139
votes
Mathematical games interesting to both you and a 5+-year-old child
One evening at the dinner table, when my oldest daughter was 3 or 4, I was in a teasing mood, and I called her a goose. She didn't want to be a goose, so she refuted the claim, "I am not a goose!" ...
126
votes
What happened to Suren Arakelov?
There are memoirs by Mikhail Zelikin in Russian. He knew Arakelov personally and quite explicitly describes what happened to him.
Через несколько дней произошло следующее. Был арестован Солженицын....
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114
votes
PhD dissertations that solve an established open problem
I find George Dantzig's story particularly impressive and inspiring.
While he was a graduate student at UC Berkeley, near the beginning of
a class for which Dantzig was late, professor Jerzy ...
99
votes
Mathematical games interesting to both you and a 5+-year-old child
The game "Set" seems to fit the bill. It's a card came where there are cards that show images which have four different features, each of which comes in three possibilities:
number (1, 2, or 3 ...
93
votes
Accepted
Why isn't integral defined as the area under the graph of function?
Actually, in the following book the Lebesgue integral is defined the way you suggested:
Pugh, C. C. Real mathematical analysis.
Second edition. Undergraduate Texts in Mathematics. Springer, Cham, ...
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73
votes
Mathematical games interesting to both you and a 5+-year-old child
Another topological game: Sprouts.
Rules:
The game is played by two players, starting with a few spots drawn on a sheet of paper. Players take turns, where each turn consists of drawing a line ...
70
votes
Accepted
Ideas for introducing Galois theory to advanced high school students
I have now twice taught Galois theory to advanced high school students at PROMYS. This is a six week course, meeting four times a week, for students who already are comfortable with proofs and, in ...
57
votes
PhD dissertations that solve an established open problem
I am quite surprised that nobody has mentioned Grothendieck's thesis. Apparently Laurent Schwartz gave Grothendieck a recent paper listing a number of open problems in functional analysis at one of ...
49
votes
Mathematical games interesting to both you and a 5+-year-old child
Knot or not?
The topological game involves a projection of a knot, drawn onto paper, such as:
Player 1 picks an intersection and assigns a crossing (which segment of the curve is "above" and which "...
46
votes
Great graduate courses that went online recently
Algebraic Geometry in the Time of COVID (AGITTOC) happened during late summer of 2020. Ravi Vakil gave what he called "pseudolectures" following his "Rising Sea Notes". He didn't ...
41
votes
Are hypergeometric series not taught often at universities nowadays, and if so, why?
[Q1] Gert Heckman from Nijmegen University teaches a course on hypergeometric functions (here are the lecture notes, first taught at Tsinghua Univ.).
[Q2]
In the foreword, Heckman hints at why this ...
40
votes
Short papers for undergraduate course on reading scholarly math
I think this is a delightful paper:
Hull, Thomas C. "Solving cubics with creases: The work of Beloch and Lill." The American Mathematical Monthly 118, no. 4 (2011): 307-315.
(PDF download.)
It ...
38
votes
Examples of common false beliefs in mathematics
I do not think anybody mentioned this example:
If $M$ is a $C^k$-smooth manifold then the tangent space $T_pM$ is isomorphic to the space of derivations of germs (at $p$) of $C^k$-smooth functions on ...
38
votes
PhD dissertations that solve an established open problem
Godel's Completeness Theorem, was part of his PHD thesis.
It was definitely an active field of research, but I don't know to what degree the problem was an open one, in the way we understand it today....
37
votes
PhD dissertations that solve an established open problem
The thesis of Martin Hertweck answered the at that time 60-years-old
isomorphism problem for integral group rings in the negative, by constructing
a counterexample. That is, a pair of non-isomorphic ...
34
votes
PhD dissertations that solve an established open problem
Scott Aaronson's thesis, Limits on Efficient Computation in the Physical World, refuted some popular wisdom.
In the first part of the thesis, I attack the common belief that quantum computing ...
33
votes
How to explain to an engineer what algebraic geometry is?
Perhaps you're going about this the wrong way. Instead of trying to describe what most algebraic geometers do today, try to describe a problem, or set of problems, that is reasonably concrete and ...
32
votes
What kid-friendly math riddles are too often spoiled for mathematicians?
Here's a few, two I got to solve myself as a kid and one (a trickier one, in my opinion) that was spoiled for me.
There are $1000$ lights all in a line and turned on. At time $n$, person $n$ comes by ...
31
votes
PhD dissertations that solve an established open problem
Lisa Piccirillo, who recently obtained her PhD from the University of Texas, Austin, showed that the Conway knot is not slice, answering a relatively famous open problem in topology. You can read a ...
31
votes
Great graduate courses that went online recently
Most of them are not so recent as you are asking for, but in my opinion is still worthy to look at them.
Claudio Arezzo's lectures on differential geometry
https://www.youtube.com/playlist?list=...
30
votes
Mathematical games interesting to both you and a 5+-year-old child
Dots and Boxes
Is a pencil-and-paper game for two players. It's quite simple to explain but quite hard to play. Five year olds should be able to learn it and with some training maybe also being good ...
30
votes
How to explain to an engineer what algebraic geometry is?
Abhyankar's book Algebraic Geometry for Scientists and Engineers doesn't give a short answer, but many long ones, with explicit examples of determining the geometric nature of the solutions of ...
30
votes
Why isn't integral defined as the area under the graph of function?
If $f: \mathbb{R} \to [0,\infty)$ is Borel (or Lebesgue) measurable, then for each rational $a > 0$ define $X_a = f^{-1}([a,\infty)) \times [0,a)$. Then each $X_a$ is measurable and their union is ...
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29
votes
Accepted
How should you explain parallel transport to undergraduates?
This may not reallly be an answer that you like, but I think that, maybe you misunderstood what Ben McKay was trying to describe. Here is a more explicit, extrinsic description that may help:
Suppose ...
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28
votes
Mathematical games interesting to both you and a 5+-year-old child
There is an infinite indexed family of family-friendly, $\geq2$-player, perfect-information, draw-free-if-finite, cheap-to-construct, two-player combinatorial1, solved, sequential games. They are ...
28
votes
PhD dissertations that solve an established open problem
John von Neumann's dissertation seems to be an example with just the right timing.
But at the beginning of the 20th century [in 1901, to be precise], efforts to base mathematics on naive set theory ...
26
votes
PhD dissertations that solve an established open problem
A question, a book, and a couple of dissertations; the most relevant, I think, is the thesis by Petkovšek. Hopefully this is an acceptable MO answer. First, the question comes from Knuth in The Art of ...
26
votes
What kid-friendly math riddles are too often spoiled for mathematicians?
To make this suitable for MO rather than math.SE, perhaps we can define a "too often spoiled" puzzle to be one that can be recognized instantly by a mathematician even with what looks like ...
25
votes
Mathematical games interesting to both you and a 5+-year-old child
What about Spot It! (US), also known as Dobble (Europe)?
We are given a deck of 55 cards. Each card contains 8 different symbols, such that any pair of cards in the deck has precisely one symbol in ...
25
votes
Are hypergeometric series not taught often at universities nowadays, and if so, why?
I think you are correct that a university course on hypergeometric functions is rare. Instead, a course on ordinary differential equations may include a section on hypergeometric functions, as an ...
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