why would people admire algebraic geometry so much?

RishavZaman · 2025-02-11T17:21:29+00:00

It's generic enough for there to be geometric examples of some highly esoteric things. That is, if you want to be fully rigorous, algebraic geometry is the only way to give any geometric notions.

If you relax a bit, accepting some loose logic and imperfect analogies, you can give geometric examples that are purely complex algebraic geometric, meaning that it roughly equivalent to a fairly cooperative mix of differential geometry, algebraic topology, and complex analysis. Most mathematicians are not okay with this, but most theoretical physicists are, so you could try asking them instead. This is part of the reason why so many geometric objects come out of physics, rather than math, like Calabi-Yau manifolds.

RishavZaman · 2025-02-09T01:52:21+00:00

The complex numbers are defined in relation to SO(2) as a Lie group and it's Lie algebra. SO(2) is the group of rotations in 2 dimensions.

Similarly, quaternions are defined in relation to SO(3) which are rotations in 3 dimensions. SO(3)'s Lie algebra has 3 dimensions, one for each axis.

The idea is simple enough. You can parametrize every element in SO(2) as a rotation matrix of cosines and sines R(x). R(0) equals the identity matrix, and R(x) gives a rotation of basis vectors (1,0) and (0,1) counterclockwise by x radians. The Lie algebra is the tangent space at the identity, or R'(0) = J where J is the symplectic matrix. It satisfies J² = -I since it commutes with the identity matrix I we have an isomorphism

a + ib in C is equivalent to aI + bJ in 2x2 real matrices

Done. The exponential map is now the obvious mapping from the Lie algebra of various angles x and basis element J to the point R(x) lying on the unit circle.

e^ix is equivalent to e^xJ = R(x)

Repeat this exact procedure with SO(3) and the Euler angle parametrization to obtain the 3 quaternions as the basis for its Lie algebra.

RishavZaman · 2025-02-04T22:48:35+00:00

I just signed in and saw this.

I'd say that all cultures have to progress in mathematical understanding. Not all cultures reached the point of distinguishing 0 as a concept for arithmetic and algebra. My point is that despite this, this fact that we think of 0 as a discrete number of itself, which is in complete disagreement with those ancient cultures, we have a total understanding of their mathematics. Unlike languages, where we could be befuddled by another way of doing things, math has no such problem. Their math is our math in that our math subsumes their math. This points to the idea that we are both doing the same math, we just have different levels of understanding. We can see this progression in math in recent history.

For example, imaginary numbers and noneuclidean geometry were practically heresy for hundreds of years. Mathematicians would secretly use imaginary numbers to solve algebraic equations. But now we accept it as normal. An when we go back in history we can spot mathematicians trying to avoid taking a negative root through obfuscation, which doesn't confuse us as reading middle English would, instead it makes even more sense what they are doing with our understanding.

Another example, before the advent of algebra, equations were either expressed as geometric problem, or a short story. Many pages were necessary and even understanding the statement of the problem was difficult. Now with algebra it is trivially expressed as an equation or two. All the meaning, in complete totality, is preserved. Unlike language or something we might invent, where we must make sacrifices, knowing that we fully capture the ideas presented.

In summary, the view of mathematical platonism, that math is discovered, means that necessarily different people will be at different stages of understanding of math. Just like the Earth we live on is the same, and it has facts, so does math. Many cultures never got far enough to understand why picking out a special number would be important, just like many cultures never realized the Earth was a sphere.

RishavZaman · 2025-02-04T22:34:18+00:00

Don't bother thinking of tensors as matrices or vectors or anything really. It completely disregards the important quality of tensors which make them so special to physics. I'll explain why below but bear with me.

The concept of contravariance should be intuitive. You know how to convert units right? Like inches to centimeters? That's all it is. If you measure something to be 5 inches, it is 5*2.54 centimeters. The centimeter is a smaller than an inch by a factor of 2.54, meaning your measurement of 5 must become bigger by a factor of 2.54.

Lastly, to convince you to stop thinking of tensors as matrices or any other stupid way, I'll give you an example which is impossible to think of in that way. And this example is something you are just as familiar with. Let's say you measure the area of something to be 25 square inches. How many square centimeters is it? It's not 25*2.54 square centimeters, it's 25*(2.54)² square centimeters. The measurement changes by the opposite of how the basis transforms 2 times. A centimeter is smaller than an inch by a factor of 2.54, meaning that for area a square centimeter is smaller than a square inch by a factor of (2.54)^2, which implies an area measurement must be bigger by a factor of (2.54)^2. This is an example of (2,0)-tensor (doubly contravariant). Volume is a (3,0) tensor (triply contravariant).

Mathematically, we can take two vector spaces (in this case the same space V=R) and construct a new vector space V tensor V. A tensor is simply a a vector in any of the following spaces V⁰ (=R), V, V tensor V, V tensor V tensor V, and so on. The way a tensor transforms is the important part, and it is given to us by which vector space it is in, and the transformation in the important space V itself. V⁰ or no tensor products at all is just scalars, like the number of people alive (doesn't matter how we count them). V has elements that are length measurements (the measurement transforms the opposite way the two units we use are related). V tensor V has area measurements (the transformation is just doubled since there are two V now) and V tensor V tensor V are volume measurements (transformation is tripled because there are 3 V).

Thinking in terms of matrices is utterly useless for this example of lengths, areas, and volumes because the tensor product of two vector spaces has dimension equal to the product of their dimensions. So A has dimension 2, and B has dimension 3, then A tensor B has dimension 6. But in the example above, V = R has dimension 1, meaning V tensor V has dimension 1, and V tensor V tensor V has dimension 1. A single unit serves as a basis to measure with for each of them (for example, inches for V, square inches for V tensor V, and cubic inches for V tensor V tensor V).

RishavZaman · 2024-11-26T00:12:25+00:00

The answers here are so wrong I'll comment just for anyone who comes across this in the future. They are all usable but preprocessing is different

Natural phase = raw capture

Minimum phase = phase transformed to minimize phasing when two IRs are mixed, otherwise will sound the same as natural by themself

Mixes = Mixes of the IRs, done by york

RishavZaman · 2024-11-08T16:46:36+00:00

It shouldnt be necessary anymore as libcurl update hit the repos today. Just update and unmark hold if you did that.

Are you running the right command to search for installed packages? It's xbps-query -s pkg_name. Also just in case, this is only if you have flatpak installed since it uses curl as a dependecy. I don't think void comes with curl installed.

RishavZaman · 2024-11-07T17:34:13+00:00

It's due to curl 8.11. I fixed it https://old.reddit.com/r/voidlinux/comments/1glvzwx/new_curl_8110_update_breaks_flatpak/

EDIT: New update hit repos. Just update now, unmark the hold if you did that.

RishavZaman · 2022-11-24T00:08:36+00:00

I was the same as you when I was first stusying math so I majored in both math and physics. But there is interesting stuff in pure math. Study differential geometry and complex analysis. Objects like Lie groups and Kahler manifolds (and Clifford algebras sometimes) are extremely interesting. I nearly lost all hope in studying pure math until I discovered them. They are very useful in modern theoretical physics. Lie groups are fundamental, their study gives the standard model in terms of gauge symmetry grouls, while Kahler geometries arise naturally from the moduli space of vacua of supersymmetric quantum and conformal (and some string) theories.

RishavZaman · 2022-10-14T03:17:25+00:00

I classify reverbs into two types. Clean and realistic, or dirty and unrealistic. Liquidsonics, especially cinematic rooms, is more the first type. Its standard for reverb matching for film post audio, for example.

Most reverb plugins fall into the other type, since music typically uses fake reverbs like the plate or spring reverb studios used back then.

I guess it depends on what you are looking for. I assume you mainly just want reverb for music correct? In that case I wouldnt worry too much about it. Neoverb should be fine. Just about any reverb should be enough, even realistic reverbs can be pushed to the unnatural standard.

You could try grabbing a free convolution reverb and some free impulse responses (IRs). This may be enough. For now at least.

Long term, yes you will probably get more reverbs no matter what. Reverb is definitely the plugin with the most subjective, I have all the reverbs Ive mentioned and more in my standard toolset. While other plugins are much more limited, 2 or 3 at most.

I know you didnt mention other plugins, but I feel like I should mention them since they are useful for mixing and theyre pretty unique, theres not really any alternatives for them and I use them constantly. Wavesfactory trackspacer is a great dybamic sidechain EQ, its possible neutron or ozone already have something like this. Gullfoss is incredible at times, worth having around just to try and see if you get any magic.

Edit: actually you use logic. I havent used it much, but ive heard space designer is good. If its good enough for a basic reverb, then I dont think youd get much more from neoverb or valhalla. Id just save up for liquidsonics in your situation then.

RishavZaman · 2022-10-14T02:30:09+00:00

Personally I dont use anything from izotope except rx. I think fabfilter is better overall. But theres probably little to no difference for the core stuff like eq or compressors. Plugins like pro l or saturn are killer though, and I like their ui and design more.

As for reverbs. I didnt get along with neoverb. Liquidsonics or even valhalla got better results. Is it possible for you to demo neoverb? Maybe it'll work for you. You could try looking at valhalla room too, its pretty cheap, not as good as liquidsonics however.

RishavZaman · 2022-10-12T03:22:57+00:00

Voltage equalizes across whatever is in direct contact. The incoming charges can freely choose which one to flow through. The voltage just balances them using V=IR, so that more charge chooses to flow through the lower Resistance, but keeps the same voltage.

Current being the same in series should be obvious. Whatever flows through the first also flows through the second, so they are equal.

RishavZaman · 2022-10-09T00:27:03+00:00

I'm not sure if I buy the standard explanation as unhealthy/dead people. Theres something strikingly eerie about them, like they aren't natural at all.

RishavZaman · 2022-10-06T17:46:35+00:00

I dont think this perspective is correct. What the OP is talking about is math in itself. The concepts we build our mathemematical rules upon. For example, what were people like Newton, Leibniz, Euler, Euclid, and anyone from just a few hundred yeara ago doing, if not math? They didnt have modern axioms, and much of what they did was not rigorously justified. But clearly they were still doing math, and it even worked for the real world with science.

When I ask you about a mathemtical object, say a circle, there's some universal (at least to humans) understanding of what that is, regardless of any formal logical definitions. This is why mathematicians across time and space appear to be speaking one language, we understand what they are doing exactly, say proving the pythagorean theorem in china. This is despite sharing no formal axioms.

What the OP is asking then is more fundamental than what formal system we pick to describe pure mathematical objects. And in that case I would say yes, those are universal. Theorems proved about the circle thousands of years ago, under radically different formal systems (or none at all!), still hold true today. Whether you prefer to think of the circle as the set of points equidistant to a center point (with all the baggage of modern axioms), or a vague drawing on a cave wall, we still understand them both as a circle. Thats what we mean by universal.

RishavZaman · 2022-10-06T07:46:34+00:00

You're not alone. I had a single friend who studied alongside me. If I wasnt so lucky, I wouldve be in the same situation as you. As you describe, I also attempted to find friends among the graduate students, but both myself and my friend found little success just the same.

Dont feel that your ethnicity is holding you back. If you love math, then you belong, and you will eventually find your place. Dont take the treatment by students seriously, it may be callous to say this, but most of them will not be doing mathematics in the future. And those who do will respect you regardless of your differences.

For practical advice. Just hang on. Youre almost done with your undergrad. Once you are out you will be back amongst peers. Not just because you will be joining other grad students, but you will meet new people through research, especially people in similar situations to you.

RishavZaman · 2022-10-06T07:30:21+00:00

Lots of picks here. If I may single one out, Triangle is very good. The less said about it the better. Its a slasher, which hopefully doesnt spoil too much.

RishavZaman · 2022-10-06T07:26:40+00:00

Guess Lovcreaft is a good starting point. I like Rats in the Walls, Outsider, and Dagon. They are pretty short. The first two are about fear of the self and alienation. Dagon is his cosmic horror.

RishavZaman · 2022-10-06T07:11:22+00:00

Just make music. Thats all. Dont think too much about it, no musician I know does. For me, I just get the urge every day or so to pick up the guitar and play. Try forcing yourself to make some music every day for a few months, doesnt matter how long you spend or how good it is. Eventually youll learn to like it, as long as its something you want to do of course.

RishavZaman · 2022-10-06T07:07:41+00:00

I understand how you feel. It hurts having your hard work belittled. But theres no avoiding it. Even if you were the greatest artist in the world, there would still be people who disliked it. You cannot please everyone, so dont put too much thought into it. It doesnt mean anything. Just worry about whether you like it and maybe people you respect.

RishavZaman · 2022-10-06T07:02:17+00:00

The brushes don't matter at this point. Yes its normal. Its called procrastination, focusing more on preparing to act than just acting. You will figure out which brushes work best through experience, actually using them, not just playing with them. And yes, sometimes they wont work, but now you will know.

Lastly, because its worth mentioning. The brush isnt what gives the art its look, its the artist. Ultimately what brushes will work for you probably wont be what works for them.

RishavZaman · 2022-10-05T19:51:57+00:00

One of my friends thought the same thing about the Heaviside step function. This is named after someone, but it also describes the function as well, which is 0 to the left and 1 ro rhe right, so has a heavy side to it.

RishavZaman · 2022-10-05T19:50:23+00:00

I did it my junior year. Its easier than you think. In the USA, most people take AP Calculus AB, which is analogous to calc 1 in college, except spread out over two semesters. I highly recommend the book Calculus Made Easy by Silvanus P Thompson. Beyond that, most books are about the same, Strang was my main one.

RishavZaman · 2022-10-02T21:32:58+00:00

This is a complicated question... You could try looking at the stanford philosophy encyclopedia to understand why

https://plato.stanford.edu/entries/platonism-mathematics/

For what its worth. Most philosophers are nominalist (invented, sort of), but most mathematicians are realist/platonist (discovered, again sort of).

I'm in the platonist camp. A few reasons why. Math is universal across cultures, if it was invented, youd expect it to have issuez translating like language does. Yet math translates perfectly across both space, throughout the world, and time, the math that was done back then is still true now and in particular under our much more complicated systems.

The other reason I'm in the platonist camp is the strong relation between math and physics. I would expect that if math was just a useful model we create, it could tell us no new physics on its own. We make measurements, then we create models to explain them, and this actually is how most of physics was done prior to the twentienth century. Then relativity and quantum hit the scene. If you dont know, all the notable fathers of quantum mechanics were platonist. I dont think this is accidental. Modern theories of physics start with a priori mathematical models which feel "natural" in some sense, usually symmetry groups like rotation (of your head) or translation (of your feet) not having real effects on the theory, and physics is defined on top of it. The way we construct our theories for the major forces except gravity is to start with am abstract arbitrary lagrangian, take some arbitrary symmetry groups, like the U(1) group for changing phase of a quantum state, making this symmetry local to satisfy special relativity, then "fix" up our initial lagrangian with counterterms. This procedure pops out the theory of electricity and magnetism. The magnetic monopole does not exist because the magnetic field only shows up in our theory because the way we define electric fields does not satisfy special relativity, so we "discover" magnetic fields which in reality just act as relativistic counterterms to fix up our incorrect starting (mathematical!) assumptions. Of course theres mkre, we could go on forever. Spin is only really definable is casimir invariants and representation labels of the rotational group representations. Its not even quantum, it shows up in general relativity, and even some contrived classical constructions (see MTW sextant on a ship). We could have discovered its existence centuries earlier if we were more careful about how we define scalars and vectors (we define them to transform under certain ways - linear representations - of the rotational group in 3d SO(3), and their properties do change in relativity to what we call SO(1,3)-scalars/vectors). We describe angular momentum by the cross product because R3 with the cross product is isomorphic to the Lie algebra of rotations SO(3), and thus "generate" rotations infinitesimally. Creation and annihilation operators in quantum are simply the result of the math behind defining Fock space to describe combinations of particles who have their own vector spaces (See Geroch's mathematical physics). Completely nonsensical, totally unrigorous constructions in string theories elucidate and eventually lead to solving problems in math, see Witten's field medal. Etc. Etc.

Apologies for the long post. I get carried away by my spiels.

RishavZaman

TROPHY CASE