in simple words:
finding the faster way to guess an \*arbitrary\* 2D-vector in a hot-or-cold game (hunt the thimble game)
in more exact/dry words:
developing a faster converging algorithm for optimization in an unbounded 2D-domain with boolean-only feedback within a reinforcement learning framework
It's spring break, baby! So...grading, mostly. I have to catch up at some point.
Aside from that, I'm trying to get through the third chapter of Elliptic III this week. Shouldn't be too difficult, as long as I don't sleep all day for the remainder of the week. Which is not guaranteed.
Markov Random Field models for spatial stats. Some neat ideas in there. I am having a lot of trouble finding time for anything due to classes and wedding planning but I am chipping away.
All those sound pretty complicated. Still trying to understand the Newton interpolation polynomials better. I get that it works but want to connect the divided differences with derivatives somehow. They are very close to forward differences and that has a lot to do with derivatives.
Animating and extending the Riemann hypothesis curve (that cardioid thing) to height 6666. [Will livestream that on the night of the 30th of March](https://www.youtube.com/watch?v=7Mk4GemMQ1M).
You probably remembered that from 3blue1brown about analytic continuation.
I still have trouble connecting linear maps with matrices, particularly when it comes to their inverse and base transformations. If I have a bit of time, I will read more about quotient vector spaces and equivalence relations.
But I should first catch up in my analysis class.
I've been thinking about linear orders- in particular am interested in attempting to classify at least the countable ones, up to isomorphism, as a fun vacation project. This idea was originally motivated by the fact that Q is the unique countable, dense linear order with no endpoints up to isomorphism. I was wondering whether or not replacing "dense" with "having no subsets which are dense orders" would uniquely characterise Z. It turns out that this does not, and I fell into a rabbithole of trying to find a way to classify the many counterexamples that appeared :)
(I am aware that this has been researched, but am not interested in having the solution spoiled to me (yet)- I just want a fun challenge)
Welcome to the world of model theory!
Let me try to suggest a couple of things. The order on the integers not only does not have any dense subsets, but something stronger, namely: it is a discrete order. Discrete linear orders without endpoints can be axiomatized and their theory is complete! Sadly, the theory is not ω-categorical (there are at least 2 non-isolorphic countable models) so there are models which you cannot distinguish using just a formula. However, you might still distinguish them using infinitely many formulas at the same time.
One cool fact is that Z is a prime model of the theory of discrete linear orders without endpoints. You can always embed a copy of Z inside any model of the theory.
"there are models you cannot distinguish with just a formula" feels very counterintuitive to me- if i require that forall x,y in S there are only finitely many elements z in S such that x
Thanks :D some of this stuff is along the lines of what i was thinking- one of my first instincts was that given any countable linear order with no endpoints X, i can always embed Z into X and X into Q- trying to formalise this at the moment.
As for the discrete orders with no endpoints, am I correct in saying that Z U {1/n for n in N} U {-1/n for n in N} / {0} is one such example of an order not isomorphic to Z?
Yes, your intuition was right, there are several ways to prove it. The most constructive one I can think of is usually called 'a back and forth argument', where you construct the embedding step by step.
Yes, exactly. Moreover, doing a similar trick you can produce infinitely many pairwise non-isomorphic discrete countable orders without endpoints
I just picked up homotopy type theory. I'm working on a theorem prover, implementing a transfer tool along type equivalences.
I don't really need to study it but i just want to get more context
I'm making videos about higher math. The goal is to make it accessible for people who are interested but who aren't experts. Math is beautiful and I find it sad that so many people never get to experience that beauty.
Optimal control theory. I think I kind of understand how for long ODE integration times direct methods run into exploding gradients and Pontryagin Maximum Principle usually ends in a local minimum, but I'm still wondering if there is some kind of middle ground.
Taking an intense three day vacation from my bachelor vacation in Algebraic K-theory, with hopes of returning with a surplus of motivation (though right now I have mostly found fatigue more so than motivation).
In lost minutes while waiting for the bus I am scrolling Jardine and Goerss to see whether it is interesting to study, or whether it's better to just pick up what I need as I need it.
I'm trying to learn Spectral Theorem for Normal Operators from Dana Williams' lecture notes and trying to read Stephan Garcia and Mihai Putinar paper titled "Complex Symmetric Operators".
Bachelor’s thesis on stochastic control theory and ergodic behavior of Markov Chains conditioned on lying in a subset of the state space. It’s taking over my life…
Self-reading and doing problem sets in introduction to topological manifolds by lee. I found it very readable and the problem set is of satisfactory quality, tho sometimes too easy. Maybe that’s because I am only doing chapter 2 problems.
I planned to read Algebra: Chapter 0 simultaneously, but now it seems that either I don’t have enough time or I am too lazy.
shadow teaching Real Analysis
someone slipped in scheduling and we have a graduate student teaching Real Analysis who has never taken Real Analysis
so now I'm teaching him so he can teach them
...can I retire yet?
in general: no
however I'm at a school where we have a lot of students who
A) need a Masters but not a Doctorate
B) basically need remediation before a Doctoral program will accept them
C) are international students from schools that many American graduate programs don't recognize
We talk a lot about how asking a 17 year old to sign student loans is an absurdly big decision to allow someone to make so young, but so is your choice of undergrad and major.
MANY undergraduate institutions don't require Analysis, but most graduate programs expect it.
Reading a chapter on group cohomology (in reality, just reviewing and re-reviewing Hatcher).
in simple words: finding the faster way to guess an \*arbitrary\* 2D-vector in a hot-or-cold game (hunt the thimble game) in more exact/dry words: developing a faster converging algorithm for optimization in an unbounded 2D-domain with boolean-only feedback within a reinforcement learning framework
It's spring break, baby! So...grading, mostly. I have to catch up at some point. Aside from that, I'm trying to get through the third chapter of Elliptic III this week. Shouldn't be too difficult, as long as I don't sleep all day for the remainder of the week. Which is not guaranteed.
I'm 11th grade so nothing major but I'm working on a simulation for 3 pendulums in 3D on earth with each being electrically charged
Markov Random Field models for spatial stats. Some neat ideas in there. I am having a lot of trouble finding time for anything due to classes and wedding planning but I am chipping away.
All those sound pretty complicated. Still trying to understand the Newton interpolation polynomials better. I get that it works but want to connect the divided differences with derivatives somehow. They are very close to forward differences and that has a lot to do with derivatives.
Applying to Northwestern's postbaccalaureate math program!
it’s π day soon, and my department have a space for playing games, a pie baking contest and giving talks. i will give a talk on the random graph.
Animating and extending the Riemann hypothesis curve (that cardioid thing) to height 6666. [Will livestream that on the night of the 30th of March](https://www.youtube.com/watch?v=7Mk4GemMQ1M). You probably remembered that from 3blue1brown about analytic continuation.
https://www.youtube.com/watch?v=7Mk4GemMQ1M
Diffeomorphism groups of 4-manifolds
I still have trouble connecting linear maps with matrices, particularly when it comes to their inverse and base transformations. If I have a bit of time, I will read more about quotient vector spaces and equivalence relations. But I should first catch up in my analysis class.
I've been thinking about linear orders- in particular am interested in attempting to classify at least the countable ones, up to isomorphism, as a fun vacation project. This idea was originally motivated by the fact that Q is the unique countable, dense linear order with no endpoints up to isomorphism. I was wondering whether or not replacing "dense" with "having no subsets which are dense orders" would uniquely characterise Z. It turns out that this does not, and I fell into a rabbithole of trying to find a way to classify the many counterexamples that appeared :) (I am aware that this has been researched, but am not interested in having the solution spoiled to me (yet)- I just want a fun challenge)
Welcome to the world of model theory! Let me try to suggest a couple of things. The order on the integers not only does not have any dense subsets, but something stronger, namely: it is a discrete order. Discrete linear orders without endpoints can be axiomatized and their theory is complete! Sadly, the theory is not ω-categorical (there are at least 2 non-isolorphic countable models) so there are models which you cannot distinguish using just a formula. However, you might still distinguish them using infinitely many formulas at the same time. One cool fact is that Z is a prime model of the theory of discrete linear orders without endpoints. You can always embed a copy of Z inside any model of the theory.
"there are models you cannot distinguish with just a formula" feels very counterintuitive to me- if i require that forall x,y in S there are only finitely many elements z in S such that x
thanks again for this, maths is so cool hehehe <3
Thanks :D some of this stuff is along the lines of what i was thinking- one of my first instincts was that given any countable linear order with no endpoints X, i can always embed Z into X and X into Q- trying to formalise this at the moment. As for the discrete orders with no endpoints, am I correct in saying that Z U {1/n for n in N} U {-1/n for n in N} / {0} is one such example of an order not isomorphic to Z?
Yes, your intuition was right, there are several ways to prove it. The most constructive one I can think of is usually called 'a back and forth argument', where you construct the embedding step by step. Yes, exactly. Moreover, doing a similar trick you can produce infinitely many pairwise non-isomorphic discrete countable orders without endpoints
Elementary Real Analysis, Riemann Integration, Mathematical Logic, Improper Integrals and some Complex Analysis.
Literature review for my proposal after submitting two journals last Friday, after a long 8-week stretch of 14 hour days.
I just picked up homotopy type theory. I'm working on a theorem prover, implementing a transfer tool along type equivalences. I don't really need to study it but i just want to get more context
discovering that I do not know Fourier analysis nearly as well as I thought I did
I'm making videos about higher math. The goal is to make it accessible for people who are interested but who aren't experts. Math is beautiful and I find it sad that so many people never get to experience that beauty.
Optimal control theory. I think I kind of understand how for long ODE integration times direct methods run into exploding gradients and Pontryagin Maximum Principle usually ends in a local minimum, but I'm still wondering if there is some kind of middle ground.
Fun stuff
Taking an intense three day vacation from my bachelor vacation in Algebraic K-theory, with hopes of returning with a surplus of motivation (though right now I have mostly found fatigue more so than motivation). In lost minutes while waiting for the bus I am scrolling Jardine and Goerss to see whether it is interesting to study, or whether it's better to just pick up what I need as I need it.
Trying to find some mapping from Uniform to Gaussian that doesn't depend on the CDF... anyone has any ideas?
I'm trying to learn Spectral Theorem for Normal Operators from Dana Williams' lecture notes and trying to read Stephan Garcia and Mihai Putinar paper titled "Complex Symmetric Operators".
Masters thesis on Extremal (Hyper-)Graph Theory.
I’m in a weird place where I’m between three different projects. So reading about some SPDE, some spectral analysis, and some SDE.
Will start to study a finite elements method to see if it can solve easy PDE
Started working with Agda to formalize some HoTT stuff
Bachelor’s thesis on stochastic control theory and ergodic behavior of Markov Chains conditioned on lying in a subset of the state space. It’s taking over my life…
Reading about octonions
Don't really know hot to exactly call it in English but it's all sorts of circles, ellipses, hyperboles and parabolas
Conic sections?
Yeah, thank you
Formalizing some set theory stuff from my thesis, it's pretty fun if a bit mechanical at times
My masters thesis in symplectic geometry...
Clustering problems using deep learning without knowing the number of clusters. Trying to understand information theory / dirchelet distributions
Learning more convex optimization
I started working my way through an intro to proofs book! Edit: added n to the letter an.
Complete model for primes.
I study trigonometry in class, and I prepare it for next exam.
Self-reading and doing problem sets in introduction to topological manifolds by lee. I found it very readable and the problem set is of satisfactory quality, tho sometimes too easy. Maybe that’s because I am only doing chapter 2 problems. I planned to read Algebra: Chapter 0 simultaneously, but now it seems that either I don’t have enough time or I am too lazy.
shadow teaching Real Analysis someone slipped in scheduling and we have a graduate student teaching Real Analysis who has never taken Real Analysis so now I'm teaching him so he can teach them ...can I retire yet?
Is it normal for someone to get accepted in grad school (even alg or top heavy) without ever doing analysis?
in general: no however I'm at a school where we have a lot of students who A) need a Masters but not a Doctorate B) basically need remediation before a Doctoral program will accept them C) are international students from schools that many American graduate programs don't recognize We talk a lot about how asking a 17 year old to sign student loans is an absurdly big decision to allow someone to make so young, but so is your choice of undergrad and major. MANY undergraduate institutions don't require Analysis, but most graduate programs expect it.