All Questions
262 questions
17
votes
6
answers
7k
views
Explaining the concept of projective space: notes for students
This is a question on teaching.
I am teaching at this moment a course in algebraic geometry for master students on a very basic level. Today (this was the fourth lecture) I discovered that only four ...
- 2,281
7
votes
8
answers
4k
views
Mathematical Advice for Interested Highschool Students
This may not be a research level math question, but I believe it is still relevant to Math Overflow.
What general resources exist for students in highschool who are very interested in Mathematics?...
19
votes
3
answers
2k
views
Research level applications of "row rank = column rank"?
No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...
- 17.6k
11
votes
1
answer
1k
views
Teaching stacks to differential geometry students
Does anyone have any experience teaching stacks over the category of manifolds to students whose background is, say, a semester-long course on manifolds? Does anyone know of any publicly available ...
- 341
0
votes
2
answers
562
views
Lines on degree 2n-3 Fermat hypersufaces
It is well known that a generic hypersurface of degree $2n-3$ in $\mathbb CP^n$ has finite number of lines. I would like to ask a couple of questions about lines on Fermat hypersurfaces and their ...
- 1,936
12
votes
9
answers
6k
views
Topics for an Undergraduate Expository Paper in Number Theory
I am teaching an undergraduate course in number theory and am looking for topics that students could take on to write an expository paper (~10 pages). No new results are expected of them. Many of the ...
CommunityBot
- 1
5
votes
3
answers
2k
views
Continuous change of basis (and on the definition of determinant) [closed]
Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...
- 23k
13
votes
1
answer
2k
views
conditional equality symbol
Is there a standard notation (perhaps $A \stackrel{\leftarrow}{=} B$) meaning "in all situations where $B$ is defined, $A$ is defined and equals $B$"?
The kind of situation in which such a notation ...
- 3,793
6
votes
2
answers
934
views
Surface Laplace-Beltrami without coordinates, exterior calculus?
Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator $\...
- 27.5k
9
votes
3
answers
1k
views
Where can I find questions motivating important ideas in math?
I would like questions that demonstrate why a mathematical tool or technique is useful, and which can be used to introduce that idea. Ideally, this would be a compilation of problems organized by the ...
CommunityBot
- 1
8
votes
4
answers
1k
views
Multivariable Calculus Lecture Ideas
I am teaching a course in multivariable calculus this semester. We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces. After that ...
CommunityBot
- 1
2
votes
3
answers
410
views
Pedagogical notes on line bundles on complex projective manifolds
I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems ...
- 15.4k
16
votes
6
answers
3k
views
How to mentor an exceptional high school student?
I have a unique and, quite truthfully, humbling opportunity. The parents of an exceptionally talented high school freshman have reached out to me and asked if I might be able to help.
This kid is ...
CommunityBot
- 1
27
votes
5
answers
7k
views
References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
- 13.2k
9
votes
7
answers
1k
views
Mathematics seminar for "non-mathematicians"
Next term I am leading a seminar for students, who will become teachers for elementary school i.e. for kids of age 6-10. The students in the seminar will have no mathematical background beyond the "...
- 12.3k
14
votes
3
answers
3k
views
Open source LaTeX lecture notes/slides/books [closed]
In the mathematics community it's quite common for professors to write their own notes for the classes they are teaching. The notes are then usually published in both PDF and PS form on the course ...
- 3,322
11
votes
1
answer
1k
views
Teaching Experience for Graduate Students. [closed]
I am currently a graduate student, who will (hopefully!) graduate in the next year (or two..). I have slowly come to realize that I enjoy teaching, and consequently want to do more of it! My main ...
- 151k
5
votes
1
answer
1k
views
Is Diagonalization worth to be taught? [closed]
When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a ...
- 12.6k
24
votes
2
answers
2k
views
Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$
Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, ...
- 33.8k
24
votes
7
answers
8k
views
How do professional mathematicians learn new things? [closed]
How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues?
- 12.7k
11
votes
4
answers
3k
views
Topological examples of profinite groups
I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first ...
- 9,627
1
vote
0
answers
430
views
Professional skills advising for math jobs [closed]
Hi,
I am a postdoc at the University of Nottingham (UK) and I am beginning to apply for Assistant Professor positions in US.
I would like to receive a feedback on the material that I am sending (...
5
votes
2
answers
2k
views
Any suggestions for a course in Mathematical Logic?
I am teaching a topics course for Mathematics majors (at Temple), and am considering Logic as the topic. I was wondering if people (a) have suggestions for an appropriate text and (b) how much might ...
- 2,754
61
votes
10
answers
10k
views
Teaching proofs in the era of Google
Dear members,
Way back in the stone age when I was an undergraduate (the mid 90's), the internet was a germinal thing and that consisted of not much more than e-mail, ftp and the unix "talk" command ...
- 20.2k
13
votes
5
answers
2k
views
How to make a lecture series useful
I have been to a number of advanced lecture courses (of between 3 and 10 lectures) over the years, given (in principle) by experts to graduate students and experts in neighbouring fields. Examples of ...
- 50.6k
27
votes
17
answers
9k
views
Using slides in math classroom
I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the ...
- 1,037
4
votes
0
answers
795
views
Almost linear ODE: how node becomes a spiral
Most introductory ODE books contain a discussion of almost linear systems, and there are two cases when the behavior of an almost linear system near an equilbrium point can differ from the behaviour ...
CommunityBot
- 1
3
votes
2
answers
957
views
Simple definition of the Hausdorff measure using squared paper
I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.
For simplicity, I was hoping to give a more intuitive ...
- 44.4k
3
votes
3
answers
2k
views
What to teach in a second graduate course in algebra? What textbook to use?
There is a standard syllabus for a first graduate course in algebra. One teaches groups,
rings, fields, perhaps a little bit of Galois theory, perhaps a little bit of
category theory, perhaps a ...
- 822
12
votes
11
answers
2k
views
Giving a math talk with no blackboard or projector
I need to give a math talk to a group of undergraduates. I am asking for advice because this talk will take place at a department picnic and there will be no blackboard or projector. I would like to ...
- 1,701
7
votes
5
answers
2k
views
Commutative algebra final project
I'm looking for a topic for a final project in commutative/homological algebra, for first year master's students (in a decent European university). During the course, they will cover the following ...
CommunityBot
- 1
2
votes
4
answers
6k
views
Undergraduate Derivation of Fundamental Solution to Heat Equation
It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a ...
- 3,322
3
votes
0
answers
431
views
Concrete questions that turn into math problems [closed]
I'm writing an article about the way we teach math, trying to find out why so many people are discouraged from learning, and have no interest for math and logic.
At some point, I want to show that ...
- 131
12
votes
5
answers
2k
views
Introducing Cryptology to Undergraduates
This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures on,...
- 3,867
10
votes
8
answers
2k
views
Undergraduate Probability Topics
I am teaching undergraduate probability this semester, and I am looking for some suggestions about inspiring applications that could be reasonably covered over the course of two one-hour lectures or ...
CommunityBot
- 1
19
votes
9
answers
5k
views
Mathematics and autodidactism
Mathematics is not typically considered (by mathematicians) to be a solo sport; on the contrary, some amount of mathematical interaction with others is often deemed crucial. Courses are the student's ...
CommunityBot
- 1
22
votes
4
answers
5k
views
What is the best way explain to undergraduates that all 1-dimensional manifolds are orientable?
Let's suppose that $M$ is a connected $1$-dimensional smooth manifold (Haussdorf and paracompact). We know that there are exactly two types, up to diffeomorphism (even up to homeomorphism), namely $\...
- 3,540
8
votes
1
answer
4k
views
Who is this guy : Z.A. Melzak (wrote Companion to Concrete Mathematics) ? [closed]
Author : Z.A. Melzak
Book Title : Companion to Concrete Mathematics.
Publication : Dover renewed 2004 2 volumes in one. Copyright 1972/1976.
I found this book extremely nice.
To whet your appetite ...
- 1,306
4
votes
2
answers
869
views
Terminology question on covering spaces
I'm teaching an elementary class about fundamental groups and covering spaces. It was very useful to use "fool's covering spaces" of a space $X$, defined as
functors $\Pi_1(X)\to Sets$, where $\Pi_1(X)...
CommunityBot
- 1
3
votes
3
answers
1k
views
Pedagogical question concerning $\Gamma(z)$
Pedagogically speaking, I see two problems with defining
$\Gamma(z)$ (at least for real $z$) by the limit
$$\Gamma(z)=\lim_{m\to\infty}\frac{m! m^z}{\prod_{i=0}^m (z+i)}$$
as compared with the formula
...
- 2,762
11
votes
2
answers
1k
views
Social Reading Platform for Math or LaTeX texts
Social reading is considered to be one of the big trends that could be catalysing learning by reading. Features could include:
Highlighting or annotating paragraphs or single steps in a proof for ...
3
votes
1
answer
507
views
What are some interesting grading/curving systems you have seen for a course? [closed]
It seems like every math course has something unique in how things are graded.
1) What are some interesting grading systems you have seen/used? (include curving types, etc.)
2) What are some pros ...
- 30.1k
71
votes
11
answers
9k
views
How to introduce notions of flat, projective and free modules?
In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this ...
- 12.4k
15
votes
4
answers
3k
views
How does one motivates the method of separation of variables when teaching PDE's?
I'm not sure if this question is appropriate for MO. Add comments if it is not. Thanks.
How to explain/motivate the method of separation of variables for PDEs to undergraduates? What's the real math ...
- 798
14
votes
2
answers
7k
views
What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first?
When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator $\{f\in V^*: f(u)=0\quad \forall u\in U\}$ of a ...
- 491
19
votes
1
answer
2k
views
Resources for teaching arithmetic to calculus students
Every time we teach calculus we discover that a significant portion of our students never understood arithmetic. I don't mean that they can't multiply numbers, but rather that they don't know ...
- 12.1k
27
votes
5
answers
5k
views
Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties
This really is two questions, but they are kind of related so I would like to ask them at the same time.
Question 1:
In a question asked by Amitesh Datta, BCnrd commented that it is important to ...
CommunityBot
- 1
10
votes
7
answers
2k
views
Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
CommunityBot
- 1
6
votes
3
answers
1k
views
Publishing with Undergraduates
Is doing research with a student considered to be good for a dossier? Is it okay to have few research publications but a lot of student projects? I am finishing up a grad program and am looking at ...
- 41
3
votes
6
answers
2k
views
Teach a course in 1 month
I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?
- 2,368