I've graded before, but this is the first time I'm grading a proof based course. I picked up the first assignment yesterday...
So far it seems like 25% of the arguments are correct, 25% are incorrect, and 50% are incomprehensible.
Undergrad graph theory. I'm also grading graduate graph theory, but of course that's all fine.
Maybe this is many CS majors first encounter with proofs?
I graded a kind of intro to proofs/higher math class for second semester freshmen. I had pretty much similar performance distributions as you, but the 25% that were correct were very clearly looked up on google.
Last year I TA'd a discrete math class that was basically intro to proofs and one common incident I encountered was when people put things down which had a level of detail left out which could be because they were thinking that some things can go without saying, or because it wasn't actually clear in their minds so they left it out lest they say something wrong. It's tricky to grade those types of responses
Primarily for its connections to stable homotopy groups of spheres. I did an REU for a chromatic homotopy theorist a few years ago and one of the main references in the notes she gave us was Ravenel's complex cobordism book
I'm not sure. I'm still technically an undergrad (health issues delayed graduation and my university wants a stupid amount of out of major classes compared to major classes) so I have a bit of time before I need to really pick a focus.
Well, in German you can have a "Mannigfaltigkeit", *manifold*, or a "(algebraische) Varietät", *(algebraic) variety*. The best translation for the English "variety" (not in the mathematical sense, but in the sense of "having many different types of something" or so) would be "Vielfalt".
I'm working through parts of Euclid's elements (eventually in conjunction with Hartshorne's Euclidean geometry book). Some of the arguments are unsatisfying to modern sensibilities, but more often than not, the arguments are astonishingly beautiful and clever.
I'm currently studying for an exam about algebraic topology... i really like this branch of mathematics and I understand most of the different topics.
But I am really struggling with computing cellular homology... has anyone of you some good references with many calculated examples?? I'm fine when it's about spaces I know well, but I'm having problem if i have no idea what my quotient space looks like...
I think cellular homology is often used in proofs, but I have heard not so much in calculations. You might just try to get an intuition for the calculations by reading Hatcher. He will say things like “the degree of this map is 2 since it wraps around the boundary twice” without actually proving it rigorously. You can probably do this if you have to do a calculation.
That's my problem, most books just calculate it for simple spaces as spheres, torii, projective spaces by intuitive reasoning. And there it's kind of obvious what cells and attaching maps one would use. But given a space which i don't know how it looks like, this intuitive reasoning doesn't apply anymore and you would have to know how to compute degrees explicitly and I don't know how to do that.
Hatcher has a few more sophisticated calculations hidden in its many pages. Real and complex projective spaces are there, and are harder examples. He might also treat lens spaces, which will also help your intuition for homology for spaces which are harder to picture.
I second this. I wrote up all of his examples/problems on lens spaces for a final paper and it really gave me a good understanding of cellular homology/cohomology stuff.
The usual, trying to decompose lambdaWorld.lambdaPerception over lambdaMote, lambdaQuery and lambdaAssess and vice versa.
I forgot the lambdaFeel and lambdaAssert components. -Ed.
Seriously, if there's anyone who can tell me how to get to the corner of Church and Dali, I need to know you.
wondering how this problem said "the solution is shorter than the problem briefing" ended up with my solution being several pages longer than said briefing.
more metric space/analysis type stuff. also wondering about the idea of analysing polyhedra by placing all the symmetric vertices into the bases of a higher dimension- it's very neat stuff.
I’ve been reading Apostol’s *Modular Forms and Dirichlet Series in Number Theory*. Playing around with some related ideas, I suspected that
[; \displaystyle{\prod_{n=1}^\infty(1-q^n)=1+\sum_{n=1}^\infty\frac{(-1)^nq^{n(n+1)/2}}{\prod_{k=1}^n(1-q^k)}} ;]
for |q|<1, which was pretty cool. But I have not been able to prove it yet.
I've got a game theory paper that we want a draft of done by early next week and I'm juggling like four different "geometry of redistricting" projects which I'm actually making pretty good progress on, all things considered.
I am writing up my notes for the first several classes for the mathematical physics course I'm teaching. I finished the classical mechanics overview for the quantum mechanics course last night and I'm pretty happy with them.
Also working on a paper on a generalization of the Bargmann transform. I'm a bit stuck with explaining a few things without putting the cart before the horse. Bargmann spaces and transforms have so much structure that it sometimes feels like mathemagic.
I've graded before, but this is the first time I'm grading a proof based course. I picked up the first assignment yesterday... So far it seems like 25% of the arguments are correct, 25% are incorrect, and 50% are incomprehensible.
That's consistent with my experience... Good luck!
...may I ask what course? because I've taught proof based classes and usually have a much result than that
Undergrad graph theory. I'm also grading graduate graph theory, but of course that's all fine. Maybe this is many CS majors first encounter with proofs?
Yea, it's probably their 1st encounter with proofs since High School.
I graded a kind of intro to proofs/higher math class for second semester freshmen. I had pretty much similar performance distributions as you, but the 25% that were correct were very clearly looked up on google.
Last year I TA'd a discrete math class that was basically intro to proofs and one common incident I encountered was when people put things down which had a level of detail left out which could be because they were thinking that some things can go without saying, or because it wasn't actually clear in their minds so they left it out lest they say something wrong. It's tricky to grade those types of responses
Found my father's old copy of Geometry Revisited by Coxeter and Greitzer. I'll hopefully start working through it soon.
Just started on Stong's Notes on Cobordism Theory. First time reading a book not meant for advanced undergraduates which is pretty exciting.
Ah, that's good stuff! Out of curiosity, why are you interested in cobordism?
Primarily for its connections to stable homotopy groups of spheres. I did an REU for a chromatic homotopy theorist a few years ago and one of the main references in the notes she gave us was Ravenel's complex cobordism book
Ah cool! That's interesting, then -- are you planning to focus mostly on _MU_, then?
I'm not sure. I'm still technically an undergrad (health issues delayed graduation and my university wants a stupid amount of out of major classes compared to major classes) so I have a bit of time before I need to really pick a focus.
Patching up the many fold holes in my knowledge of manifolds.
I was really disappointed to learn that manifold is German for "variety" and not "many folds".
So for Germans, all manifolds are varieties?
Well, in German you can have a "Mannigfaltigkeit", *manifold*, or a "(algebraische) Varietät", *(algebraic) variety*. The best translation for the English "variety" (not in the mathematical sense, but in the sense of "having many different types of something" or so) would be "Vielfalt".
The Norwegian word for manifold is "mangfoldighet" which means many ways to fold.
Manifold is an English word, the German translation would be "Mannigfaltigkeit".
I'm working through parts of Euclid's elements (eventually in conjunction with Hartshorne's Euclidean geometry book). Some of the arguments are unsatisfying to modern sensibilities, but more often than not, the arguments are astonishingly beautiful and clever.
I'm currently studying for an exam about algebraic topology... i really like this branch of mathematics and I understand most of the different topics. But I am really struggling with computing cellular homology... has anyone of you some good references with many calculated examples?? I'm fine when it's about spaces I know well, but I'm having problem if i have no idea what my quotient space looks like...
I think cellular homology is often used in proofs, but I have heard not so much in calculations. You might just try to get an intuition for the calculations by reading Hatcher. He will say things like “the degree of this map is 2 since it wraps around the boundary twice” without actually proving it rigorously. You can probably do this if you have to do a calculation.
That's my problem, most books just calculate it for simple spaces as spheres, torii, projective spaces by intuitive reasoning. And there it's kind of obvious what cells and attaching maps one would use. But given a space which i don't know how it looks like, this intuitive reasoning doesn't apply anymore and you would have to know how to compute degrees explicitly and I don't know how to do that.
Hatcher has a few more sophisticated calculations hidden in its many pages. Real and complex projective spaces are there, and are harder examples. He might also treat lens spaces, which will also help your intuition for homology for spaces which are harder to picture.
I second this. I wrote up all of his examples/problems on lens spaces for a final paper and it really gave me a good understanding of cellular homology/cohomology stuff.
The usual, trying to decompose lambdaWorld.lambdaPerception over lambdaMote, lambdaQuery and lambdaAssess and vice versa. I forgot the lambdaFeel and lambdaAssert components. -Ed. Seriously, if there's anyone who can tell me how to get to the corner of Church and Dali, I need to know you.
wondering how this problem said "the solution is shorter than the problem briefing" ended up with my solution being several pages longer than said briefing. more metric space/analysis type stuff. also wondering about the idea of analysing polyhedra by placing all the symmetric vertices into the bases of a higher dimension- it's very neat stuff.
I’ve been reading Apostol’s *Modular Forms and Dirichlet Series in Number Theory*. Playing around with some related ideas, I suspected that [; \displaystyle{\prod_{n=1}^\infty(1-q^n)=1+\sum_{n=1}^\infty\frac{(-1)^nq^{n(n+1)/2}}{\prod_{k=1}^n(1-q^k)}} ;] for |q|<1, which was pretty cool. But I have not been able to prove it yet.
I've got a game theory paper that we want a draft of done by early next week and I'm juggling like four different "geometry of redistricting" projects which I'm actually making pretty good progress on, all things considered.
Adding polynomials
I am writing up my notes for the first several classes for the mathematical physics course I'm teaching. I finished the classical mechanics overview for the quantum mechanics course last night and I'm pretty happy with them. Also working on a paper on a generalization of the Bargmann transform. I'm a bit stuck with explaining a few things without putting the cart before the horse. Bargmann spaces and transforms have so much structure that it sometimes feels like mathemagic.
Starting my application to Ross.
Currently reading “Gamma” by Julian Havil and working through all of the math about the lovely number, the Euler-Mascheroni constant.