia. In terms of what you should be doing.. it's hard to say. What is your goal after undergrad? Do you want to work in industry or research? So far, what kind of courses have you gravitated towards more? Also, if you're unsure, you should totally pick a broad spectrum of courses ranging from the more applied to the more proof-based and see what catches your fancy. Maybe even doing a research internship over the summer (though this is rarely enough time to REALLY get a feel for things imo) or/ and an industry internship might help you decide on which path you're more keen on. All that said, foundational courses in probability and statistics, convex optimization and analysis will probably go a long way in setting you up for ML research. Especially if you're keen on doing theory - the more practice you have with proof-based stuff, the better - measure theory, functional analysis, topology, a rigorous proof oriented course in algorithms would all be great to cover (don't mean to overwhelm you - these are just good to do over a period of time, there's of course no need to rush through these, and you definitely don't need to know ALL of this to get started with research). @@cleoingles2827
@@PhDVlog777 Totally understand :) Fwiw, I prefer staying anonymous too. So happy having this conversation in private, even using pseudonyms if that's what you'd prefer. But totally cool if you're just not comfortable doing this at this stage, no sweat
you might be interested in problems surrounding buoyancy. it was recently discovered that there are other convex bodies (beside euclidean balls) which float in stable equilibria in every orientation a few years ago, but theres lots of convex geometry involved
The first problem ironically feels the most accessible to think about despite being the most difficult. My intuition is to do a proof by contrapositive, taking K to not be a Euclidean ball and showing that the constant volume condition isn't met. Maybe one thing to consider is that the diameter of these sets isn't constant, and then maybe there's a way to relate the diameter of these sets to their volume? I'm not sure, fun problem tho.
I'm still a college senior (I graduate in like 2 months)... I'll be starting grad school from this Fall, and my future advisor has already asked me about my future intentions of pursuing a PhD... I've already been assigned a title for my Master's thesis, and probably will get my PhD thesis title too if I just agree to do it in the same place... I have mixed emotions about this, and already feel overwhelmed...
This PhD program in pure math seems like a brutal, relentless grind. Putting up with the endless pressure is something I could never imagine doing, let alone the personal, social and financial sacrifices along the way. I admire your stoicism, bro. I sure hope it's worth it all when you finally get to the end of the long, rocky road. Best of luck to you!
I can't imagine committing to a PhD for 5-6+ years of my life, without at least having a general idea of what I want my thesis to be about. Instead, leaving it up to chance and whatever advisor I get assigned. Just seems shocking to me that this is a common sentiment: “You pretty much just do what your advisor wants you to do”. I'm sure this is true to some extent, but to the point of going into a completely different field, tragic... This might be different for other STEM fields, but I did my undergrad in CS and have done research before. When I got into it, I had a very specific field that I wanted to work within, and I was able to do 2 years of grant funded research in that field. I've considered doing a PhD and if I did, I would be quite specific on the area of research. Not to say I'd know exactly the topic of my thesis, but it'd be for sure within my specified field.
just a wild guess for the #1. Consider the Ball of the same dimension d but way greater than K, let B shrink to K till it hit K's surface. Then if there are cavities, show the possibility of finding two mu s/t condition of constant volume will be violated...
Grassmannian isn't a group, but it is a smooth manifold that you can realize as the quotient of a group action. Namely, consider the transitive action of O(n) on G(n,k) (transitive because you can rotate any k-plane into any other k-plane). Then the stabilizer is O(k) x O(n-k), so G(n,k) is a smooth manifold diffeomorphic to O(n)/O(k) x O(n-k).
I’m currently about to begin my masters in applied math this fall, I’m a 4+1 student in mathematics (combined BS/MS). Right now I’m looking at PhD programs in statistics to prepare to enter after my masters. I’m really interested into the Applied side of things but research and theoretical foundation is of strong interest to me too. The PhD program I’m looking at ensures I get a good background in graduate level real analysis and advanced probability theory, along with the relevant training in statistical methodology. That way I can have a bit of pure math training too. Hopefully the topic I’m able to research for my dissertation are a little bit more broad. I suppose like you said it would depend on my advisor.
This is out of left field. But for the first problem, I want to know if there is a subset of n-vectors that are unique to discs. It wouldn’t be trivial to solve after that but i think it would be much easier.
I think, prove all possible perimeters are unique, so must form a line. That line is finite, despite having a constant value, so the relationship between the perimeters is irrational. No consumable prediction about their shape can exist, and they diverge, so that shape isn't closed. I think if you try to remove the volume of the shape and only wonder about the perimeters you are losing the rational connection between all sides and cannot form conclusions about whether they form a sphere. Thanks for the video, fun insight into a math program!
So far it has been much less stressful, but I have been feeling directionless lately. This isn't good since I need to average two papers per year to finish my dissertation.
I have a question. How do some people get PHDs at insanely young age. Do they just find some groundbreaking solution to some open problem and they get awarded PHD sort of honorarily? @@PhDVlog777
Just letting you know, man. You shouldn't be posting what you're working on online. A previous coworker of mine had his material outright copied by another student because the first professor that was his doctoral advisor handed over that material directly to another after he requested another advisor. It was obviously the result of being petty, but the point is, you should be cautious about what you're revealing. You're putting in too much time and effort to risk it.
Yeah, I have a feeling that maybe nobody has had this talk with him yet, I was also surprised that he's sharing this. It's cool and all that he's sharing it, but if someone scoops your research before you can publish and you have sunk years of your PhD into it, that can and has ended people's grad school journeys. I come from the perspective of chemistry, and it seems to be the norm that you do not present or otherwise discuss the specifics of your research-in-progress to anyone other than direct collaborators until you have submitted a manuscript for publication to stake your claim, or are already established in your niche.
He's basically just giving the statement of some open problems that seem to be publicly known. It would be different if he showed lots of progress he made on a problem. Unless this video stoked the curiosity of some random professional mathematician who was looking for different problems to work on.
Is 3 years of class taking the norm (pun intended?) in the US? Seems a long time. I'd have expected a year or so of class taking post undergrad before diving into research ...
MS degree is usually about 2 years. In a PhD the coursework requirement I've heard is roughly equivalent to MS I'm sure it varies dramatically. The MBA or basket weavers will have a much easier time of grinding coursework than someone studying Physics
I was 2 years full-time coursework, then 3rd year was part-time coursework and part time research (plus teaching). From there research only plus teaching. In the US.
I would say that a mathematician in applied mathematics would have a better chance of conducting research experimentally more so than people studying pure math. So yes but it would be difficult finding a job like that with a math background
@@PhDVlog777 sir I am interested in math but also some physics courses.is a mathematician contribute into physics research .is there any area who is intersection between math and physics for a math students point of view.
@@amanbhagwani6937 Yes, it's called mathematical physics. Roger Penrose, David Hilbert, and Emmy Noether are some prominent contributors to the field. Especially look up Noether's Theorem.
Difficult to say, but it would not be entirely surprising. For instance, from a certain general perspective, the salient part is that "grounded" objects such as "Euclidean balls" can be characterized by restricted sets of information - indeed, in certain contexts, only limited sets of information would be feasibly obtained due to significant limitations on experimental parameters. For instance, it may just so happen that, hypothetically, there are interests in the investigation of the distribution of certain quantities (traditionally, such quantities include masses or charges, but the future of physics may see many more alternative kinds of "fundamental" quantities), and in being able to successfully recover a Euclidean ball via the experimentally determined sets of information as given in the hypotheses would then mean that certain traditional methods which hold for Euclidean balls can immediately be applied. To be more specific, notice how the hypothesis of the first problem can be characterized via a boundary condition as imposed on a ring, that it is not entirely too surprising that, in certain contexts, experimental data are gathered on a "ring" (i.e., see circular particle accelerators), and that one requires an intersection with an orthogonal space should not be too unimaginable either (consider that in mathematical physics, many vector quantities are typically realized orthogonally and thus the popularization of the cross-product in applications of vector calculus/analysis in mathematical physics). Then, by obtaining appropriate experimental data that allows the determination of a Euclidean ball, many more straightforward computation methods can then be applied over the surface of Euclidean balls, as in methods that characterize geodesics (as an example, despite the use of connections so as to characterize geodesics more generally in differential geometry, abstract spaces, as opposed to "grounded" spaces, tend to make computations somewhat ambiguous, sometimes). Note, if geodesics can be computationally appraised in a more straightforward fashion, then optimization problems can proceed in a more straightforward fashion, and so one can envision, say, the more efficient planning of routes over regions of space in contexts of logistics and what not (say, logistics over the span of the solar system) - variational methods in contexts of potential theory is, of course, already applied in the planning of space travel.
Your question rests on the assumption that anyone can predict the future. Nobody knows where things will be applicable. The point is to discover math, that's what makes it pure. Nobody knew where prime numbers would be applicable, but now they're very fundamental in our every day lives.
bro, I don't want to sound like a hater, but third year and still taking classes in not a good plan. Just focus on research. Job market is extreamly competitive right know. Another advise: do you really want to share the problems that your advisor give you to solve during you PhD? there are strong people out there, and you're given them ideas...
Really cool! I'm doing a PhD in Theoretical CS (in particular I'm interested in aspects of the mathematical foundations of machine learning, and more broadly optimization/ probability). One of the things I've been thinking about is designing loss functions for machine learning problems. Turns out there's a fair amount of convex geometry involved in the underlying proofs. I was wondering if you'd be open to a chat sometime? Feel like we're looking at similar problems from different lenses!
Hey! What did your undergrad look like? I'm currently working on a CS degree with a minor in mathematics because I'd like to work with machine learning. I'm curious if it might be better to go for a math major with a minor in CS or something else. What sort of classes were you taking in your undergrad?
That's super cool! I am an undergrad math who wants to do the same thing!
My undergrad was in math and stats. Lots of emphasis of probability, stats and some pure math. Ranged from more applied stat modelling courses to stuff like real analysis and measure theoretic probability. I enjoyed the stuff on the proof-heavy end of the spectrum considerably more than the applied modelling courses. Also worked in industry afterwards, and just really missed actual proof-based math, so returned to academia.
In terms of what you should be doing.. it's hard to say. What is your goal after undergrad? Do you want to work in industry or research? So far, what kind of courses have you gravitated towards more? Also, if you're unsure, you should totally pick a broad spectrum of courses ranging from the more applied to the more proof-based and see what catches your fancy. Maybe even doing a research internship over the summer (though this is rarely enough time to REALLY get a feel for things imo) or/ and an industry internship might help you decide on which path you're more keen on. All that said, foundational courses in probability and statistics, convex optimization and analysis will probably go a long way in setting you up for ML research. Especially if you're keen on doing theory - the more practice you have with proof-based stuff, the better - measure theory, functional analysis, topology, a rigorous proof oriented course in algorithms would all be great to cover (don't mean to overwhelm you - these are just good to do over a period of time, there's of course no need to rush through these, and you definitely don't need to know ALL of this to get started with research). @@cleoingles2827
I will consider it. I am still cautious about reaching out to viewers even though the STEM community is good people lol. But I will give it a think :)
@@PhDVlog777 Totally understand :) Fwiw, I prefer staying anonymous too. So happy having this conversation in private, even using pseudonyms if that's what you'd prefer. But totally cool if you're just not comfortable doing this at this stage, no sweat
you might be interested in problems surrounding buoyancy. it was recently discovered that there are other convex bodies (beside euclidean balls) which float in stable equilibria in every orientation a few years ago, but theres lots of convex geometry involved
Hmmmmmmmmm... ;)
The first problem ironically feels the most accessible to think about despite being the most difficult. My intuition is to do a proof by contrapositive, taking K to not be a Euclidean ball and showing that the constant volume condition isn't met. Maybe one thing to consider is that the diameter of these sets isn't constant, and then maybe there's a way to relate the diameter of these sets to their volume? I'm not sure, fun problem tho.
I'm still a college senior (I graduate in like 2 months)... I'll be starting grad school from this Fall, and my future advisor has already asked me about my future intentions of pursuing a PhD... I've already been assigned a title for my Master's thesis, and probably will get my PhD thesis title too if I just agree to do it in the same place... I have mixed emotions about this, and already feel overwhelmed...
This PhD program in pure math seems like a brutal, relentless grind. Putting up with the endless pressure is something I could never imagine doing, let alone the personal, social and financial sacrifices along the way. I admire your stoicism, bro. I sure hope it's worth it all when you finally get to the end of the long, rocky road. Best of luck to you!
Post AI world without employees nor industry are for these guys early expected by Hermann Hesse in Das Glasperlenspiel.
The pressure and isolation sucks but that math itself is fun.
I can't imagine committing to a PhD for 5-6+ years of my life, without at least having a general idea of what I want my thesis to be about. Instead, leaving it up to chance and whatever advisor I get assigned. Just seems shocking to me that this is a common sentiment: “You pretty much just do what your advisor wants you to do”. I'm sure this is true to some extent, but to the point of going into a completely different field, tragic...
This might be different for other STEM fields, but I did my undergrad in CS and have done research before. When I got into it, I had a very specific field that I wanted to work within, and I was able to do 2 years of grant funded research in that field. I've considered doing a PhD and if I did, I would be quite specific on the area of research. Not to say I'd know exactly the topic of my thesis, but it'd be for sure within my specified field.
just a wild guess for the #1. Consider the Ball of the same dimension d but way greater than K, let B shrink to K till it hit K's surface. Then if there are cavities, show the possibility of finding two mu s/t condition of constant volume will be violated...
the potential issue I see here is that the cavity may take very symmetrical structure, so that all mu^perp intersected with the cavity look the same?
Free PhD?? That’s all you had to tell me! (None of this makes sense to me)
How do you deal with loneliness? Loneliness made me insane, which I've started accepting just now.
Grassmannian isn't a group, but it is a smooth manifold that you can realize as the quotient of a group action. Namely, consider the transitive action of O(n) on G(n,k) (transitive because you can rotate any k-plane into any other k-plane). Then the stabilizer is O(k) x O(n-k), so G(n,k) is a smooth manifold diffeomorphic to O(n)/O(k) x O(n-k).
I’m currently about to begin my masters in applied math this fall, I’m a 4+1 student in mathematics (combined BS/MS). Right now I’m looking at PhD programs in statistics to prepare to enter after my masters. I’m really interested into the Applied side of things but research and theoretical foundation is of strong interest to me too. The PhD program I’m looking at ensures I get a good background in graduate level real analysis and advanced probability theory, along with the relevant training in statistical methodology. That way I can have a bit of pure math training too. Hopefully the topic I’m able to research for my dissertation are a little bit more broad. I suppose like you said it would depend on my advisor.
This is out of left field. But for the first problem, I want to know if there is a subset of n-vectors that are unique to discs.
It wouldn’t be trivial to solve after that but i think it would be much easier.
I think, prove all possible perimeters are unique, so must form a line. That line is finite, despite having a constant value, so the relationship between the perimeters is irrational. No consumable prediction about their shape can exist, and they diverge, so that shape isn't closed. I think if you try to remove the volume of the shape and only wonder about the perimeters you are losing the rational connection between all sides and cannot form conclusions about whether they form a sphere. Thanks for the video, fun insight into a math program!
can you do a video on what it’s like reading a research paper and how someone who has never read one to go about it
Good luck!
you've probably mentioned this, but what college are you current at?
sounds amazing i don't understand what you said though!!!
Hey, how’s the research phase been compared to the homework/quals phase? Is it much less stressful/chill?
So far it has been much less stressful, but I have been feeling directionless lately. This isn't good since I need to average two papers per year to finish my dissertation.
I have a question. How do some people get PHDs at insanely young age. Do they just find some groundbreaking solution to some open problem and they get awarded PHD sort of honorarily?
@@PhDVlog777
@@PhDVlog777 'two papers per year to finish my dissertation' really?
is it specific to your field or is this the norm?
Which is why the late Freeman Dyson was always openly opposed to the PhD program
Grassmannian manifold! Just saw it in my analysis course HW (which is actually a course studies analysis on manifolds.)
hey guys i have solved the free phd problem but i am not in a phd programm how do i proceed?
I guess you write the paper and published it.
Just letting you know, man. You shouldn't be posting what you're working on online. A previous coworker of mine had his material outright copied by another student because the first professor that was his doctoral advisor handed over that material directly to another after he requested another advisor. It was obviously the result of being petty, but the point is, you should be cautious about what you're revealing. You're putting in too much time and effort to risk it.
Yeah, I have a feeling that maybe nobody has had this talk with him yet, I was also surprised that he's sharing this. It's cool and all that he's sharing it, but if someone scoops your research before you can publish and you have sunk years of your PhD into it, that can and has ended people's grad school journeys. I come from the perspective of chemistry, and it seems to be the norm that you do not present or otherwise discuss the specifics of your research-in-progress to anyone other than direct collaborators until you have submitted a manuscript for publication to stake your claim, or are already established in your niche.
He's basically just giving the statement of some open problems that seem to be publicly known. It would be different if he showed lots of progress he made on a problem. Unless this video stoked the curiosity of some random professional mathematician who was looking for different problems to work on.
Is 3 years of class taking the norm (pun intended?) in the US? Seems a long time. I'd have expected a year or so of class taking post undergrad before diving into research ...
It is definitely not the norm. The department likes us to take classes but not for so long. 3 years is max typically.
MS degree is usually about 2 years. In a PhD the coursework requirement I've heard is roughly equivalent to MS
I'm sure it varies dramatically. The MBA or basket weavers will have a much easier time of grinding coursework than someone studying Physics
I was 2 years full-time coursework, then 3rd year was part-time coursework and part time research (plus teaching). From there research only plus teaching. In the US.
Wow, nice work 😊😊
And here I am on RUclips university
I have a question from you .can a mathematician become a experimentalist scientist .can he and
She do hard core experiment?like other scientist .
I would say that a mathematician in applied mathematics would have a better chance of conducting research experimentally more so than people studying pure math. So yes but it would be difficult finding a job like that with a math background
@@PhDVlog777 sir I am interested in math but also some physics courses.is a mathematician contribute into physics research .is there any area who is intersection between math and physics for a math students point of view.
@@amanbhagwani6937 Yes, it's called mathematical physics. Roger Penrose, David Hilbert, and Emmy Noether are some prominent contributors to the field. Especially look up Noether's Theorem.
@@amanbhagwani6937 theoretical physics is basically the physics version of pure math
Can just take math classes in college because I’m joining the army and all I want to do is learn math.
kanye reference
Cool
Tomorrow i am going to get the PhD😂
Excuse me,which university do you study?
Bro I am amazed honestly but where are these problems applicable in future
Why would they need to be?
It’s basic science anything there can be super relevant in the future, not applied
Difficult to say, but it would not be entirely surprising. For instance, from a certain general perspective, the salient part is that "grounded" objects such as "Euclidean balls" can be characterized by restricted sets of information - indeed, in certain contexts, only limited sets of information would be feasibly obtained due to significant limitations on experimental parameters.
For instance, it may just so happen that, hypothetically, there are interests in the investigation of the distribution of certain quantities (traditionally, such quantities include masses or charges, but the future of physics may see many more alternative kinds of "fundamental" quantities), and in being able to successfully recover a Euclidean ball via the experimentally determined sets of information as given in the hypotheses would then mean that certain traditional methods which hold for Euclidean balls can immediately be applied.
To be more specific, notice how the hypothesis of the first problem can be characterized via a boundary condition as imposed on a ring, that it is not entirely too surprising that, in certain contexts, experimental data are gathered on a "ring" (i.e., see circular particle accelerators), and that one requires an intersection with an orthogonal space should not be too unimaginable either (consider that in mathematical physics, many vector quantities are typically realized orthogonally and thus the popularization of the cross-product in applications of vector calculus/analysis in mathematical physics). Then, by obtaining appropriate experimental data that allows the determination of a Euclidean ball, many more straightforward computation methods can then be applied over the surface of Euclidean balls, as in methods that characterize geodesics (as an example, despite the use of connections so as to characterize geodesics more generally in differential geometry, abstract spaces, as opposed to "grounded" spaces, tend to make computations somewhat ambiguous, sometimes).
Note, if geodesics can be computationally appraised in a more straightforward fashion, then optimization problems can proceed in a more straightforward fashion, and so one can envision, say, the more efficient planning of routes over regions of space in contexts of logistics and what not (say, logistics over the span of the solar system) - variational methods in contexts of potential theory is, of course, already applied in the planning of space travel.
Your question rests on the assumption that anyone can predict the future. Nobody knows where things will be applicable. The point is to discover math, that's what makes it pure. Nobody knew where prime numbers would be applicable, but now they're very fundamental in our every day lives.
potentially any field with high dimensional geometries (big one being AI)
You guys are struggling to prove that a ball exists?
Bruh 😅
bro, I don't want to sound like a hater, but third year and still taking classes in not a good plan. Just focus on research. Job market is extreamly competitive right know. Another advise: do you really want to share the problems that your advisor give you to solve during you PhD? there are strong people out there, and you're given them ideas...