Is there an $E_1$-definition of primality? Here, $E_1$ denotes the set of arithmetic formulas starting with a bounded existential quantifier, followed by a quantifier-free formula. Is there an $E_1$-formula $\phi$ such that $\phi(n)$ holds
iff $n$ is prime? If yes, it is likely to be rather complicated to obtain, as this apparently implies that PRIMES is in P.
Clearly, there is such a $U_1$-formula (i.e. one starting with a bounded universal quantifier instead). And of course there's this famous prime polynomial, but this gives a $\Sigma_1$-statement where the variables correspond to exponentiations and factorials, so certainly there can be no polynomial bounds for them.


A set that can be covered by arbitrarily small intervals Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-small if there is a collection $\{I_i\}$ of intervals such that the length of each $I_i$ equals $s_i$ and the union $\bigcup I_i$ contains $X$. And $X$ is said to be small if it is $S$-small for any sequence $S$.
Obviously every countable set is small. Are there uncountable small sets?
Some observations:

A set of positive Hausdorff dimension cannot be small.
Moreover, a small set cannot contain an uncountable compact subset.


On finite subsets of set of integers, which lies in its sum-set , whose sum of elements equals $0$ Let $n>1$ be an integer and $S \subseteq \mathbb Z$ be such that $|S|=n$ and  $S \subseteq S+S:=\{a+b : a,b \in S\}$ ; then does there exist $T \subseteq S$ with $1 \le |T| \le n/2$ such that $\sum_{a \in T}=0$ ?


Modular form, number of divisors The Fourier expansion of Eisenstein series $E_k$ $(k \ge 4)$, which are modular forms, as well as the quasimodular $E_2$, involves powers-of-divisors $\sigma_{k-1}(n) = \sum_{d|n} d^{k-1}$.
Is there some generalization of a modular form (quasimodular, mock modular, etc.) where you can find some similar Fourier expansion of the form $$f(z) = 1 + C \sum_{n=0}^{\infty} \sigma_0(n) q^n, \; q = e^{2\pi i z}$$ involving just the number-of-divisors function $\sigma_0(n)$? I haven't been able to find this.


Continuous change of basis (and on the definition of determinant) 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 similarly for the second basis. Prove that the first basis can be continuously transformed into the second one, while remaining linearly independent at all times, if and only if the two bases have the same orientation.
The "only if" direction is easy, because the determinant, which must change continuously, cannot change from positive to negative without going through zero. I'm looking for a proof of the "if" part.
More broadly, I'm looking for comments on the issue of defining the determinant in a nice way. The definitions I've seen say something like:
"The determinant is a quantity that has some nice properties. For one, the determinant is zero if and only if the corresponding matrix is singular. Furthermore, its absolute value equals the volume of the parallelepiped spanned by the vectors. And the sign corresponds to the orientation of the vectors. And what is the "orientation" of a tuple of vectors? Well, it's defined as the sign of the determinant!"
The above claim, if correct, might lead to a more natural (and less circular) definition of orientation, and also of the determinant.
Also, is it necessary to define the n-dimensional volume of a parallelepiped as the absolute value of the determinant (as I have seen in some places)? Can't they be shown to be equal via elementary arguments? Consider the "cut-and-paste" proof that the area of a parallelogram equals the area of a rectangle with the same base and height. I think a similar n-dimensional cut-and-paste can show that
$$\mathrm{vol}(u_1, u_2, \cdots , u_n) = \mathrm{vol}(u_1 + k u_2, u_2, \cdots, u_n),$$
and similarly for the other elementary properties of the determinant. But I haven't thought it through.
Thanks in advance!


When is $L^{2}(X,\mathscr{B},m)$ spearable If $X$ is a metric space, $m$ is a Borel probability space on $(X,\mathscr{B})$ where $\mathscr{B}$ is the $\sigma$-algebra generated by open sets on $X$, can we prove that the space　$L^{2}(X,\mathscr{B},m)$ is spearable?


measure zero in R but not in R^2 I want to find some subset of R^2 which its intersection with every vertical line is measure zero if we see it as a subset of R and it is not measure zero in R^2?


Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$ Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a line in $\mathbb{R}^2$ which contains the image $f(\mathbb{R})$ of $f$?
Yes, this question seems rather elementary, but I have already spent (or lost?) too much time on this devilish problem, and I have communicated this question to sufficiently many people to know that it is far from trivial...


Inner product of linear bounded operators between Hilbert spaces Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces.
Can we equip $L(X,Y)$ with a natural inner product? I think it should look like
$\langle S, T \rangle = \sup_{x \in X} \dfrac{ \langle S x, T x \rangle_Y }{ \|x\|^2_X }$
where $S$ and $T$ and are from $L(X,Y)$. I have not found such a construct in standard text books on Hilbert spaces, therefore I would like to learn whether this is the way to do it.


When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals) Sorry if this is a silly question.  I was wondering, under what axioms of set theory is it true that if $\alpha$,$\beta$ are cardinals, and $2^\alpha=2^\beta$, then $\alpha=\beta$? Do people use these conditions to prove interesting results?
This question is prompted from a recent perusing of Johnson's "Topics in the Theory of Group Presentations", where in the first few pages he "proves" free groups of different rank are non-isomorphic:
the number of mappings from a free group of rank $\omega$ to the group $\mathbb{Z}/2\mathbb{Z}$ is $2^\omega$, which would be invariant under isomorphism; and then he assumes the topic of my question: $2^\alpha=2^\beta$ implies $\alpha=\beta$.
But I remember reading something a few years ago about a student of R.L. Moore (I only remember his last names was Jones) "proving" the Moore Space conjecture, and using that $\alpha > \beta$ implied $2^\alpha > 2^\beta$, but that this was incorrect.
Anyway, I realized I don't know anything about when this is true or false, so I thought I'd ask.


maximum size of intersecting set families Suppose $n$ is a big number and $k\geq 2$. How many sets $S_1,\dots,S_m\subset [n]$ can we find such that
(1) $|S_i| = k$ for all $i$,
(2) $|S_i\cap S_j| \leq 1$ for all $i\ne j$.
What's the maximum possible value of $m$?
(I just need to know the growth order of $m$ depending on $n$ and $k$. For instance, when $k=2$, we have $m = \binom{n}{2} \sim n^2$.)
I tried to look it up in the literature, but it looks like this is different from the classical intersecting family that I have an upper bound on the size of intersection of a pair of sets instead of a lower bound.


Is $\text{Sym}(\omega)/\text{(fin)}$ embeddable in $\text{Sym}(\omega)$? Let $\omega$ denote the set of natural numbers, let $\text{Sym}(\omega)$ be the collection of bijections $\psi:\omega\to\omega$, and let $\text{(fin)}$ be the set of members of $\text{Sym}(\omega)$ having finite support. Formally, $$\text{(fin)} = \{\psi\in\text{Sym}(\omega): (\exists k\in\omega)(\forall i\in\omega\setminus k)\psi(i) = i\}.$$
Is $\text{Sym}(\omega)/\text{(fin)}$ isomorphic to a subgroup of $\text{Sym}(\omega)$?


Estimate for products of integers that are relatively prime with $N$ Let $N$ be a positive integer. Are there known estimates for the product of all numbers that are smaller than $N$ and relatively prime with $N$? One can assume that $N$ is free of squares, if this helps.


Given the vertices of a convex polytope, how can we construct its half-space representation? Let us say I have the vertices of a polytope $V = \{v_1,\dots,v_k\} \subset \mathbb R^n$. Is it possible to write $V$ as intersection of half-spaces using the information from the vertices, i.e., can I write the polytope in the form $Ax \leq b$ where $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$?
The columns of $A$ are not necessarily the vertices of the given polytope. An example, consider a polytope in $\mathbb R^2_+$ with vertices $\{(0,1),(1,1),(2,0),(0,0)\}$. It can be observed that the corresponding half space representation is $Ax\le b$, where
$$A=\begin{pmatrix} 0 & 1 \\\ 1 & 1 \\\ -1 & 0\\\ 0 & -1\end{pmatrix}$$
and $b = (1,2,0,0 )^T$. Thank you.


Mathematics Book on Yang-Mills Equation I am planning to read two papers - Atiyah-Bott's paper on Yang-Mills equations on Riemann surfaces and Hitchin's Self-Duality equations on Riemann Surface. Can someone please suggest some book where basics of Yang-Mills equations are discussed?
Thanks.


Complete theory with exactly n countable models? For $n$ an integer greater than $2$, Can one always get a complete theory over a finite language with exactly $n$ models (up to isomorphism)?
There’s a theorem that says that $2$ is impossible.
My understanding is this should be doable in a finite language, but I don’t know how.
If you switch to a countable language, then you can do it as follows. To get $3$ models, take the theory of unbounded dense linear orderings together with a sequence of increasing constants $\langle c_i: i < \omega\rangle$. Then the $c_i$’s can either have no upper bound, an upper bound but no sup, or have a sup. This gives exactly $3$ models. To get a number bigger than $3$, we include a way to color all elements, and require that each color is unbounded and dense. (The $c_i$’s can be whatever color you like.) Then, we get one model for each color of the sup plus the two sup-less models.


Dirichlet series of the reciprocal radical function Define $ rad(n):=\prod_{p|n}p $ and $a_n:=\frac{n}{rad(n)}.$ For example $a_n=1$ whenever n is a squarefree integer. The associated Dirichlet series $$F(s):=\sum_{n} \frac{a_n}{n^s}=\prod_{p} (1+\frac{1}{p^s} \frac{1}{1-p^{1-s}})$$ has abscissa of convergence $s_0=1.$  Are there any results regarding the distribution of $a_n$, e.g. whether $\sum \limits_{n \leq x} a_n \ll x (\log x)^A $ for some  real constant $A>0$  ?


Is chern classes of holomorphic vector bundles enough to generate Hodge cycles Let $X$ ba a smooth projective variety of dimension $n$. Hodge Conjecture states that every Hodge cycle in $Hdg^k(X,\mathbb{Q})$ comes from a Chern class of codimension $k$ in  $CH^k(X,\mathbb{Q})$. Now the $k$-th Chern class of holomorphic vector bundles generates a subgroup $CH^k_{vec}(X,\mathbb{Q})$. Is it possible that
 every Hodge cycle in $Hdg^k(X,\mathbb{Q})$ comes from $CH^k_{vec}(X,\mathbb{Q})$? Is there any counterexamples or results?


Optimizing the condition number Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns $v_1, \dotsc, v_m$ is as small as possible -- notice that this is trivial if $S$ does not span $W,$ since the condition number is always $\infty,$ so we can assume that $S$ does span. The question is: is there a better algorithm than the obvious $O(N^{m+c})$ algorithm (where we look at all the $m$-element subsets of $S?$). and how much better? One might guess that there is an algorithm polynomial in the input size, but none jumps to mind.


Why is the motivic category defined over the site of smooth schemes only? Fix a base scheme $S$. Stable and unstable motivic categories over $S$ are defined as certain categories of higher stacks on the Nisnevich site $Sm_S$ of smooth schemes over $S$. Why smooth?
As a casual observer, I see two broad classes of possible reasons:

Technical reasons: Perhaps the Nisnevich topology doesn't make sense over non-smooth schemes, or maybe $\mathbb{A}^1$ or $\mathbb{G}_m$-locality become more complicated over non-smooth schemes.
Goals of the construction: Perhaps the stable and unstable categories make perfect sense over non-smooth schemes, but maybe motivic cohomology or algebraic K-theory simply don't extend to stacks over the resulting category.

In either case, I wonder: is it clearly impossible, by some modification of the construction of these categories, to get things to work over non-smooth schemes? Would this be desireable?


an easy example of valuation ring which is not noetherian？ Is there an easy example of valuation ring which is not noetherian？


A question about simple arcs in higher dimensional Euclidean spaces. Let E(n) be n-dimensional Euclidean space. It is known that there exist subsets of E(n) which are
simple arcs and have positive n-dimensional Lebesgue measure when n=1 or 2. Does this continue to
be true for arbitrarily large n? If not, what is the largest n for which it holds and is there a
simple proof of this fact? Intuitively, I feel that there should be no upper bound, but cannot see
how to prove it.


Unprovable sentence about integers Is there any natural* statement S about the natural integers such that if PA contains no contradictions then neither PA+S nor PA+not S contains a contradiction?
If unknown, where can I read about the philosophical views on it?
*By natural I mean not a logical trick or non-constructive existence proof such as Rosser's sentence, but a clear statement such as the Collatz conjecture.


Weyl group action on the Lie algebra Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping algebra?


Group where Out(G) acts differently on conjugacy classes and irreps? $\def\Conj{\mathrm{Conj}}\def\Irrep{\mathrm{Irrep}}\def\Out{\mathrm{Out}}$Let $G$ be a finite group, let $\Conj(G)$ be the set of conjugacy classes of $G$, let $\Irrep(G)$ be the set of isomorphism classes of complex irreps of $G$ and let $\Out(G)$ be the outer automorphism group of $G$. Then $\Out(G)$ acts on both the sets $\Irrep(G)$ and $\Conj(G)$. (Since $\mathrm{Aut}(G)$ acts in an obvious way, and the inner automorphisms act trivially.)
The permutation representations $\mathbb{C}^{\Irrep(G)}$ and $\mathbb{C}^{\Conj(G)}$ are isomorphic (because evaluation of characters gives a perfect pairing between them, and both representations, being permutation representations, are self dual). Is there a group where the sets $\Conj(G)$ and $\Irrep(G)$ don't have an $\Out(G)$-equivariant bijection?


Quantifying Gödel's Incompleteness Theorem and Sparsity of Examples We know from Gödel's First Incompleteness Theorem that there are true statements in the natural numbers that have no proof. Obviously we know of many that do ("theorems").
Are results known about the relative sparsity of these two sets? For instance, perhaps one could order statements by the number of characters N, and look at true statements. Do we know if "most" true statements are theorems, or the opposite, and does the fraction depend on N? Are Gödel's examples rare jury-rigged freaks, or do they actually become the rule?


Best known Upper bound on Twin Primes I know that there is a result from J Wu that the number of twin primes less than a given magnitude $N$ does not exceed
$$\frac{2aCN}{\log^2{N}}$$
Where $C=\prod \frac{p(p-2)}{(p-1)^2}$ and $a$ is something like $3.4$.  Is this a direct result of the Selberg Sieve, or is there additional knowledge on the distribution of Twin Primes used?


Is the regularity of finitely generated rings decidable? Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?
I mean by regular that the localization at every prime ideal is a regular local ring.
The question arose from my interest in the desingularization problem. To have a desingularization algorithm of arithmetic schemes, one first needs to know the regularity of a given scheme.
The definition of the regularity is point-wise. Naively one has to check the regularity point by point of $\mathrm{Spec}\, R$ for a given ring $R$. It is not an algorithm in the sense that it never halts if $R$ is regular.
If $R$ is defined over a prime filed, $\mathbb{Q}$ or $\mathbb{F}_p$, then one can use the Jacobian criterion: first compute the Jacobian ideal and then check if it is trivial by computing its reduced Gröbner basis. In positive characteristic, one may also use Kunz's criterion in terms of Frobenius maps. As far as I know, there is no such a global criterion for rings over $\mathbb{Z}$. Serre's criterion by the finiteness of global dimension looks global at the first glance. But one needs to know the projective dimensions of infinitely many modules.
So, my guess is that the answer to the question would be NO. Does someone know the answer or related works?


Is it provable in $\mathsf{ZF}$ that there is a group structure on any set $X$? Given a set $X$ is it provable in $\mathsf{ZF}$ that there is a binary operation $\ast: X\times X\to X$ such that $(X,\ast)$ is a group?


Maximal number of times distance $1$ can occur among $n$ points in the plane For $n\in\mathbb N$, let $f(n)$ be the maximal number of times distance $1$ can occur among $n$ points in the plane:
$$
f(n) = \max_{ \{ x_1,\ldots,x_n \} \subset \mathbb R^2} \# \big \{ i<j : \| x_i-x_j \| =1 \big \}.
$$
What is the asymptotic growth of $f(n)$?
Is $\lim_{n\to \infty} f(n)/n=\infty$?

The function $f(n)$ starts very much like A047932:
0, 1, 3, 5, 7, 9, 12, 14, 16, 19, 21, 24, ...
but eventually grows faster.


partially commutative like monoids Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ (empty word) whenever $\{a,b\} \notin E(G)$ and $c \in I$ is arbitrary.
I have the following questions about the monoid $M(G)$.

Is this monoid $M(G)$ well studied in the literature?
What are some algebraic combinatorics or general combinatorial significance of this monoid?

Thanks for your time and have a good day.


Modular forms of weight 1/3 I know that theta series are modular forms of weight 1/2 and I have read about modular forms of other fractional weights. Does anybody have an explicit example of a modular form of weight e. g. 1/3 or 1/4?
I thought that in analogy with theta series one could define for example $\sum_{n\in \mathbb Z}q^{n^4}$, but I don't know how to continue.


Elementary proof for $n^2>p>n$ for all $n>1$ Is there any elementary way of proving that for all natural numbers $n>1$ there exists a prime $p$ such that $n<p<n^2$. And I mean elementary, not using the Prime Number Theorem or Bertrand's Postulate.


$\mathcal{M}_{g,n}$ a scheme for $n \gg 0$? I think that for $n \geq 3$, the Deligne-Mumford moduli stack $\mathcal{M}_{0,n}$ is a scheme. Is it more generally true that for every $g$, the Deligne-Mumford moduli stack $\mathcal{M}_{g,n}$ is a scheme for $n \gg 0$? (My intuition is that the presence of many disjoint marked points "kills" automorphisms.)


Reference request: Urbanik's work on random integrals and Orlicz spaces Several important papers on Lévy processes are referring to the following paper:

K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces,
  Bulletin de l'Académie Polonaise des Sciences, Série des sciences
  mathématiques, astronomiques et physiques, 15 (1967), p. 161-169

I couldn't find this reference in the traditional databases, either on the website of the journal. Is it possible to find it on the internet?


Maxwell's equations and differential forms Is there a textbook that explains Maxwell's equations in differential forms?
What I understood so far is that the $E$ and $B$ fields can be assembled to
a 2-form $F$, and Maxwell's equations can be written quite nicely
with the Hodge $*$ and the exterior deriative $d$.
Going further the equations can be derived as Euler-Lagrange (or Yang-Mills?) equations from a connection of a fibre bundle.
I am searching for a book that describes how the geometric entities are mapped to the physical entities with a focus on mathematical exactness.


The sum of the signs of conjugacy classes in the symmetric group S_n Let $r$ be the number of conjugacy classes of the symmetric group $S_n$ whose sign is $1$, i.e.
\begin{equation}
r := \#\{c \in \text{Conj} (S_n): \text{sgn} (c) = 1 \}.
\end{equation}
Let $s$ be the number of conjugacy classes of $S_n$ whose sign is $-1$, i.e.
\begin{equation}
s := \#\{c \in \text{Conj} (S_n): \text{sgn} (c) = -1 \}.
\end{equation}
I want to prove that $r - s$ is the number of self-conjugate partitions of $n$.
In other words, I want to prove that
\begin{equation}
\sum_{c \in \text{Conj} (S_n)} \text{sgn} (c) = \#\{\lambda: \lambda \ \text{is a self-conjugate partition of } n\}.
\end{equation}


What is the defining property of reductive groups and why are they important? Having read (skimmed more like) many surveys of the Langlands Program and similar, it seems the related ideas apply exclusively to groups that are "reductive".
But nowhere, either in these surveys or elsewhere, have I been able to find a simple and compelling definition of what it means for a group to be reductive and why this property is important and why Langlands was naturally led to frame his conjectures for them and not, say, for any group.
Any suggestions or links?


What to do with antique/older mathematics books? Throw away or something else? My father, who held 4 post graduate degrees and was a lifetime student, passed away recently.  He has an entire bookcase full of older mathematics books, including some on related topics such as computer science.  I already threw away some books related to computer programming, including those on languages that no one uses anymore, like APL.  But then it occurred to me that maybe someone would want some of these older books and I should not just throw them away.  Does anyone here have any suggestions as to what I should do with them?  thank you


A measurable set such that its intersection and difference with every interval have the same measure Let $\Omega = [0,1]$. I want a Lebesgue measurable set $S$ with the following property.
$$ \ell(S \cap I) = \ell(I \backslash S)$$ for every subinterval $I$ of $[0,1]$, where $\ell(A)$ is the Lebesgue measure of $A$.
A friend recently told me that Lusin's theorem says that such a set does not exist. I don't seem to find a result I can quote (and learn from) that says the same though. Is it true that such a set does not exist?
Thanks.


Is there a link between $H_2(G,\mathbb{Z})$, the Schur Multiplier of a group, and the "other" Schur multipliers of a group? The name for the the following 2 mathematical objects:

$$H_2(G,\mathbb{Z})$$ and
$$\{K:G\times G\longrightarrow\mathbb{C}\ |\ \forall T\in B(l^2(G))\text{we have that}~S:G\times G\longrightarrow\mathbb{C}\text{defined by} \\ S(g,h)=K(g,h)T(g,h)\text{also represents an element of}~B(l^2(G))\}$$
where $T$ and $S$ are seen as infinite matrices in the the canonical basis of $l^2(G)$

is the same: Schur multiplier of a group. Why? Is there a strong connection between them? I'd say it comes from the fact that originally $H_2(G,\mathbb{Z})$ was defined as $H^2(G,\mathbb{C}^*)$ for finite groups, which has to do with projective representations, and representations are related to the second object. But infinite groups interest me more.
I think Herz first defined and gave the name to the second object, but I don't know why he chose this name which already exited in the literature, unless there is a strong link between them. (the paper is in French and I can't read it).


Lagrange's four squares theorem in other fields Is something known about analogues of Lagrange's four squares theorem in number fields other than $\mathbb{Q}$?
I'm more interested in the case of finite extensions of $\mathbb{Q}$.
For example, is it true that any positive number in $\mathbb{Q}(\sqrt{3})$ can be represented as sum of several squares from the same field?


D-modules and Algebraic Solutions of PDEs  I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to  using D-modules to  study PDEs (and systems of PDEs). When I was doing a perusal on "A primer of algebraic D-modules by  S. C. Coutinho" the justification on the importance of D-modules; they provide an algebraic tool towards the solution of differential equations. This is the story I always hear!  Do someone have a reference or more information about D-modules and algebraically solution of PDEs ?.


If ZFC has a transitive model, does it have one of arbitrary size? It is known that the consistency strength of $\sf ZFC+\rm Con(\sf ZFC)$ is greater than that of $\sf ZFC$ itself, but still weaker than asserting that $\sf ZFC$ has a transitive model. Let us denote the axiom "There is a transitive model of $T$" by $\rm St(\sf ZFC)$.
If $M$ is a transitive model of $\sf ZFC$ of size $\kappa$ then we can easily generate transitive models of any smaller [infinite] cardinal by using the downward Löwenheim–Skolem theorem and the Mostowski collapse. Note that we can use the latter because the former guarantees that the model uses the real $\in$ relation, so it is well-founded.
But the upward Löwenheim–Skolem makes use of compactness, which can easily generate models which are not well-founded, and therefore cannot be collapsed to a transitive model.
So while $\rm Con(\sf ZFC)$ can prove that there is a model of $\sf ZFC$ of any cardinality, can $\rm St(\sf ZFC)$ do the same, or do we have a refinement of the consistency axioms in the form of bounding the cardinality of transitive models?
It is tempting to take a countable transitive model and apply forcing and add more and more sets, but forcing doesn't add ordinals and the result is that we cannot increase the size of the model without bound.


A discrete operator begets even/odd polynomials Given a function $f(x)$ define the forward shift operator by $Ef(x)=f(x+1)$ and the discrete derivative $\delta f(x)=(E-1)f(x)=f(x+1)-f(x)$.
Given a partition $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_k)$, where $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_k\geq1$ and $k>0$, define the operator
$$L_{\lambda}=\frac{(E^{\lambda_1}-1)\cdots(E^{\lambda_k}-1)}{\delta}.$$
Let $(x)_n=x(x-1)\cdots(x-n+1)$ be the falling factorial.

Question.  If $\lambda\vdash n$ then is it true $\Phi_n(x)=L_{\lambda}(x)_n$ is either an even or an odd polynomial, with non-negative integer coefficients? It appears to be so.

Example. If $\lambda=(n)$ then $L_{\lambda}(x)_n=\frac{(x+n)_{n+1}-(x)_{n+1}}{n+1}$ indeed satisfies the claim (check!).


Proof that Newton expansion over derivatives has the properties of an integral Let's consider a Newton expansion over consecutive derivatives of a function:
$$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
Can it be proven that such expansion, when converges, has the properties of an antiderivative? What are the conditions for that if it is not always the case?


Valuation Criterion of Properness, (Irreducible) Varieties Greetings,
(I suspect this question has nothing to do with the Valuation Criterion of Properness, but I don't know for sure - feel free to modify my title)
This question arises in section 2.4 of Fulton's book on Toric Varieties - in the proof that $\phi_* : X(\Delta') \to X(\Delta)$ is proper iff $\phi^{-1}(|\Delta|)=|\Delta'|$.
Let's say I give you a variety map $f: X \to Y$.  To prove it's proper I must take any dvr R with fraction field $K$ and any commutative diagram where one path is $Spec(K) \to Spec(R) \to Y$ and another path is $Spec(K) \to X \to Y$ (the map $X \to Y$ is $f$), and tell you why there exists a unique map $Spec(R) \to X$ that makes both 'triangles' commute.  (I don't know how to make pretty diagrams on math overflow, apologies.)
The claim that is made is:  Let $U \subseteq X$ be your favorite open subset.  If $X$ is irreducible, then to prove the above we may assume that $im(Spec(K) \to X) \subseteq U$.
Will someone please explain why this claim holds?
Robert


About direct products of groups Let $G$ be a group. Suppose that $G\simeq G\times G\times G$ (here $\simeq$ is an isomorphism of groups). Is it true that in this case $G\simeq G\times G$? Of course, this question is slightly artifitial. But it is interesting whether someone knows the answer.


A question about a manifold in an $n$-dimensional Alexandrov space with curvature bounded below Suppose $M$ is an $n$-dimensional Alexandrov space with curvature bounded below(maybe with boundary), subspace $A\subset M$ is an $n$-dimensional manifold without boundary. Then whether every point in $A$ is a manifold point in $M$?
Here, a point in $M$ is a manifold point if it has an open neighbourhood in $M$ such that it is homeomorphic to $\mathbb{R}^n$.
Thanks!


Why are they called reductive groups? The reductive groups play a central role in the Langlands correspondence. Why are these groups called reductive? Does this name suggest something conceptual about these groups?


Burnside ring and zeroth G-equivariant stem for finite G Let $G$ be a finite group. The theorem that the Burnside ring $A(G)$ is isomorphic to the zeroth stable stem $\pi^{G}_0(S)$ is usually said to originate from Segal. I search for a reference of a proof this theorem, which is not obscured by tom Diecks generalisation to compact Lie groups. Who knows a good reference? I can not even find the right paper of Segal. Where did the original proof appear?


Does identical scalar curvature imply isometric? Suppose I have a (smooth) manifold with two different metrics $g,h$, where their respective scalar curvatures $R_g, R_h$ are the same. What, if anything, can I say about the relationship between the two. In particular, are they isometric? I care about complete (asymptotically flat) manifolds in particular, but would be interested in any related results.


Can one characterize the category of finite-dimensional vector spaces? Let $K$ be a field. Does the category of finitely generated $K$-modules have a nice characterization, for example as the unique abelian category satisfying a certain simple condition? For example, we know that:

Every short exact sequence is split.
The Euler characteristic of every bounded exact sequence is zero.

Are either of those enough to characterize the category?


Which power series in $\mathbb{Z}_p[[T]]$ are rational functions? Consider the power series ring $\mathbf{Z}_p[[T]]$, where $\mathbf{Z}_p$ denotes the $p$-adic integers. I'll call a function $f(T) \in \mathbf{Z}_p[[T]]$ a rational function if I can write it as:
$$f(T) = \dfrac{g(T)}{h(T)} $$
where $g, h \in \mathbf{Z}_p[[T]]$ are polynomials. (Note that $g(T)/h(T)$ is shorthand for $g(T)$ times the multiplicative inverse of $h(T)$.)
My question is: is it possible to characterize which elements of $\mathbf{Z}_p[[T]]$ are rational functions? If I am given $f(T) \in \mathbf{Z}_p[[T]]$, and I write it as $f(T) = \sum a_nT^n$, can I tell whether $f(T)$ is a rational function just by looking at the coefficients $a_n$?


Second order linear ODE question with boundary conditions I'm working on a stochastic differential equations research problem and I have come across this second order ODE, my gut tells me it has an analytic or close to analytic solution, but I just can't find it. Any help would be appreciated.
$$y''(x)-(A+B\,\sin 2x)\, y'(x)-\lambda y(x)=0\quad$$
for $x\in(0,\pi)$ with boundary condition $y(0)=y(\pi)$.
I substituted $z(2x)=y(x)$ and then  $\theta=2x$, and obtained
$$4z''(\theta) -2\,A(1+\gamma\,\sin \theta)\,z'(\theta) -\lambda z(\theta)=0$$
for $\theta\in(0,2\pi)$ with boundary condition $z(0)=z(2\pi)$, where $\gamma:=B/A$.


Kahlerness of the projectivized cotangent bundle Let $X$ be a smooth not necessarily compact complex manifold which admits a Kahler metric. Is it true that its projectivized cotangent bundle also admits a Kahler metric? If not, are there sufficient conditions for that?


Deformation of stable curve with regular total space Let $k$ be a field, let $X/k$ be a stable curve. Is it always possible to find a deformation $\mathcal{X}/k[[t]]$ such that $\mathcal{X}$ is regular?
(Sorry for the confusion, this is a duplication of one of my previous post....The answer to this question is yes, by Theorem B.2 in Brian Conrad’s Appendix to “Specialization of linear systems from curves to graphs”
by Matthew Baker.)


A criterion for metrizable topological spaces Let $X$ be a topological space. True or false?
$X$ is metrizable if and only if it contains  a sequence of metrizable spaces $\{X_n\}$ with $X=\bigcup X_n$!


In a fibration, can a deformation retraction of the base be lifted to the total space? Given a fibration $p:E \rightarrow B$ and if $A$ is a deformation retract of $B$. Is it true that $p^{-1}(A)$ is a deformation retract of $E$?. If this is not true, can some conditions be imposed on $p$, $E$ or $B$ to make that statement true?. If so, the theorem is still valid for a strong deformation retract?


How to compute the Eigen values of diagonal plus a rank one matrix? I'm trying to find information on the eigenvalues of an n×n matrix $A$ such that
$A=D+J$
Where $D$ is some complex valued diagonal matrix, and $J$ is a rank one matrix, $J = uu^T$.
How to compute the eigen values of $A$ in this case?


Integer Points on the Elliptic Curve $y^2=x^3+17$. I came across the problem "find all integer solutions to $y^2=x^3+17$."
I've tried several things, without any success, and I was hoping that someone could help out.  (Some ideas or a reference for where to find it are both appreciated)
By numerical calculation I have found that the following integer points $(x,y)$ lie on the curve
$(-1,4)$, $(-2,3)$, $(2,5)$, $(4,9)$, $(8,23)$, $(43,282)$, $(52,375)$, $(5234,378661)$
and this is probably all of them.
Thanks


Why/does 'low-dimension' topology end with dimension 4? Put another way, assuming it is somewhat fair to say that we (not I, but those who know better--part of my question is whether my stated assumption is in fact warranted) have in some sense a qualitatively better handle on manifolds up to and including dimension 4 than we do for the case of 5+ dimensions, is it just a coincidence (i.e., 'non-mathematical' factors) that it seems 'natural' for us to think of space-time in terms a four-dimensional manifold?
Edit: apologies for my naivety; I do not mean to imply that my assumption is solid. I probably overestimated the legitimacy of the question I was asking.


Zero set of Hölder function Assume $f$ is a Hölder function on $[0,1]$ such that the set of its zeros $Z(f)$ does not include any interval. Is the measure of $Z(f)$ zero?


Classification of complex irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ Is there a classification of complex irreducible representations of the group $\operatorname{GL}_n(\mathbb{F}_q)$, where $\mathbb{F}_q$ is a finite field with $q$ elements?


Is $p_1p_2\ldots p_n +1$ a prime number for infinitely many $n\in \mathbb{N}$? Let $p_1,p_2,\ldots,p_n,\ldots,$ be the sequence of prime numbers. Are there infinitely many $n\in \mathbb{N}$ such that the natural number $p_1p_2\ldots p_n +1$  is a prime number?


Algorithmically handling the Spin groups in larg(ish) dimensions Question: Is there a reasonably efficient algorithmic representation of $\mathit{Spin}_n$?  By this I mean, a way to store its elements and operate on them (multiply, inverse, maybe compute exponentials from the Lie algebra, and also some kind of distance to the origin; essentially, those operations we can do in $\mathit{SO}_n$) that does not involve an exponential complexity in $n$?
Discussion:

One way to represent elements of $\mathit{Spin}_n$ is by their action on spinors, i.e., on the spin representation; but then they are matrices of size $2^{\lfloor n/2\rfloor}$, which grows exponentially with $n$.  This is a rather unsatisfactory state of affairs for an object which stores merely one more bit of information than an element of $\mathit{SO}_n$ (so the latter can be represented by a matrix of size merely $n\times n$).  The question is whether we can do with less, and how.

If the issue is merely to store an element $g$ of $\mathit{Spin}_n$, we can indeed do better: fixing once and for all a maximal torus $T \subseteq \mathit{Spin}_n$, say the inverse image in $\mathit{Spin}_n$ of the torus of $\mathit{SO}_n$ consisting of rotation matrices which are block diagonals of $2\times 2$ rotation matrices with angles $\theta_1,\ldots,\theta_{\lfloor n/2\rfloor}$, we can describe $g$ by giving a rotation $r \in \mathit{SO}_n$ which¹ conjugates $g$ into $T$ and an element $t = r g r^{-1}$ of $T$, itself represented by $\lfloor n/2\rfloor$ rotation angles $\theta_1,\ldots,\theta_{\lfloor n/2\rfloor}$, each defined mod $4\pi$, but mod adding $2\pi$ to any even number of these angles; so the data of $r$ and the $\theta_i$ defines $g$ (this representation is not unique, but it's not hard to decide when two are equal).  The problem with this representation is that I can't see any way to multiply them; so a specific form of my question might be: how do we multiply elements of $\mathit{Spin}_n$ written in the “standard” form I just described?

A priori we need to take $r \in \mathit{Spin}_n$, but given $r \in \mathit{SO}_n$, the two $\tilde r$ which lift it to $\mathit{Spin}_n$ differ by a central element, so $\tilde r g \tilde r^{-1}$ is the same in either case, and by abuse of notation I call this $r g r^{-1}$.

Note: In the above, $\mathit{Spin}_n$ is implicitly taken to be the compact real form of the Lie group.  I omit any discussion concerning how to represent real numbers in a computer: one possible way to make the question more precise is to say that I am, in fact, really talking about $\mathit{Spin}_n$ over the field of real algebraic numbers (which can then be represented exactly by computer), and/or that the complexity is taken to be a black-box complexity in terms of algebraic operations on the coefficients.  But beyond the real numbers, I'm also interested in comments or answers on how to represent elements of (the various algebraic forms of) $\mathit{Spin}_n$ over any field.


Can one gives an immersion of an exotic sphere ? Can one gives an immersion of an exotic sphere (John Milnor's exotic spheres) S^7 in an S^8(1)? (asking for some examples).Thanks!
Ps:Here "S^(R)"means n-sphere of radius R.


Order of the symplectic group over $\mathbb{Z}/4\mathbb{Z}$ Let $p$ be a prime number and $q$ some power of it. It is well-known that the order of the symplectic group $\text{Sp}_{2g}(\mathbb{F}_q)$ over the finite field $\mathbb{F}_q$ equals $q^{g^2}\prod_{i=1}^g(q^{2i}-1)$.
Is there (a similar) expression for the order of the finite group scheme $\text{Sp}_{2g}(\mathbb{Z}/4\mathbb{Z})$? What if we replace $\mathbb{Z}/4\mathbb{Z}$ by $\mathbb{Z}/n\mathbb{Z}$ for any integer $n$?


On sequences of rational functions Let $\{f_n\}_{n=0}^\infty$ be a sequence of rational functions of the following form: $$ f_n(z) = \sum_{m=1}^\infty \frac{C_{m,n}}{z-m}$$ with $C_{m,n} \in \mathbb{Z}$, $C_{1,n} = 1$, and for each $n \in \mathbb{N}$ there exists some $M(n)$ such that for every $m \geq M(n)$ we have $C_{m,n} = 0$. Suppose that $f_n$ converges uniformly in any open disk of radius $ < 1$ around the origin in the complex plane, and denote by $f \colon D_1(0) \to \mathbb{C}$ the limit.

Is it necessarily possible to extend $f$ to a holomorphic function on
  $\mathbb{C} \setminus S$ where $S \subseteq \mathbb{C}$ is a (possibly
  infinite) set of isolated points.


Uniform Faltings Suppose I give you positive integers $g\geq 2$ and $N.$ Is it always possible to find an absolutely irreducible curve of genus $g$ over $\mathbb{Q}$ which has at least $N$ rational points? For that matter, what if $g=2?$ I assume that the answer is YES, but what do I know?


Simultaneous diophantine approximation Let $r(x)$ be the function $x$ mod $1$, i.e. $x$ minus its floor.
Now let $m$ be a given positive integer, and $c$ a vector in $\mathbb{R}^m$ whose components are linearly independent over $\mathbb{Q}$, where (without loss of generality) the first component is $c_1=1$. Is the set of points $(r(c_2n),\ldots,r(c_mn))$, for $n\in\mathbb{N}$, dense in the $(m-1)$-dimensional unit cube? (It is known that the origin is a limit point, under weaker assumptions.)
If not, is anything known about vectors $c$ for which this is the case?


Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Let $X_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Also are there examples where $\aleph_0<X_G/\mathord\sim<2^{\aleph_0}$ where $N \sim M \iff G/N \cong G/M$?


Definition of Prime Numbers The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I learned that (in order to make prime factorization unique) 1 is not considered to be prime, making 2 the first prime.
Questions:

When and by whom was the characterization of primes as natural numbers that are only divisible by 1 and by themselves without remainder, given?
When and by whom was 1 deprived its prime status?
Does a definition of primes exist that rules out 1, and that does not refer to prime factorization or its uniqueness? (Would “a natural number is a prime number if it is only divisible by 1 and by itself and if it doesn’t divide bigger prime numbers” be acceptable?)


Smallest permutation representation of a finite group? Given a finite group G, I'm interested to know the smallest size of a set X such that G acts faithfully on X. It's easy for abelian groups - decompose into cyclic groups of prime power order and add their sizes. And the non-abelian group of order pq (p, q primes, q = 1 mod p) embeds in the symmetric group of degree q as shown here: www.jstor.org/stable/2306479.
How much is known about this problem in general?


A simple closed curve on a surface How to describe a simple closed curve on an oriented surface of genus g? I know the answer only for the torus. It would be nice to find an article or a book where proof can be found.


Why is Givens better than Householder? When trying to transform only the lower triangular part of a matrix ( transforming only the even elements to 0), why is Givens recommended instead of Householder? Thank you!


A fun game related to knot theory I recently learned the following rather fun game: a group of people is standing up roughly in circle, facing each other. Then participants randomly join hands, in such a way that nobody holds its own hand, and that everybody hold hands with two distinct person.
The result is clearly a link, and the goal of the game is to untangle it, i.e. participants have to move while still holding hands until they can stand up in (possibly several !) circle(s) holding hands with their direct left and right neighboors. Obviously people are allowed to not face the center of the circle (or to be upside down, I guess) so that the first Reidemeister move is allowed, but I guess it's not really important.
Clearly it requires quite some physical and intellectual skills, a lot of collaboration between the players, and a rather long time, which makes it a fun activity, and I recommend trying this at home with friends or at work with fellow mathematicians.
The person who explained the game sweared that it "always works", although it's fairly easy for someone having heard of knot theory to come up with an example where it doesn't. For example the trefoil is easily obtained already with 3 players, but it's also clear that this correspond to a rather particular choice.
In fact, this one time we actually did not manage to untangle the link though we managed to simplify it a lot. Of course even if the link is mathematically trivial it might be physically challenging to actually untangle it, but let's ignore that.
Basically, this game can be formalized as a process which generate links randomly, and I'm curious whether something interesting can be said about it. I must admit that since this question did not actually come up in my research I haven't given much thought about it.. Obvious questions are: can every link be obtained this way ? If not do they correspond to a known family of links ? Is there any way to support the organizer's claim, ie is there a way to estimate the probability of getting a trivial link ?


book about string theory a la Von Neumann Can we summarize string theory (in its actual state) in some principles and fundamental equations like electromagnetism, general relativity, quantum mechanics and classical mechanics ?
I am looking for a book for string theory for mathematicians similar to the book of Von Neumann for QM.


Off-diagonal holomorphic extension of real analytic functions on $\mathbb{C}^n \times\mathbb{C}^n$ I am struggling trying to understand an statement in a paper I am reading:
Let $M$ be a complex manifold of dimension $2n$. Let's consider a function $\xi$: $M$ $\rightarrow$  $\mathbb{C}$  whose real and imaginary parts are real analytic functions. Let $diag$($M$,$M$) be the diagonal of $M\times M$. Then there is a holomorphic extension $\Xi$ of $\xi$ where $\Xi$ is defined in a open neighbourhood of $diag$($M$,$M$).
I already checked some references about real analytic functions but I could not find anything useful about holomorphic extensions of real analytic function in several complex variables. Even in the case $M$ = $\Omega$ a domain of $\mathbb{C}^n$ I could not figure out how to extend $\xi$($z_1$,.....,$z_n$) to a holomorphic function with the double of complex variables $\Xi$($z_1$,..,$z_n$,$\lambda_1$,..,$\lambda_n$) when we are off-diagonal.
Thanks


Isometric imbedding of finite metric space into standards spaces Is it true that any metric space consisting of $n$ points can be isometrically imbedded into $n-1$ dimensional Euclidean space? Hyperbolic space?
(For $n=3$ this is true.) If not, what are necessary/sufficient conditions?
The case $n=4$ is the first unknown to me case.
What happens with imbeddings into the unit sphere, where certainly one needs some extra conditions on the metric space, like perimeter of each triangle is at most $2\pi$.


norm of operator between matrix algebras equipped with trace norm Let $M_i$ stands
for the algebra of $d_i\times d_i$ matrices with $\|T\|=d_i
\|T\|_1=d_i (trace{(T^\ast T)}^{\frac{1}{2}})$, and $M_{ij}$
stands for the algebra of $d_i d_j\times d_i d_j$ matrices with
$\|T\|=d_i d_j \|T\|_1$.  The norm-decreasing maps
$$\rho_{i,j}:M_i\hat{\otimes} M_j \rightarrow M_{ij}; $$
$\rho_{i,j}(A\otimes B)=\left(\begin{matrix}A_{11}&\ldots&A_{1i}\\\vdots&\ddots&\vdots\\A_{i1}&\ldots&A_{ii}\end{matrix}\right) \otimes \left(\begin{matrix}B_{11}&\ldots&B_{1j}\\\vdots&\ddots&\vdots\\B_{j1}&\ldots&B_{jj}\end{matrix}\right) = \left(\begin{matrix}A_{11}B&\ldots&A_{1i}B\\\vdots&\ddots&\vdots\\A_{i1}B&\ldots&A_{ii}B\end{matrix}\right)$
gives a bijection between $M_i\hat{\otimes} M_j$ and $M_{ij}$. What's the norm of $\rho^{-1}_{i,j}$?


Geometric meaning of Ricci flow What is the geometric meaning, for a metric in function of the time that is a solution of the Ricci flow ($g'(t)=-2Ric(t)$), compared to one that is not?
EXPLANATION
I'm interested to understand, being that not all metrics satisfy the equation, $g'(t)=-2Ric(t)$, what differences there are, from the geometrical point of view, between a metric that is a solution of the Ricci flow, and one that is not.
Because, for example, there may be a family of metrics within which only one is the flow solution and the others are not solution...which "quality" (pass me the term) has from the geometrical point of view this metric  that the others do not have?


Probability of positive definiteness of a random matrix Given an $n \times n$ symmetric random matrix whose entries have distribution $N(0,1)$, how to calculate the probability of positive definiteness of this matrix?


Bass' stable range condition for principal ideal domains Do you know a characterization of commutative rings $R$ whose every prime factor ring of $R$ is a principal ideal domain?


Infinite graphs with large degree but no perfect matching Is there an example of an infinite connected, simple, undirected graph $G = (V,E)$ such that every vertex has $|V|$ neighbors, but $G$ does not have a perfect matching (that is, a set $M\subseteq E$ of pairwise disjoint edges such that $\bigcup M = V$)?


Products of Conjugacy Classes in S_n The conjugacy classes of the permutation group $S_n$ are indexed by partitions like $[6]$ and $[2,2,2] = [2^3]$ describing the cycle type.  What happens when you take products of two whole conjugacy classes?  I saw in a paper,
$$[6][2^3] = 6[3,1^3] + 8[2^2,1^2]+5[5,1]+4[4,2]+3[3^2]$$
Which I take to mean if you multiply a 6-cycle (abcdef) and a product of disjoint 3-cycles (pq)(rs)(tv), you can get

a three-cycle (abc),
two two-cycles (ab)(cd),
a five-cycles (abcde),
a four-cycles and a two-cycle (abcd)(ef),
two three cycles (abc)(def)

With certain multiplicities. Is it predictable what kinds of conjugacy classes you get?  Is there an interpretation of this as the intersection cohomology of some moduli space?


Applications of symplectic geometry to classical mechanics It is claimed that classical mechanics motivates introduction of symplectic manifolds.  This is due to the theorem that the Hamiltonian flow preserves the symplectic form on the phase space.

I am wondering whether symplectic geometry has applications to classical mechanics.  Was this connection useful for classical mechanics? Were methods of symplectic geometry relevant for it via, say, the above theorem?


Math paper authors' order It seems in writing math papers collaborators put their names in the alphabetical order of their last name. Is this a universal accepted norm? I could not find a place putting this down formally.


Connected but no path-connected components Is there an infinite Borel subset of plane which is connected but whose only path connected components are singletons?
I know that a Bernstein set is a non-Borel example of such a set. Thanks!


Torsion in homology or fundamental group of subsets of Euclidean 3-space Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not torsion-free?
Context: The analogous question has a negative answer in dimension 2.  This is a theorem of Eda's (1998).  In dimension 4 and higher, the answer is positive as the real projective plane embeds.  If the subset of 3-space has a regular neighbourhood with a smooth boundary, a little 3-manifold theory says the fundamental group and homology groups are torsion-free.
edit: Due to Autumn Kent's comment and the ensuing discussion, torsion in the homology has been ruled out provided the subset of $\mathbb R^3$ is compact and has the homotopy-type of a CW-complex (more precisely, if Cech and singular cohomologies agree).


Asymptotics on number of bounded prime gaps It's been over 2 years since the groundbreaking paper by Yitang Zhang in which he has shown that infinitely many prime pairs are by some constant $H$, with $H\leq 70000000$. Over the course of the following year the bound on $H$ has been greatly reduced to $H\leq 246$. However, I have never seen a result which would say anything more about number of such pairs, and I probably wouldn't believe if someone said to me that the methods used don't provide any estimates.
My question here is:

What is the best known lower bound on the number of prime pairs separated by less than (or equal) a given number $n$? I am particularily interested in $n=246$ (best for which we know there is infinitely many) and some big $n$, like original $70000000$ (for which we might know stronger bounds).

Thanks in advance.


Smooth functions with zeros of infinite order on a closed set It follows from Whitney extension theorem that for every closed set $ C \subseteq \mathbb{R}^n $ and for every $ k \geq 1 $ there exists a function $ f \in C^k(\mathbb{R}^n) $ such that $ C = \{x : f(x)=0 \} $ and $ D^if(x) =0 $ for every $ x \in C $ and $ i = 1, \ldots , k $.
Is it possible to replace $ k $ with $ \infty $ in the statement above?


Given a specific function $f$, how to compute the left-inverse of $f$ in the sense of $\approx$? For a non-negative function $\varphi$ defined on $[0,\infty)$, the left-inverse $\varphi^{-1}$ of $\varphi$ is defined by setting, $\forall t\geq 0$,
$$\varphi^{-1}(t):=\inf\{u\geq0:\varphi(u)\geq t\}.$$
For two functions $\varphi$ and $\psi$, we say that $\varphi\approx\psi$, if there exist constants $c_1,c_2>0$, such that $\forall t\geq0$,
$$c_1\varphi(t)\leq\psi(t)\leq c_2\varphi(t).$$
Let $p\geq1$, define $f(t):=t^p\ln(e+t)$, $t\geq0$ (this is an example appearing in the paper about equations). How to compute the $f^{-1}$ in the sense of $\approx$? I guess its
answer is $\forall t\geq0$, $$f^{-1}(t)\approx\left[\frac{t}{\ln(e+t)}\right]^{1/p}\quad \text{or}\quad f^{-1}(t)\approx \min\left\{t^{1/p},\left[\frac{t}{\ln(e+t)}\right]^{1/p}\right\},$$ but I do not know how to compute $f^{-1}$ in the sense of $\approx$, I need some help.


Monotonicity of Loewner ellipsoid? Given two $0$-symmetric convex bodies $K \subset L \subset \mathbb{R}^n$, is it true that the Loewner ellipsoid of $K$ is contained in the Loewner ellipsoid of $L$?
I have just finished proving a lemma stating that the Loewner ellipsoid depends continuously on parameters and the proof is a bit more elaborate than I first expected.  I suddenly realized that I was unconsciously assuming that the Loewner ellipsoid is not monotone (otherwise the lemma would be trivially true), but that I did not have a ready example showing that this was the case.
I profit to ask a second question: is there a reference for the fact that the Loewner ellipsoids of a continuous family of convex bodies form a continuous family?
Rephrase in terms of normed or Finsler bundles and Euclidean structures if you want to be very rigorous.
My proof of this fact involves "looking under the hood" at the proof of uniqueness of the Loewner ellipsoid and using Berge's maximum theorem for set-valued maps. It's natural (after all this is just a problem in mathematical programming), but I was expecting a triviality.


The $2\pi$ in the definition of the Fourier transform There are several conventions for the definition of the Fourier transform on the real line.
1 . No $2\pi$. Fourier (with cosine/sine), Hörmander, Katznelson, Folland.
$ \int_{\bf R} f(x) e^{-ix\xi} \, dx$
2 . $2\pi$ in the exponent. L. Schwartz, Trèves
$\int_{\bf R} f(x) e^{-2i\pi x\xi} \, dx$
3 . $2\pi$ square-rooted in front. Rudin.
${1\over \sqrt{2\pi}} \int_{\bf R} f(x) e^{-ix\xi} \, dx$
I would like to know what are the mathematical reasons to use one convention over the others?
Any historical comment on the genesis of these conventions is welcome.
Who introduced conventions 2 and 3? Are they specific to a given context?
From the book of L. Schwartz, I can see that the second convention allows for a perfect parallel in formulas concerning Fourier transforms and Fourier series. The first convention does not make the Fourier transform an isometry, but in Fourier's memoir the key formula is the inversion formula,  I don't think that he discussed what is now known as the Plancherel formula. Regarding the second convention, Katznelson warns about the possibility of increased confusion between the domains of definition of a fonction and its transform.


Integers mod $p$ "together" algebraically closed Let $f(x)$ be a nonconstant polynomial over $\mathbb Z$.
Must $f$ have a zero in $\mathbb F_p$ for some prime number $p$?
More generally, let $f_1,\dots,f_k$ be such polynomials, must there exist a $p$ such that each $f_i$ has a zero in $\mathbb F_p$?


introductory book on spectral sequences I have studied some basic homological algebra. But I can't send to get started on spectral sequences. I find Weibel and McCleary hard to understand.
Are there books or web resources that serve as good first introductions to spectral sequences? Thank you in advance!


Simplicial sets of categories as models for $(\infty,1)$-categories Disclaimer: I am not knowledgeable at all about the subject of this question, so my apologies if I say something incorrect or too imprecise.

In my understanding, there are several models for $(\infty,1)$-categories: Joyal-Lurie's quasi-categories, simplicially enriched categories, Segal catgories, etc. . They are all, for the most part, models for the same mathematical objects and can be used interchangeably to develop higher category theory.
Here is something that sounds like it might be a model. Assume all categories below are as small as necessary.

Definition 1: A (special) simplicial set of categories $C_\bullet$ is a functor $C:\Delta^{\text{op}} \to \bf{Cat}$, denoted by $[k] \mapsto C_{[k]}$, satisfying the following pair of properties.

The map $\text{Mor}(C_\bullet) \to \text{Obj}(C_\bullet) \times \text{Obj}(C_\bullet)$ given by sending a morphism $f:x \to y$ to the pair $(x,y)$ is a map of simplicial sets. (Not sure if this is automatic)
The simplicial set $\text{Map}(x,y)$, defined as the union over all $k$ of the morphisms $f_{[k]}$ whose image under any degeneration map $C_{[k]} \to C_{[0]}$ is a morphism $x \to y$, is a Kan complex.

My question is just the following one.

Question: Does Definition 1 (or a corrected/similar definition) provide a known model for $(\infty,1)$-categories? If so, where can I find a reference?

It would be great if this reference related this model to one of the more popular ones, such as quasi-categories.


A description of the fundamental class of the group cohomology of the fundamental group of an orientable surface of genus $g$ I asked this question on Mathematics Stack Exchange some months ago but I got no answer.
Suppose one has an orientable compact surface $S$ of genus $g\ge 2$, $x\in S$, and $G=\pi(S,x)$ the fundamental group. There is a well-known description of the group with generators and relations as $$G=\langle a_1,b_1,\dots, a_g,b_g \ \mid \ [a_1,b_1]\cdots[a_g,b_g]=1\rangle$$
where $[a,b]=aba^{-1}b^{-1}$ as usual.
Now, it is know that the homology $H_2(S,\mathbb{Z})\cong \mathbb{Z}$ and that $$H_2(G,\mathbb{Z})\cong H_2(S,\mathbb{Z})$$
using group cohomology.
On the other hand the group cohomology can be descrived using the so called bar resolution, whose elements  $z$ are in $\mathbb{Z}[G]\otimes \mathbb{Z}[G]$ such that $\delta_2(z)=0$, where $\delta_2(g\otimes h)= g+h-gh$, where $\delta_2$ goes to $\mathbb{Z}[G]$ (and then quotienting by the image of a $\delta_3$ analogous to $\delta_2$).
My question is: what is the generator of $H_2(G,\mathbb{Z})$ expressed as an element in $\mathbb{Z}[G]\otimes \mathbb{Z}[G]$ in terms of the generators $a_i$, $b_i$, $i=1,\dots,g$ given above?
I am studying group cohomology using K.S. Brown book "Cohomology of groups".


Proof of a folkloric result about PI-algebras I am not not an specialist in PI-algebras, but I can say I have a rather good understanding on the subject.
It is, of course, interesting to discover if an algebra $A$ is a PI-algebra. But it is also an interesting problemto determine when a certain algebra cannot satisfy any polynomial identity!
For instance, the Weyl algebra $A_n$ is simple, hence primitive. If it were a PI-algebra, by Kaplansky Theorem, it would be finite dimensional over its center. However, we know that $A_n$ is an infinite dimension vector space, and its center restrict to the scalars.
Fix a base field of characteristic 0., and let $\mathfrak{g}$ be a finite dimensional Lie algebra. I result I've heard a number of times is that the enveloping algebra can be a PI-algebra if, and only if $\mathfrak{g}$ is abelian - in which case $U(\mathfrak{g})$ is just the polynomial algebra.
Does someone have a reference for this? Or can indicate a sketch of a proof? Or, provide a counter-example to show that this folklric fact is, in fact, false?


Euler systems over abelian number fields Im confused with the following statement:
Coleman’s conjecture concerning circular distribution imply that Euler systems over abelian number fields arise in “an elementary” way from the theory of cyclotomic units.
What is that “elementary way”?
I would appreciate if someone could help.


Explicit calculation algorithm for distance function I study differential geometry. Although there is a lot of study on the local theory, a global description lacks some explicit explanations. I mean, the study of surfaces describes curves, tangent plane, covariant derivative, the geodesic equation. However, I was not able to find a systematic manner to calculate the distance function rather than solve the geodesic equation with begin and end points and integrate its length with extreme as control points. Can you see any manner to find the distance function on an algebraic connected and complete (geodesic may pass everywhere) surface of type $z = f(x, y)$?


Inverse problem for zeta functions of curves over finite fields We know, thanks to A. Weil, that such a function is rational, and the numerator has all of its zeros on the circle $z \overline{z} = q,$ where $q$ is the order of the field. The question is: can every polynomial in $\mathbb{Z}[x]$ with the zeros on the circle arise in this way? Is there some obvious and/or conjectural obstruction?


Asymptotics of the right singular vectors as the number of rows diverge Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \Sigma_m V_m^T$ as the singular value decomposition, where $V_m \in \mathbb{R}^{n \times n}$ are the right singular vectors.
What can be said about $V_m$ as $m \to \infty$? Is anyone aware of literature or techniques discussing this?


How to construct a continuous finite additive measure on the natural numbers I want to find some condition to construct a continuous finitely additive measure on the natural numbers, i.e. $f:P(\mathbb{N})\rightarrow [0,1]$ such that $f(\{n\})=0$, and $f$ is an additive measure.
I know in ZFC we can use an ultrafilter $U$ and define $f$ by $f(A)=1\Leftrightarrow A\in U$, but this is too trival.
How about ZF? or some other condition? like large cardinal.


Textbook request for class field theory I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier analysis on local fields, Fourier transform, multiplicative characters, and local zeta functions.
I know the book "Algebraic Number Theory" by Cassels and Fröhlich. I studied its chapter on global fields, and I know the definition and first few properties of adeles and ideles. Then I tried to read the chapter on Tate's thesis but I find it very difficult. Actually, I am not so good in analysis and I don't have a clear conception of the Haar measure.
It will be very helpful if someone suggests textbooks appropriate as an introduction to the subject of class field theory and where the above topics are well explained with all the details.


Is there a Hodge structure for smooth proper varieties over $\mathbb{C_p}$? For smooth proper varieties over $\mathbb{Q_p}$, we have several comparison theorems in p-adic Hodge theory, in particular a p-adic Hodge structure.
Now for $\mathbb{C_p}$, is there any such results for smooth proper varieties over $\mathbb{C_p}$?
If the answer is negative, then what is the reason?


Does "Every infinite set is splittable" imply $\mathsf{AC}$? We say an infinite set $X$ is splittable if there are $X_1, X_2\subseteq X$ with $X_1\cap X_2 = \emptyset$, $X_1\cup X_2 = X$ and there are bijections $\varphi:X_1\to X_2$ and $\psi:X_1\to X$.
Does the statement "Every infinite set is splittable" imply $\mathsf{AC}$?


Partial Sum of the Binomial Theorem The binomial theorem states $\sum_{k=0}^nC_n^kr^k=(1+r)^n$. I am interested in the function \begin{equation}
\sum_{k=0}^mC_n^kr^k, \quad m<n
\end{equation}
for fixed $n$ and $r$, and both $m$ and $n$ are integers. Are there any notable properties for this function? Any literature references?
In particular, do any good closed-form approximations exist for this partial sum of the binomial theorem?


Efficient algorithm for solving a convex quadratic program Let $A \in \mathbb{R}^{n \times m}$ and $b \in \mathbb{R}^n$. Suppose $m \ll n$. How to solve this quadratic program efficiently?
$$\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$


On primes of the form 2^n + k For any odd number $k$, can we always find an $n$ so that $2^n > k$ and $2^n + k$ is a prime number?
Any comment or remark is welcomed!


Projectives and Injectives in Functor Categories Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories?
Here is a precise question. Let $C$ be a small category, whose total morphism set has cardinality $\alpha$. Let $A$ be an abelian category with enough projectives (dually, injectives) and coproducts (products) up to cardinality $\alpha$. The functor category $A^C$ is clearly abelian.
Question Does $A^C$ have enough projectives (dually, injectives)?
A reference would be ideal but an explanation would be very welcome too. I am getting fricasseeed this morning by this question: surely, the issue should be whether one can cover an object in $A$ by a projective object functorially but I have no idea how to spell it out...


Fitting $\frac1n\times\frac1{n+1}$ rectangles into the unit square Consider the set of rectangles $r_n | n \in \Bbb N$ such that rectangle $r_n$ has shape $\frac1n\times\frac1{n+1}$.  The total area composed by one copy of each $r_n$ as $n$ ranges from $1$ to infinity is $1$.  Call that set of rectangles $S$.
In Concrete Mathematics it is speculated that you can fit all the rectangles in $S$ into a unit square, without overlap of any interior area. It is also speculated (by the second author) that they can't all be fit, and it is presented as a research problem.
I have a computer-based approach which could potentially decide the issue if you can't fit them (but won't provide a proof if they can fit).  In particular, it seems to indicate that if you add the restriction that the rectangles (ordered by area) are forced to alternate long-side-vertical and long-side-horizontal, you come to an impasse.
But since the volume I saw it in is 25 years old, I'm wondering, before implementing and error-checking and optimizing the method for arbitrary placement, whether this question has, in the intervening years, been resolved.
So my question is

Is it known whether $S$ can be packed into a unit square, without overlap?


Is every square root of an integer a linear combination of cosines of $\pi$-rational angles? For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of the cosines of rational multiples of $\pi$?
Products of linear combinations of cosines of rational multiples of $\pi$ are themselves such linear combinations, so it only needs to be true of primes. But I do not know, for example, a representation of $\sqrt 7$ in this form.


Extension of base field for modules of groups and cohomology Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V_K) \cong K \otimes_k H^n(G,V)$?
If no, what about with some extra assumptions (for example $G$ is finite and $k$ is finite)?
My main motivation is in computing some cohomology/ext groups for $KG$-modules, where $K$ is algebraically closed and $G$ is finite. When $k$ is a finite field, there are ways to compute these with GAP for example.


How big can a commutative algebra of $n \times n$ matrices be? What's the maximum possible dimension of a commutative subalgebra of the algebra of $n \times n$ complex matrices?
There's a theorem of Burnside saying that any commutative subalgebra of a matrix algebra can be upper triangularized.  My friend Bruce Smith pointed out that for $n$ even we can get a commutative subalgebra of dimension $(\frac{n}{2})^2 + 1$.  For $n = 4$ its elements look like this:
$$\left( \begin{array}{cccc}
a & 0 & b & c \\
0 & a & d & e \\
0 & 0 & a & 0 \\
0 & 0 & 0 & a
\end{array}\right)$$
and the same trick works in any even dimension.  For $n$ odd we can get dimension $\frac{(n-1)(n+1)}{2} + 1$ using a similar idea, with a rectangle rather than a square of nonzero entries in the upper right corner.
Can one do better?  Someone must have figured this out.


Inducing a Monoidal Structure using an Equivalence of Categories Given an equivalence of categories  $C \equiv D$, such that $C$ has a monoidal structure, is it clear that we can use the equivalence to induce a monoidal structure on $D$. Is there a standard reference for this?


What is the geometric meaning of homology group and cohomology group? I am reading some algebraic topological book, where they said that the p-th homology group tells us how many p-dimensional holes inside the set. How should I understand this? I know that in the planar case, the 1-th order homology group tells us exactly how many 1-holes the set has, but I do not have a good feeling about the p-holes, could anybody explain this by some very simple examples?
Also they said the cohomology groups are dual to homology group, is there also a good geometric meaning of cohomology group? I heard some duality result, like the Alexander duality between homology group and cohomology group, but I do not know the real meanings.


tr(ab) = tr(ba)? It is well known that given two Hilbert-Schmidt operators $a$ and $b$ on a Hilbert space $H$, their product is trace class and $tr(ab)=tr(ba)$. A similar result holds for $a$ bounded and $b$ trace class.
The following attractive statement, however, is false:
Non-theorem:
Let $a$ and $b$ be bounded operators on $H$. If $ab$ is trace class , then $ba$ is trace class and $tr(ab)=tr(ba)$.
The counterexample is $a=\pmatrix{0&0&0\\0&0&1\\0&0&0}\otimes 1_{\ell^2(\mathbb N)}$, $b=\pmatrix{0&1&0\\0&0&0\\0&0&0}\otimes 1_{\ell^2(\mathbb N)}$.
I'm guessing that the following is also false, but I can't find a counterexample:
Non-theorem?:
Let $a$ and $b$ be two bounded operators on $H$. If $ab$ and $ba$ are trace class, then  $tr(ab)=tr(ba)$.


Convoluted Cantor-like measure which has a continuous component Let $\mu$ be a finite measure on $\mathbb R$ which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable
$$
 \sum_{k\ge 1}3^{-k}X_k
$$
where the $X_k$ are IID Bernoulli (0-1 valued) random variables, because $\mu$ only charges numbers which ternary expansion contains only $0$s and $1$s.
Is it always possible to find $n\ge 1$ such that the $n-$th convolution product of $\mu$, denoted by $\mu^{\otimes n}$, has a non-zero component with respect to Lebesgue measure? With the example above, the answer is yes with $n=2$ because if one takes $X_n'$ independent copies of the $X_n$, then $\mu^{\otimes 2}$ is the law of
$$
\sum_{k\ge 1}3^{-k}(X_k+X_k'),
$$
which charges all ternary expansions.
On the other hand, my experience with ugly measures is that the answer should be no, but I can't find a counter-example...


Diffeomorphism of 3-manifolds Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder whether the surgeons' key problem has been solved. Is every simple homotopy equivalence between smooth, closed 3-manifolds homotopic to a diffeomorphism?
In related vein, it follows from J.H.C. Whitehead's theorem that a map of closed, connected smooth 3-manifolds is a homotopy equivalence if it has degree $\pm 1$ and induces an isomorphism on $\pi_1$. Is there a reasonable criterion for such a homotopy equivalence to be simple? One could, for instance, ask about maps that preserve abelian torsion invariants (e.g. Turaev's).


How to compute direct images for a blowing up? Let $X$ be a smooth algebraic variety, $Z\subset X$ a smooth closed subvariety, and
$\pi:\tilde{X}\to X$ the blowing up of $X$ along $Z$. Let $E\subset\tilde{X}$ be the exceptional divisor, and $n$ an integer. Are there "formulas" for the sheaves $R^i\pi_*(\mathcal{O}_{\tilde{X}}(nE))$?
Are there simple formulas?


When Max(R) is Hausdorff space? Let $R$ be reduce commutative ring with identity (a commutative ring such that $a^n$=0 ($a\in R$) implise $a=0$) and $Max(R)$ be the set of all maximal ideals of $R$. The hull-kernel (or Zariski topology) topology on $Max(R)$ is the topology obtained by taking the collection of sets $U(a) =\{m\in Max(R) : a\not\in m\}$ or arbitrary $a\in R$ as a base for the open sets. when is $Max(R)$ with hull-kernel topology Hausdorff space?


How to prove that $\phi: \;\mathrm Mod(S_g)\to \mathrm Sp(2g, \mathbb{Z})$ is an epimorphism? How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an epimorphism? My idea was to work with generators, but I was not able to prove it this way.
I would love to get detailed answers in order to understand this better.


Can there exist different smooth, proper schemes over the p-adics with the same generic fiber? Can there exist smooth, proper $X_1,X_2/\mathbb Z_p$ such that their generic fibers are isomorphic but their reductions mod $p$ are not? Are there examples if we insist that the special fibers are distinct even over $\overline{\mathbb F}_p$?
An obstruction is that the two reductions should have the same etale cohomology (by proper base change) and tame geometric etale fundamental group (by Grothendieck's comparison theorem).


How does one show the existence of discrete and complementary series for SL(2,R)? In his book on $\mathrm{SL}(2,\mathbb{R})$, Lang shows that any nontrivial irreducible unitary representation of this group is infinitesimally isomorphic to an irreducible admissible subrepresentation of an induced representation (Theorem 8, p. 123). This implies, regarding the classification of irreducible unitary representations of $\mathrm{SL}(2,\mathbb{R})$, that the so-called principal and mock discrete series exist since the related induced representations are unitary. However, this is not the case with discrete and complementary series. Is there an easy way to show the existence of those two types of representations? Lang seems to suggest so, at least for the complementary series, as he mentions the possibility of a unitarization by completing the space of $K$-finite vectors with respect to a certain scalar product (p. 123). I do not understand how this works. (One gets a different space after completion, thereby losing the original action of the group. What is the new action then?)
To rephrase: given that there exists a (nonunitary) irreducible admissible representation of $\mathrm{SL}(2,\mathbb{R})$ in a certain infinitesimal equivalence class like, say, discrete series of lowest weight $2$, can one find in a more or less straightforward way an irreducible unitary representation belonging to the same class?


Infinite tensor product of Hilbert spaces Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading the article as it mostly involved elementary tensors, but I am curious to know the exact construction of such a tensor product and the inner product that is given on the infinite tensor product. It would be of great help to me if anyone gives a reference or an explanation of such a construction. Thanks in advance!


Skew fields inside quaternion division algebras Suppose that $Q$ is a quaternion division algebra with center $k$, where $k$ is an arbitrary commutative field (let's say with $\operatorname{char}(k) \neq 2$ if necessary). Assume that $D$ is an arbitrary skew field (which a priori has nothing to do with $Q$ nor with the base field $k$), and assume that there is an injective ring homomorphism $\varphi \colon D \hookrightarrow Q$.

Is it true that $D$ is either a commutative field or a quaternion division algebra again?

My first guess was that this should be obviously true, but failing to see an obvious argument, I wonder whether it's true at all...


A stronger version of Fermat's last theorem Motivated by Fermat's last theorem, one may wonder the following conjecture is true or not.
The equation $x_1^m+\cdots+x_n^m=1$ has nonzero rational solutions iff $n\geq m$.
Here a nonzero rational solution means nonzero $y_1,\cdots,y_n\in\mathbb{Q}$ satisfying the above equation.
When $n=2$, the above conjecture is confirmed by Fermat's last theorem.


Conformal maps in higher dimensions In dimension 2 we know by the Riemann mapping theorem that any simply connected domain ( $\neq \mathbb{R}^{2}$) can be mapped bijectively to the unit disk with a function that preserves angles between curves, ie is conformal.
I have read the claim that conformal maps in higher dimensions are pretty boring but does anyone know a proof or even a intuitive argument that conformal maps in higher dimensions are trivial?


Introductions to modern algebraic geometry I was wondering if people have any feelings on the pros and cons of various introductions to algebraic geometry (among the contenders would be Ravi Vakil's notes, Hartshorne, Mumford, Harris, Eisenbud-Harris, and I am sure many others I am not aware of).


Evaluation of an interesting Integral Supposedly the answer is 1 but I have no idea how to evaluate this thing analytically.
$$f(n) = \frac{2}{\pi} \int_{0}^{\infty} 2\cos(x) \cdot \frac{\sin(x)}{x} \cdot \frac{\sin(x/3)}{x/3} \cdot \cdots \cdot \frac{\sin(x/(2n+1))}{x/(2n+1)} dx$$
Any help would be appreciated.


Is there a manifold with fundamental group $\mathbb{Q}$? It is known that the fundamental group of a locally path connected, path connected compact metric space is finitely presented or uncountable. Furthermore the fundamental group of every manifold is countable so the fundamental group of every compact manifold must be finitely presented.

Q1: Is there a non-compact manifold whose fundamental group is isomorphic to $\mathbb{Q}$, the group of rational numbers?

Q2: Or more general, for given countable group $G$, is there a manifold whose fundamental group is isomorphic to $G$?


Starting PhD at the age of 25 For over an year I have otherwise been pretty active on MathOverflow but for this question I would like to remain anonymous.
I would be starting my grad school in Fall 2011 in the US while in the 25th year of my life. I will be joining a grad school which is ranked by most lists within their top 10.
I have been working and hope to continue to work in areas in theoretical physics which have a strong interface with mathematics, especially geometry.
I understand that most people start their PhD at the age of 22 (or even below!).
I would like to know how does it affect my career, now that I will be getting my PhD around the age of 30. I am very worried and extremely depressed that this is possibly too late to start a PhD. I guess most scientists become faculty by the age of 30 when I would be getting my PhD!
I would be happy to get any feedback/advice about starting PhD so late in life.

Also can one get a PhD in theoretical physics/mathematics in less than 5 years?
I have made this question community wiki.


Elementary Luroth theorem proof? Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such methods as algebraic extensions and complex analysis. Is there any way to prove this fact with only elementary methods?


Higher homotopy groups and ramified covering maps It  is  known  in  elementary  algebraic  topology that a  covering  map  induces  an isomorphism  of  higher  homotopy  groups.
Is  there  any  relation  of  the  higher  homotopy  groups  of  the  total  space  of  a ramified  covering in the  sense  of  topology (a map which  is  a  covering  map  outside  of  a  nowhwere dense  set) and  its  base space? I  am  interested  in  base  spaces  which  are   manifolds  with mild singularities.


Relationship between fragments of the axiom of choice and the dependent choice principles The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and  $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there is a function $f:\kappa\to S$ such that for every $\alpha<\kappa$, $f\upharpoonright\alpha R f(\alpha)$. The axiom of choice fragment ${\rm AC}_\kappa$ states that every family of size $\kappa$ has a choice function. There are several classical theorems (see Jech's "Axiom of Choice", chapter 8) concerning the relationship between the dependent choice principles and fragments of the axiom of choice.
Theorem 1: Over  ${\rm ZF}$, ${\rm AC}$ is equivalent to $\forall\kappa\,{\rm DC}_\kappa$.
Theorem 2: Over  ${\rm ZF}$, $\forall \kappa\,{\rm AC}_\kappa$ implies ${\rm DC}_\omega$.
Theorem 3: It is consistent with  ${\rm ZF}$ that $\forall \kappa\,{\rm AC_\kappa}$ holds but ${\rm DC_{\omega_1}}$ fails (theorem 8.9).
Theorem 4: It is consistent with  ${\rm ZF}$ that ${\rm AC}_\kappa$ holds for some cardinal $\kappa>>\omega$ but ${\rm DC}_\omega$ fails (theorem 8.12).
Jech proves theorems 3 and 4 using permutation models (and then discusses how to obtain ${\rm ZF}$-models with the same properties). But I am wondering whether there are direct symmetric model constructions for these two results. Either a reference for the arguments or the arguments themselves would be appreciated.


Eigendecomposition of a summation of matrices Can anyone tell me if there's a way to relate the eigendecomposition of the result of a summation of matrices with the eigendecomposition of those matrices?
More specifically:
If I have a matrix $K = \sum\limits_{m=1}^M a_m K_m$, $a_m \in R$.
How can I relate its eigendecomposition, $K = V \Lambda T^T$, with the eigendecomposition of the matrices in the summation, i.e. $K_m = V_m \Lambda_m V_m^T$?


Injectivity of cardinality of power set For two sets $A$ and $B$. Suppose $|2^A| = |2^B|$ (cardinality of power sets of $A$ and $B$). Does this imply $|A|=|B|$?
(It is easy to see that $|A|=|B|$ if we assume generalized continuum hypothesis. Do we have the same result without it?)


Direct proof of injectivity of $L_\infty$ I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result.

$L_\infty$ as commutative unital $C^*$ algebra is isometrically isomorphic to $C(K)$ for some compact $K$.
Every $C(K)$ space which is a dual space is isometrically injective.

However the proof for $\ell_\infty$ is quite simple. Let $i:X\to Z$ be isometric embedding and $T:X\to \ell_\infty$ be a bounded operator. Let $e_n:\ell_\infty\to\mathbb{C}:x\mapsto x(n)$ be coordiante functionals, then consider bounded functionals $f_n:\mathrm{Im}(i)\to \mathbb{C}:z\mapsto e_n(T(i^{-1}(z)))$ extend them by Hahn-Banach theorem to get functionals $g_n:Z\to\mathbb{C}$. The desired operator is $
\hat{T}:Z\to\ell_\infty: z\mapsto(g_1(z), g_2(z),\ldots)$
My question:
Does there exist a direct proof that $L_\infty$ is isometrically injective, a proof similar to the arguments used for the $\ell_\infty$ space? The problem in mimicking proof for $\ell_\infty$ arose from the fact that I can't find family of functionals $(E_n:n\in\mathbb{N})\subset L_\infty^*$ similar to coordinate functionals $(e_n:n\in\mathbb{N})\subset\ell_\infty^*$.
Thank you.


Euler's Totient Function Let $\phi(\cdot)$ be the Euler totient function, and let $n=p_1^{k_1}\cdots p_s^{k_s}$ be the prime factorization of $n\in \mathbb{N}$. The well-known Euler's product formula states that $\phi(n)=n(1-\frac{1}{p_1})\cdots(1-\frac{1}{p_s})$. For some fixed positive integer $a\leq n$, let $\phi(n,a)$ denote the number of positive integers which are less than $a$ but coprime to $n$. My question is that whether there is some formula for computing $\phi(n,a)$.


Computing $\prod_p(\frac{p^2-1}{p^2+1})$ without the zeta function? We see that $$\frac{2}{5}=\frac{36}{90}=\frac{6^2}{90}=\frac{\zeta(4)}{\zeta(2)^2}=\prod_p\frac{(1-\frac{1}{p^2})^2}{(1-\frac{1}{p^4})}=\prod_p \left(\frac{(p^2-1)^2}{(p^2+1)(p^2-1)}\right)=\prod_p\left(\frac{p^2-1}{p^2+1}\right)$$
$$\implies \prod_p \left(\frac{p^2-1}{p^2+1}\right)=\frac{2}{5},$$
But is this the only way to compute this infinite product over primes? It seems like such a simple product, one that could be calculated without the zeta function.
Note that $\prod_p(\frac{p^2-1}{p^2+1})$ also admits the factorization $\prod_p(\frac{p-1}{p-i})\prod_p(\frac{p+1}{p+i})$. Also notice that numerically it is quite obvious that the product is convergent to $\frac{2}{5}$: $\prod_p(\frac{p^2-1}{p^2+1})=\frac{3}{5} \cdot \frac{8}{10} \cdot \frac{24}{26} \cdot \frac{48}{50} \cdots$.


Congruence modulo 4 for a generating function leads to perfect squares? Consider the number of integer partitions $p(n)$ of $n$ whose generating function is
$$\sum_{n\geq0}p(n)\,x^n=\prod_{k\geq1}\frac1{1-x^k}.$$
Also, the number of partitions into distinct parts $Q(n)$ of $n$ whose genertaing function is
$$\sum_{n\geq0}Q(n)x^n=\prod_{k\geq1}(1+x^k).$$
Expand the ratio of these two generating functions so that
$$\sum_{n\geq0}a(n)x^n=\prod_{k\geq1}\frac{1+x^k}{1-x^k}.$$

QUESTION. Why is $a(n)\equiv 2\,\, (\text{mod}\, 4)$ iff $n$ is a perfect square, for $n\geq1$?


Criterion for smooth functions Let $f:\mathbb{R}→\mathbb{R}$ a real-valued function and $m,n∈\mathbb{N}^∗$ coprime, i.e. greatest common divisor of m and n is 1, and define $f^m:=f\cdot f\cdot\ldots\cdot f.$
Show that
$$f^m,f^n\in C^\infty\Rightarrow f\in C^\infty.$$
I tried to use the fact that every closed subset of a manifold can be expressed as a level set of some smooth real-valued function, but I did not conclude anything.


Are there non-reflexive modules isomorphic to their bi-dual? Let $M$ be an $R$-module. We say that $M$ is reflexive if the natural map $M\rightarrow M^{**}$ is an isomorphism.
I'd like to know if there exists a module isomorphic to its bi-dual but not reflexive, do you know an example?


non-trivial convergent sequence I have reached a deadlock to find a  example to show that a compact Hausdorff space does not need to have a  no non-trivial convergent sequence.(except $\beta\omega$)

can you give me a example of a compact Hausdorff space with no non-trivial convergent sequence?


Do $k$ specific prime factors uniquely determine the continuous composite sequence of length $k$? 猜想:不存在两个长度为k且一共含有k个不同素因子的连续合数序列的素因子集合相等。例如
24 25 26 27 (2 3 5 13)
其余长度为4的连续合数序列要么素因子个数大于4  要么素因子集合不等于{2 3 5 13}  这个连续合数猜想的重大特定情况下的猜想。难度可与哥德巴赫猜想匹敌，非天才误入。再比如2 3 5 唯一决定了8 9 10 除此之外再无其他。总之经过计算机验证没有找到反例。

Attempt at translation of the main aspects:
Let $S$ be a set of positive integers. Denote by
$$ P(S) := \{ p: p \text{ is a prime factor of some } s\in S\} $$
Example: $P(\{24,25,26,27\}) = \{2,3,5,13\}$.

Conjecture: if $S_1, S_2$ are two sets of $k$ consecutive composite positive integers, such that $P(S_1) = P(S_2)$ and $|P(S_1)| = k$, then $S_1= S_2$.

Computer searches have not yielded counterexamples.


Understanding the relations without the knowledge of Plucker relations Consider the grassmannian $\mathrm{Gr}(2,5)$. We know there is an embedding of $\mathrm{Gr}(2,5)$ into $\mathbb{P}^9$ by using the 10 Plucker coordinates, and they satisfy 5 Plucker relations. And, so the coordinate ring of $\mathrm{Gr}(2,5)$ is the polynomial ring in these Plucker coordinates, modulo the ideal generated by these 5 Plucker relations.
But, say as a novice I don't know about the Plucker relations, dimension or degree, then how do I prove that these 5 relations are a minimal set of generators for the ideal of relations? How do I prove that there are no other relations?
Edit I'm looking for a down-to-earth reasoning. Something like- I start with some relation, and then I'm able to prove that it belongs to the ideal generated by these 5 Plucker relations.


How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations? As the question title asks for, how do others visualize the Riemann-Roch theorem with complex analysis or geometric topology considerations? That is all Riemann would have had back in the day, and he reminds me of Thurston in the sense he just drew a picture and then the proof popped out. Nowadays most formulations of Riemann-Roch are couched in algebraic language and do not invoke any geometric intuition for the complex plane, so I am asking here. Bonus points for pictures.


Are these vector bundles, trivial bundle? We identify the  vector  space tensor  product $\mathbb{R}^{m} \otimes \mathbb{R}^{n}$  with $\mathbb{R}^{mn}$
Let  $X$ be the  space  of  all  non zero simple tensors $X=\{a\otimes b \mid a\in \mathbb{R}^{n} \setminus \{0\}, \;b\in \mathbb{R}^{m} \setminus \{0\}\}$.
Let  $\pi:\mathbb{R}^{mn}\setminus \{0\}  \to  \mathbb{R}P^{(mn-1)} $  be  the  natural  projection. Put $PX=\pi (X)$
Is   the  tautological line  bundle  restricted to $PX$, a trivial bundle?
Is the  following  bundle  $(E,X,q))$  a trivial bundle  over $X$:
$E=\{(x\otimes y, T) \mid T:E_{x} \to E_{y} \;\; \text{is  a  linear  map }$  where $E_{x}= \{v\in \mathbb{R}^{n} \mid  v.x=0\} $
$X$  is  the  space simple  tensors  and $q$ is  the obvious projection.


Notations - Hardy and Sobolev Spaces After some confusion on my part, I wanted to know is there a profound mathematical reason why both Hardy spaces and Sobolev spaces are denoted by $H^p$(1). Is it just coincidence? Does it have any historical meaning?
Note: Granted that a Sobolev space of $k$ derivative all in $L^p$ will not be denoted this way, but usually when $p=2$ then it is omitted, and we are left with $H^k$.


Does the ball maximize the "kissing probability" of symmetric convex bodies? Given a symmetric convex body $K \subset \mathbb{R}^n$ (i.e., a bounded symmetric convex set with non-empty interior), I am interested in the following quantity
$$p_K := \Pr_{x_1, x_2 \sim K}[x_1 \in K + x_2]
\; ,$$
i.e., the probability given two uniformly random and independent vectors $x_1,x_2$ in $K$ that $x_1 \in K + x_2$,
which you might call the "kissing probability" of $K$, in analogy with the kissing number. (I know it's not a great name.) An equivalent definition is
$$
p_K := \mathop{\mathbb{E}}_{x \sim K}[\mathrm{vol}(K \cap (K+x))/\mathrm{vol}(K)]
\; .
$$
For example if $K$ is the $n$-ball, then $p_K \approx (3/4)^{n/2}$, and if $K$ is the $n$-cube, then $p_K = (3/4)^n$.
My main question is whether this quantity is maximized by the ball. More generally, has $p_K$ been studied? Can we get a decent upper bound on $p_K$?

Some context:
This quantity arises relatively naturally in the study of sieving algorithms for lattice problems. At a very high level, these algorithms work by sampling a very large number of more-or-less random vectors from a convex body $K$ and then looking for pairs of vectors $x_1, x_2$ such that $x_1 \in x_2 + K/(1+\varepsilon)$, taking their difference, and repeating the procedure on the differences. The running time of these algorithms is more-or-less governed by the number of points that need to be sampled in order to guarantee that nearly every point can be paired with another, which is more-or-less $1/p_K$.
So, if one can find a convex body with $p_K \gg (3/4)^{n/2}$, then one might hope to find a faster algorithm for lattice problems, which would be quite significant.


Is there a high-concept explanation for why characteristic 2 is special? The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for $2$. So in these examples characteristic $2$ is a messy special case.
On the other hand, certain types of combinatorial questions can be reduced to linear algebra over $\mathbb{F}_2,$ and this relationship doesn't seem to generalize to other finite fields. So in this example characteristic $2$ is a nice special case.
Is anything deep going on here? (I have a vague idea here about additive inverses and Fourier analysis over $\mathbb{Z}/2\mathbb{Z}$, but I'll wait to see what other people say.)


Homotopy classes of maps and cohomology classes (Hatcher, AT, Thm 4.57) Hatcher's AT Theorem 4.57 is used in both the algebraic topology construction of Seifert surfaces, and the (similarly flavored) proof that given a compact 3-manifold (with or without boundary), we can associate a properly embedded surface $S$ to any $[a] \in H_2(M, \partial M;
\mathbb{Z})$ (this is used to define the Thurston Norm).
Theorem 4.57 states (direct quote): there are natural bijections $T: \langle X, K(G,n) \rangle \to H^n(X; G)$, for all CW-complexes $X$ and all $n > 0$, with $G$ any abelian group. Such a $T$ has the form $T([f]) = f^*(\alpha)$ for a certain distinguished class $\alpha \in H^n(K(G,n); G)$ (Hatcher, Algebraic Topology, Page 393). (Notation: $\langle X, K(G,n) \rangle$ is the set of basepoint-preserving homotopy classes of maps from $X$ to a $K(G,n)$).
The proof (to me, at least) is somewhat complicated, and is constructed by creating a cohomology theory, and uses some basic formalisms (that is, once you get through the 8 pages of algebraic topology, the proof of 4.57 falls out in 1 paragraph).
My question: the $T$ above is a bijection; suppose we assume $X$ is something nice, like a smooth manifold, and $K(G,n)$ is also relatively simple, perhaps $S^1$, which is a $K(\mathbb{Z}, 1)$ -- is there some topological or geometric way to see how a function $f \in \langle X, K(G,n)\rangle$ is built in association with $[a] \in H^n(X,\mathbb{Z})$?
A more general (and fluffy) question: again assuming $X, K(G,n)$ are "nice", as above, is there some intuition for why this theorem holds? (I understand that this is perhaps a somewhat unreasonable question).
Thank you!


Can $\mathbb{R}^2$ be covered by disjoint sets homeomorphic to the union of the segments $[(0,0), (0,1)], [(0,0), (1,1)], [(0,0), (1,0)]$? This question was asked at the french ENS oral examination. I do not really know how to approach it. I think the answers no.
What I've gathered so far :
Lets call $T$ the subset of $\mathbb{R}^2$ in the title (for obvious reasons). If the union exists, by Baire's theorem it must be uncountable. Let $E$ be the set of maps $T \to \mathbb{R}^2$ of which the images disjointly cover $\mathbb{R}^2$.
Since the domain is compact every map is uniformly continuous : by uncountability, for all $\epsilon > 0$ there is an $\eta$ such that an uncountably infinite number of maps $f$ verify the property $|x - y| \leq \eta \implies |f(x) - f(y)| < \epsilon$.
By uncountability, I also think an uncountably infinite number of the maps above should also have their image in a well chosen compact region of the plane, denoted $K$.
Let $E'$ be an uncoutably infinite subset of $E$ where all the maps verify the two properties ("uniform" uniform continuity and image in a given compact set $K$). For any finite subset of $T$, by using successive extractions I should be able to find a sequence of maps $(f_n)$ in $E'$ such that for any point $x$ in this finite subset the images $f_n(x)$ converges to a point $y \in K$. By using uniform uniform continuity I hoped to find that the maps $f_n$ would be arbitrarily close to one another and necessarily cross.
However, the more I think about this approach the less likely I think it is to work.
Would anyone have an idea ?


Generalized Sphere Kissing Problem I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching a central unit ball.
That is, center the unit ball $B \subset \mathbb{R}^3$ at the origin and define $k_{s}(\eta)$ to be the maximum number of balls of radius $\eta$ which can kiss $B$ and form a solid packing (non-overlapping interiors). How can you upper bound $k_{s}(\eta)$ for $\eta >0$? A remarkable case of note is $k_{s}(1) = 12$ which corresponds to the solution of the Newton-Gregory problem.


When are there no maps from a variety of high dimension to a variety of low dimension? It's easy to show that the only maps from $\mathbb P^{n+d} \to \mathbb P^n$ are the constant maps for $d \geq 1$. Given two smooth, projective varieties $X,Y$ of dimensions $n+d,n$ as above, are there any nice, general conditions under which the only maps between them are constant? It's not always true as the example $X = Z\times Y \to Y$ shows.
By Noether normalization, we can assume that $Y = \mathbb P^n$ so I believe we are looking for conditions on $X$ so that any $n+1$ divisors on $X$ intersect non trivially.


Can Hodge symmetry and invariance of Hodge numbers in smooth families be proven purely algebraically? Let $k$ be an algebraically closed field of characteristic 0.
I am wondering if there are proofs of the following facts that do not use the analytic topology over $\mathbb{C}$:

Let $X$ be a smooth projective variety over $k$. Then $h^{p,q}(X)=h^{q,p}(X)$ for all $p,q$.
Let $f:X\to Y$ be a smooth proper morphism of varieties over $k$. Then $h^{p,q}(X_y)$ is locally constant on $Y$.

Unlike other facts about Hodge numbers, these are only true over fields of characteristic 0.
More generally, I was wondering if there is a purely algebraic way (i.e. avoiding the analytic topology) of "doing Hodge Theory."


Example of a group satisfying Max which is not polycyclic by finite Are there examples of a group satisfying max condition which is not polycyclic by finite. I was looking into Finiteness conditions-I by D J S Robinson, and it was posed as a question there, but the book was written around 1972, so i am sure there might be an example by now.
THanks