I propose this as a companion wiki page to the one about PhD dissertations which contain a solution to an open problem in the style of big-list questions, thinking in terms of the well-known paradigm that splits mathematical research into *problem solving* and *theory building*. Theories are at times developed to solve famous open problems, but sometimes the concrete problems they solve are quickly dwarfed by the possibilities that a new theory opens.

Can you name modern mathematicians who already in their PhD theses (or earlier in their career) developed a substantial new theory or laid the foundations of a new field of research?

Let $G$ be a (discrete) group. Define $k^*(G)$ as the minimal cardinality of a set $S \subset G$ such that $C_G(S) = Z(G)$. Define $k(G) = k^*(G)$ if $G$ has trivial center (i.e. $|Z(G)| = 1$), and $k(G) = \bot$ otherwise. If $k(G) = \bot$, then the convention is that neither $k(G) \leq n$ nor $k(G) \geq n$ holds, for any cardinal $n$.

Question: Does there exist a finite group $G$ such that $k(G) \geq 3$, and more generally does every natural number $k(G) \geq 3$ occur for some finite group $G$?

I do not even know such examples for $G$ infinite, and would also be interested in such, though I do not have an immediate application for this. I am not an expert on group theory (especially finite group theory), so I do not know very effective search terms for this, and would also be interested in pointers to the literature.

What I have tried so far (though don't take my word on these):

No abelian group or a p-group or a nilpotent group is an example, since they have nontrivial centers (in the finite case), thus $k(G) = \bot$.

No finite simple group is an example, since they are all 2-generated (by CFSG), thus satisfy $k(G) \leq 2$ or $k(G) = \bot$.

$k(G \times H) = \max(k(G), k(H))$ for any groups $G, H$ (by a simple proof).

I did a quick search in GAP and seems that there are no finite groups of size up to $1151$ with this property (this is the first time I used GAP, so not sure how much proof value this has).

$k(G) = 0$ for precisely the trivial group, and $k(G) = 1$ is impossible (since any $g$ commutes with itself).

For infinite cardinal $\kappa$, $k(G) = \kappa$ where $G$ is the group of finite-support permutations on a set of cardinality $\kappa$, but of course $k(G)$ is finite (or $\bot$) for finite groups.

Arbitrarily large $k^*(G)$ are provided by wreath products $\mathbb{Z}_2 \wr \mathbb{Z}_2^d$, where $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$, but I wrote a quick proof sketch that $k(G \wr H) = 2$ whenever $|H| \geq 2$ and $k(G) = 2$ so it seems one cannot use wreath products to get examples.

For a single permutation $\pi$ on a set of size $n$, The centralizer of $\pi$ in $S_n$ never has $k(G) \geq 3$ (I prove this as Proposition 7.9 in this paper of mine, by a rather ad hoc case analysis).

In terms of the commuting graph $\Gamma(G)$ with vertices $G \setminus Z(G)$ and edges $\{(g, h) \;|\; gh = hg\}$, the question of whether $k(G) \geq 3$ is possible is equivalent to whether there exists a finite group $G$ with trivial center such that $\mathrm{diam}(\Gamma(G)) = 2$, where $\mathrm{diam}$ is the diameter, i.e. maximal minimal distance between a pair of vertices. For a finite minimal nonsolvable group the diameter is always at least $3$ according to this paper which implies $k(G) = 2$ for a minimal nonsolvable group (a different definition is used in that paper, but it should be equivalent to mine for minimal nonsolvable groups, as they have trivial center). Most literature I know about this graph and its diameter are about finding upper bounds, but I do not know if that has a relation to $k(G)$. Larger values of $k(G)$ also correspond to statements about this graph, but not about its diameter.

For context, the question arose from the study of automorphism groups of one-dimensional subshifts. I am interested in quantitative versions (or lack thereof) of the so-called Ryan's theorem, which states that the center of the automorphism group of a mixing subshift of finite type consists of only the shift maps. I ask the above question about finite groups after Lemma 7.7 here and Lemma 7.7 is my application for it. The paper of Boyle, Lind and Rudolph is a standard reference for these groups.

On a smooth maniflod $M$ of dimension $n$, a current of degree $n-p$ is a functional on the space of compactly supported differential $p$-forms which is continuos. We denote the space of currents of degree $n-p$ by $D^{'n-p}(M)$. If we consider the functionals on $D^{'n-p}(M)$ with an approperiate comapctness and continuity assumptions, then what are these functionals? Are they just differential $p$-forms, or could be more then that?

I am asking this question because I want to understand why there is no definition for the pullback of current $T$ in general (as far as I know). Let $f: M_1 \to M_2$ be a map between manifolds. Then the pullback $f^*T$ should be defined as (formally) $$\langle f^*T, u\rangle = \langle T, f_*u \rangle.$$ Here $u$ is a differential $p$-form, in particular, $f_*u$ is well-defined as a current under some compactness assumption. Thus the problem is to make sense of $\langle T, f_*u \rangle$, i.e. can a current be a functional on the space of current? (Of course, it is enough to have $T$ be a functional on the space $\{f_*u\}$, that is why pullback of current is well-defined for submersion maps).

I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (and her/his supervisor).

In my question I search for every possible open problem but I prefer (but not restricted) to receive answers about those open problems which had been unsolved for at least (about) 25 years and before the appearance of the ultimate solution, there had been significant attentions and efforts for solving it. I mean that the problem was not a forgotten problem.

I ask the moderators to consider this question as a wiki question.

I am reading Jason Lotay and Goncalo Oliveira's paper -$SU(2)^2$ invariant $G_2$-instantons, and have few questions from the same.

If we consider the space $M = S^3 \times S^3$. Then the cone metric can be written as

$g = dt^2 + g_t$, where $g_t = \sum_{i=1}^3(2A_i)^2\eta_i^+ \wedge \eta_i^+ + (2B_i)^2\eta_i^- \wedge \eta_i^-$ is the compatible metric given by the $SU(2)\times SU(2)$ invariant $SU(3)$ structure on $\{t\}\times M$.

Here $\eta_i^\pm$ are the $\eta_i^\pm$ is the standard coframe of 1-forms, while the functions $A_i(t), B_i(t)$ specify the deformation of the cone singularity. I have the following questions:

How can we see that the metric $g_t$ (and in general the SU(3) structure as given) is $SU(2)\times SU(2)$ invariant?

Why an extra $U(1)$ symmetry forces $A_2=A_3, B_2=B_3$?

Why an extra $SU(2)$ symmetry forces $A_1=A_2=A_3, B_1=B_2=B_3$?

Thanks!

Can all square integrable solutions $(\rho(t,x),j(t,x))$ of the homogeneous continuity equation $$\dot\rho(t,x)+\nabla \cdot j(t,x)=0$$ in 1+3 dimensions be approximated by solutions with compact support (in both space and time)? What are the simplest nontrivial solutions with compact support?

**Conjecture:** There is a universal constant $c$ such that for any fixed nonzero real vector $q$ of any dimension $n$ and any random vector $p$ of the same dimension $n$ with independent components uniformly distributed in $[-1,1]$, we have
$$(p^Tp)(q^Tq)\le cn(p^Tq)^2$$ with probability $\ge 1/2$.
Simulation suggests that a constant $c\approx 8$ should work.

I'd be interested in techniques for proving this and related statements. In particular, can one determine the best constant $c$?

Consider an action of a smooth linear algebraic group $G$ on a variety $X$ over an arbitrary field $k$, and the quotient stack $[X/G]$. Let $p$ be a $k$-point of $X$. If the action is transitive (i.e. $G(\bar{k})$ acts transitively on $X(\bar{k})$) then we have an isomorphism $[X/G]\cong BG_p$, where $G_p$ is the scheme-theoretic stabilizer of $p$, given as follows. In one direction, send a $G$-torsor $P$ with $G$-equivariant map $\pi: P\to X$ to $\pi^{-1}(p)$ and on the other direction send a $G_p$-torsor $Q$ to $Q\times^{G_p}G$.

This cannot be right, because if $q$ is another $k$-point of $X$ then this implies $G_p\cong G_q$, but I think we only have $(G_p)_{\bar{k}}\cong (G_q)_{\bar{k}}$ (it can happen that $p$ and $q$ are conjugate over $\bar{k}$, but not over $k$). If $[X/G]\cong BG_p$ is wrong, how do I correct it?

This game has two players, Spoiler and Solver. We start with a solved 3x3x3 rubik's cube(to make the problem easier).

Spoiler and Solver make turns making 90 degree twists. The cube is forbidden from ever repeating a position (besides the start position). This guarantees the game is finite.

If at any point (besides the beginning), the rubik's cube is in a solved state, Solver wins. If the game ends before that (because a position is entered with no valid moves), Spoiler wins.

An example game would be F,F;F,F (using basic rubik's cube notation). Solver wins this game. If a game goes through each position that is one move away from the solved state, and afterwards goes to some unsolved state, Spoiler will win (since it is now impossible to get to the solved state).

So, which player has the winning strategy?

EDIT: It may be simpler to consider the same problem with the 15 puzzle first.

I have just started learning some derived algebraic geometry. I was told that (if $ \mathrm{char}(\mathbb{K})=0 $) using commutative differential graded algebras in negative degree (for short $ \mathrm{cdga}^{\leq 0} $) is easier than using simplicial commutative rings (for short $ sComm $). I suppose that it is easier because one have to put less effort in computing fibrant/cofibrant replacements (for instance when $ I\subseteq R $ is an ideal generated by a regular sequence then we can use Koszul complex for the replacement of $ {R}/{I} $). Are there any further advantages?

Here follows the main question I am struggling with: is there any *canonical* (in some sense) way of computing fibrant/cofibrant replacement of objects in $ sComm $?

Let $G$ be a finite type commutative group scheme over a finite field $k=\Bbb F_q$ with $\Gamma=Gal(\bar k/k)$ and $l$ be a prime such that $(l,q)=1$, we can also define the Tate module $T_lG =\varprojlim_nG[l^n](\bar k)$. Then we can also consider the natural map

$Hom_k(G_1,G_2) \otimes \Bbb Z_l \rightarrow Hom_\Gamma(T_lG_1,T_lG_2)$

for any two commutative group scheme over $k$, or more generally the map

$\text{Ext}^i_k(G_1,G_2) \otimes \Bbb Z_l \rightarrow \text{Ext}^i_\Gamma(T_lG_1,T_lG_2)$ for every natural number $i$ at least when the extension groups are well-defined (This question may be related).

My first question is: when is this map surjective, injective or an isomorphism? If $G_i$ are abelian varieties and $i=0$, it's well-known that this map is an isomorphism. If $G_2=\Bbb G_m$ and $G_1$ is an abelian variety, this is also true for $i=0$ as both sides are zero, and is true for $i=1$ and $\text{dim}G_1=1$ as $\text{Ext}^1(A,\Bbb G_m)$ is the dual of $A$ and one can compute both sides explicitly (see this master thesis). However, this map can fail to be injective for trivial reasons (For instance, right side could be zero for some finite abelian groups).

So my second question is: how to describe extension groups between commutative group schemes using linear algebra datas in a systematic way? For instance, how to describe extension groups of finite group schemes like $\alpha_p$, $\mu_p$?

The last question is about growth of $\#G[l^n]$ for commutative group schemes (in order to study the Tate module). Let $k$ be any field, $G$ be a commutative group scheme over a field $k$ (not necessarily finite type) and $l$ be a prime. Assume $G[p^n]$ is a finite group scheme for every $n$, does there always exist $C >0,h \geq 0$ such that $\#G[p^n]=Cp^{nh}$ for $n >> 0$? Here the order of a finite group scheme means the dimension of its global section ring. I know the result holds for etale group schemes and abelian varieties. For example, if $G$ is constant i.e an abstract abelian group and $\#G[p^n]$ is finite for every $n$, then we know $G[p^\infty]\cong (\Bbb Q_p/\Bbb Z_p)^h\oplus T$ for some $h \geq 0$ and finite $p$-group $T$ (this result is used when studying Tate-Shafarevich group of elliptic curves).

This question already has an answer here:

- Function satisfying $f^{-1} =f'$ 4 answers

what are the solutions f :R=>R ,that verifie the equation ,for x in R, f'(x)=f^-1(x) ,where f^-1 is the reciprocal function of f.

Let $X$ be a topological vector space. Let us say that $X$ has property **P** if there exists a sequence of closed subsets $\{X_n\}$ such that

1- $X=\bigcup X_n$

2- The relative topology is both metrizable and second countable on $X_n$'s.

Q. Assume $X$ satisfying **P** property. Let $m: X\times X\to X$ be an associative multiplication. Is $m$ measurable?

Amplitudes of open A-model on a Calabi-Yau 3-fold $X$ with branes are given by the open Gromov-Witten invariants of $X$. It is known how to compute them if there is a toric action on a manifold, which preserves all the branes and the image of the curve (see Katz, Liu, Enumerative Geometry of Stable Maps with Lagrangian Boundary Conditions and Multiple Covers of the Disc and Graber, Zaslow, Open-String Gromov-Witten Invariants: Calculations and a Mirror "Theorem" for examples). It is done via equivariant localization on the fixed points of the toric action.

All the references I read, consider the toric action with integer weights (thus $S^1$), while the brane configuration I have is preserved only by certain $\mathbb{C}^{*}$ action, which rotates some of the coordinates by $\mathrm{e}^{\mathrm{i} b}$, while the others by $\mathrm{e}^{\mathrm{i/b}}$, for irrational $b$. If I try to follow the usual procedure used for the cases of $S^1$ action, I get the result which contradicts my physical expectations. I suspect that the reason is that this procedure must be modified appropriately in order to deal with irrational weights.

Could anybody recommend any reference in which such a problem is discussed?

Let $p(n)$ be a polynomial with integer coefficients.
Define $\Delta( p(n) )$, the *prime density* of $p(n)$, to be
the limit of the ratio with respect to $n$
of the number of primes $p(k)$ generated when the polynomial
is evaluated at the natural numbers $k=1,2,\ldots,n$:
$$
\Delta( p(n) ) \;=\; \lim_{n \to \infty}
\frac{ \textrm{number of } p(k), k \le n, \textrm{that are prime}}
{n}
$$
For example, Euler's polynomial $p(n)=n^2+n+41$
starts out with ratio $1$, but then diminishes
beyond $n=39$:
And it continues to diminish ...
... and by $n=10^7$ has reached $\Delta=0.22$.

**Q**. What is the largest known $\Delta( p(n) )$ over all polynomials
$p(n)$?

In particular, are there any polynomials known to have $\Delta > 0$?

Maybe these questions can be answered assuming one or more conjectures?

What is the meaning of this notation? In particular the top-cut square brackets. To put it in some context: the upper-bound of $\alpha$ is $\left \lfloor \sqrt[4]{\beta_{0}/4} \right \rfloor$

Here are the numbers whose prime factors all congruent to $\pm 1\pmod 8$: http://oeis.org/A058529

My questions are:

(1) What is the order of growth of these numbers? That is, what is the order of magnitude of the $n$th smallest among them?

(2) For the numbers whose prime factors are all congruent to $\pm 3\pmod 8$, is the order of growth the same as in (1)?

For any non-empty set $X$ let $\text{Sym}(X)$ denote the group of bijections $f:X\to X$ with composition.

Is there an infinite set $X$ and a surjective group homomorphism $\pi: \text{Sym}(X)\to \mathbb{Z}$?

Given a vector function
$$f=(f_1,\ldots,f_n)\in L^2(\mathbb R,\mathbb R^n)$$
(for some $n\in\mathbb N$), let us define
$$\Delta f:=(\Delta f_1,\ldots,\Delta f_n),$$
where $\Delta$ is the Laplacian operator, and let $Q:\mathbb R\to\mathbb R^{n\times n}$ be a potential taking values in symmetric $n\times n$ matrices. I'm interested in *vector Schrödinger operators* of the form
$$Hf=-\Delta f+Qf,\qquad f\in L^2(\mathbb R,\mathbb R^n).$$

**Question.** Is there a Feynman-Kac type formula known for $H$'s semigroup in the case where $Q(x)$ is not necessarily diagonal?

(Note: I specify abote that I'm interested in the case where $Q$ is not diagonal; if $Q=\mathrm{diag}(Q_1,\ldots,Q_n)$, then $$Hf=(-\Delta f_1+Q_1 f_1,\ldots,-\Delta f_1+Q_1 f_1),$$ in which case we can simply apply the one-dimensional Feynman-Kac formula to each component.)

Let $R \in \mathbb{R}^{n,d}$ be a random Gaussian matrix comprised of independent $\operatorname{N}(0,\frac{1}{n})$ entries, let $\textbf{w}$ and $\textbf{x}$ be vectors in $\mathbb{R}^d$, and let $\epsilon \in (0,1)$. In Shi et al. 2012, it is claimed in Lemma 10 that $$\operatorname{Pr}(1-\epsilon \leq \frac{\|R\textbf{x}\|^2}{\|\textbf{x}\|^2} \leq 1 + \epsilon) \geq 1 - 2\exp(-\frac{n}{2}(\frac{\epsilon^2}{2}-\frac{\epsilon^3}{3}))$$

I don't understand how this lower bound was obtained. It seems to me that when you want to find a lower bound on a probability in a situation like this, you should try to minimize $$\operatorname{Pr}\bigg(\frac{\|R\textbf{x}\|^2}{\|\textbf{x}\|^2} \leq 1 + \epsilon \bigg)$$ and maximize $$\operatorname{Pr}\bigg(\frac{\|R\textbf{x}\|^2}{\|\textbf{x}\|^2} \leq 1 - \epsilon \bigg)$$

If that's what Shi et al. did in this situation, I don't understand how they did it.