Most Popular
1500 questions
51
votes
1 answer
To what extent does Spec R determine Spec of the Witt vector ring over R?
Let $R$ be a perfect $\mathbb{F}_p$-algebra and write $W(R)$ for the Witt ring [i.e., ring of Witt vectors -- PLC] on $R$. I want to know how much we can deduce about $\text{Spec } W(R)$ from knowledge of $\text{Spec } R$. I am especially interested…
Hunter Brooks
- 4,683
- 1
- 33
- 28
51
votes
5 answers
What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?
I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to every "space" $X$ a triangulated category…
Dan Petersen
- 39,284
51
votes
4 answers
History of "without loss of generality"
"Without loss of generality" is a standard in the mathematical lexicon, and I am writing to ask if anyone knows where the expression was popularized. (The idea has been around since antiquity, I'm sure, but the expression itself might not be that…
David Steinberg
- 2,198
51
votes
6 answers
What does it take to run a good learning seminar?
I'm thinking about running a graduate student seminar in the summer. Having both organized and participated in such seminars in the past, I have witnessed first-hand that, contrary to what one might expect, they can be rather successful. However I…
Faisal
- 10,184
51
votes
3 answers
Can the sphere be partitioned into small congruent cells?
On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are mutually disjoint and their union is the whole sphere.…
Wlodek Kuperberg
- 7,256
51
votes
3 answers
What to do now that Lusztig's and James' conjectures have been shown to be false?
Lusztig and James provided conjectures for dimensions of simple modules (or decomposition numbers) for algebraic groups and symmetric groups in characteristic $p$. These conjectures have been considered almost certainly true and have guided a lot…
Chris Bowman
- 1,191
51
votes
3 answers
What is the difference between an automorphic form and a modular form?
This is more of a question about terminology than about math.
The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it is. Many sources use the term in different…
David Corwin
- 15,078
51
votes
9 answers
The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch?
Starting from 80-ies the ideas either coming from physics, or by physicists themselves (e.g. Witten) are shaping many directions in mathematics. It is tempting to paraphrase E. Wigner, saying about "The Unreasonable Effectiveness of Physics in…
Alexander Chervov
- 23,944
51
votes
7 answers
Why are the characters of the symmetric group integer-valued?
I remember one of my professors mentioning this fact during a class I took a while back, but when I searched my notes (and my textbook) I couldn't find any mention of it, let alone the proof.
My best guess is that it has something to do with Galois…
zeb
- 8,533
50
votes
4 answers
To which extent can one recover a manifold from its group of homeomorphisms
Question. Suppose that $M$ is a closed connected topological manifold and $G$ is its group of homeomorphisms (with compact-open topology). Does $G$ (as a topological group) uniquely determine $M$?
One can ask the same question where we regard $G$ as…
Misha
- 30,995
50
votes
6 answers
Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic
Hello,
I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the trivial bundle $S^2 \times R^2$ are not…
zygund
- 921
50
votes
5 answers
Categorical foundations without set theory
Can there be a foundations of mathematics using only category theory, i.e. no set theory? More precisely, the definition of a category is a class/set of objects and a class/set of arrows, satisfying some axioms that make commuting diagrams possible.…
user2529
50
votes
2 answers
Does Godel's incompleteness theorem admit a converse?
Let me set up a strawman:
One might entertain the following criticism of Godel's incompleteness theorem:
why did we ever expect completeness for the theory of PA or ZF in the first place?
Sure, one can devise complete theories semantically (taking…
David Feldman
- 17,466
50
votes
1 answer
Unconditional nonexistence for the heat equation with rapidly growing data?
Consider the initial value problem
$$ \partial_t u = \partial_{xx} u$$
$$ u(0,x) = u_0(x)$$
for the heat equation in one dimension, where $u_0: {\bf R} \to {\bf R}$ is a smooth initial datum and $u: [0,+\infty) \times {\bf R} \to {\bf R}$ is the…
Terry Tao
- 108,865
- 31
- 432
- 517
50
votes
2 answers
Small residue classes with small reciprocal
Let $p$ be a large prime. For any $m \in \{1,\ldots,p-1\}$, let $\overline{m} \in \{1,\ldots,p-1\}$ be the reciprocal in ${\bf Z}/p{\bf Z}$ (i.e. the unique element of $\{1,\ldots,p-1\}$ such that $m \overline{m} = 1 \hbox{ mod } p$).
I am…
Terry Tao
- 108,865
- 31
- 432
- 517