If $G$ is a locally compact group and $\Gamma $ is a discrete subgroup such that the quotient $G / \Gamma$ carries a finite left $G$-invariant Haar measure, then we say that $\Gamma$ is a lattice in $G$.

Why are lattices important? Can you give some motivating examples? What are some applications?

Coppersmith's theorem $2$ in Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities says that:

"`Let $p(x, y)$ be an irreducible polynomial in two variables over $\Bbb Z$, of maximum degree $\delta$ in each variable separately. Let $X, Y$ be bounds on the desired solutions $x_0, y_0$. Define $\tilde p(x, y) = p(x X, yY )$ and let $W$ be the absolute value of the largest coefficient of $\tilde p$. If $XY < W^{{2/(3\delta)}−\epsilon}2^{−14\delta/3}$ then in time polynomial in $(\log W, \delta, 1/\epsilon)$, we can find all integer pairs $(x_0, y_0)$ with $p(x_0, y_0) = 0$, $|x_0| < X$, and $|y_0| < Y$"'

In corollay $2$ he sharpens this to "`With the hypothesis of Theorem $2$, except that $XY ≤ W^{2/(3\delta)}$, then in time polynomial in $(\log W, 2^\delta )$, we can find all integer pairs $(x_0, y_0)$ with $p(x_0, y_0) = 0$, $|x_0|\leq X$, and $|y_0| \leq Y$"'.

If $p(x,y,z)$ is a an irreducible polynomial in $\mathbb Z[x,y,z]$ what is the precise analogous result?

I looked in Bauer and Joux's work. They extend to trivariate case but the bounds are not quite clear to me and it is quite bewildering they do not have a 'theorem' in the paper and they have variables $s_x,s_y,s_z$ and $s$ which somehow parametrizes the bounds but unlike Coppermsith's paper these boundsare not clear.

Given an $n\times n$ matrix $A$, whose elements are over $GF\left(2\right)$ and the diagonal elements are all $1$. There are $m\ (m\leq n^2-n)$ non-zero off-diagonal elements in $A$. If we are allowed to choose $k$ non-zero off-diagonal elements of $A$ and set them to $0$, then what is the minimum rank of $A$ over all $k\ (k\leq m)$ and all possible choices?

Denote the above minimum rank as $mk_2$, since the number of all possible choices (the size of the search spaces) is $2^m$, it's hard to determine $mk_2$ exactly. If we want to calculate $mk_2$ approximately, $e.g.$, find possible choices whose rank are less than $c\times mk_2$ or $mk_2+c$ ($c$ is a constant not increasing with $n$ and $m$), then what is the number of possible choices satisfying this demand?

Is there any structure in the problem of determining $mk_2$ exactly? IS there any approximation algorithm with guaranteed bound?

(If $A$ is block diagonal then the problem can be reduced by considering each block independently. )

**Definition.** A graph $G=(V,E)$ is to be $\{d_1,\dots,d_n\}$-graph if for each vertex $v\in V$ we have $\text{deg}(v)=d_i$ for some $i=1,\dots n$.

**Definition.** A connected graph $G=(V,E)$ is called $n$-connected (for n\geq 2) whenever if we remove $n-1$ vertices then the graph is still connected.

**Definition.** A $P_k$-factor of a graph $G=(V,E)$ is a spanning subgraph of $G$ such that each component of which is $P_k$, the path on $k$ vertices. We say that $G$ has a $P_k$-factorization if $E$ can be partitioned into $P_k$-factors

**Question.** Let $G=(V,E)$ be a $\{2,3\}$-graph which is also 2-connected and $|V|>5$. Does $G$ have $\{ P_3, P_4 \}$-factor?

According to Cantor's attic, Vopenka's principle is equivalent to the existence of a strong compactness cardinal for any "logic". But I can't find a definition of what a "logic" is either there or in any of the cited references.

**Question:** What is a "logic" in the sense of this statement?

Some examples are given: apparently infinitary logic $L_{\kappa\kappa}$ is a "logic", and infinitary higher(but finite)-order logic $L_{\kappa\kappa}^n$ is a "logic". Thus Vopenka's principle should imply the existence of a proper class of strongly compact cardinals, and even a proper class of extendible cardinals.

But I don't think it's supposed to be as simple as "Vopenka's principle is equivalent to a proper class of extendibles", so there must be more "logics" than these. The next thing I can think of is some sort of infinitary-order infinitary logic $L^\alpha_{\kappa\kappa}$. It would also make sense to consider structures with infinitary operations. I don't know if there's a large cardinal principle associated to strong compactness for either of these sorts of logic.

And then of course there are "logics" such as various flavors of type theory (which higher order logic starts to resemble anyway!) but just for cultural reasons, I doubt that "logic" is meant to encompass anything along these lines.

For the purposes of this question, a *categorification of the real numbers* is a pair $(\mathcal{C},r)$ consisting of:

- a symmetric monoidal category $\mathcal{C}$
- a function $r\colon \mathrm{ob}(\mathcal{C})\to\mathbb{R}$

such that:

- $r(X\otimes Y) = r(X) r(Y)$ for all objects $X$ and $Y$ of $\mathcal{C}$
- $r(\mathbb{1}) = 1$, where $\mathbb{1}$ is the monoidal unit
- $X\cong X'\implies r(X)=r(X')$

Some examples of categorifications of $\mathbb{R}$ are: finite sets and cardinality; finite-dimensional vector spaces and dimension; topological spaces (with the homotopy type of a CW-complex, say) and Euler characteristic. However, in these examples the map $r$ factors through $\mathbb{N}$ or $\mathbb{Z}$. I am interested in examples where the values of $r$ are not so restricted.

**Question:** What categorifications of $\mathbb{R}$ are there where $r$ can take all values in $\mathbb{R}$, or perhaps all values in $(0,\infty)$ or $(1,\infty)$?

I am especially interested in examples that already appear somewhere in the mathematical literature. I am also especially interested in examples where $\mathcal{C}$ is symmetric monoidal *abelian* and r(A)+r(C) = r(B) for every short exact sequence $0\to A\to B\to C\to 0$.

Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent:

- The all-one vector $j$ is contained in the conic hull of $col(A)$.
- The row sums of $A^{-1}$ are non-negative.
- $ADj > 0$, where $D$ is any diagonal matrix with trace $1$.
- The affine hull of $col(-A)$ does not intersect the non-negative orthant.

The equivalency of $(1)$ and $(2)$ follows from the equation $$Ax = j.$$ Assertion $(3)$ is deduced from Farkas' Lemma, as the existence of a positive solution to the above equation implies that there cannot exist a vector $y$ with $y'j = -1$ such that $Ay \geq 0$ (I normalized $y$ without loss of generality). The set of $y$ with sum-of-entries $-1$ is given by $\{y\;|\;y=-Dj: tr(D)=1 \;\text{and}\; D \; \text{diagonal}\}$, the affine combinations of the negative standard basis vectors. This leads to $-ADj <0$.

Finally, the matrix of images of the negative standard basis under $A$ is simply $-A$. Hence, requiring the affine hull of these images not to contain any non-negative vector should be equivalent to $(3)$.

Two sufficient conditions are that $A$ be positive monomial with zero diagonal (as at least one of the entries of $y$ must be negative), or the adjacency matrix of a regular graph. What can be said in general?

I was recently reading Bui, Conrey and Young's 2011 paper "*More than 41% of the zeros of the zeta function are on the critical line*", in which they improve the lower bound on the proportion of zeros on the critical line of the Riemann $\zeta$-function. In order to achieve this, they define the (monstrous) mollifier:

$$ \psi=\sum_{n\leq y_1}\frac{\mu(n)P_1[n]n^{\sigma_0-\frac{1}{2}}}{n^s}+\chi(s+\frac{1}{2}-\sigma_0)\sum_{hk\leq y_2}\frac{\mu_2(h)h^{\sigma_0-\frac{1}{2}}k^{\frac{1}{2}-\sigma_0}}{h^sk^{1-s}}P_2(hk). $$

This is followed by a nearly 20-page tour-de-force of grueling analytic number theory, coming to a grinding halt by improving the already known bounds from $40.88\%$ to... $41.05\%$.

Of course any improvement of this lower bound is a highly important achievement, and the methods and techniques used in the paper are deeply inspirational. However, given the highly complex nature of the proof and the small margin of improvement obtained over already existing results, I was inspired to ask the following question:

*What are other examples of notably long or difficult proofs that only improve upon existing results by a small amount?*

Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$ superconformal field theories (Wikipedia, nLab, PhysicsOverflow) sometimes called Theory $\mathcal X$. I don't really know what that would entail; apparently we have partial information about this theory, suggesting that it's difficult to describe even in a physics sense (e.g. non-Lagrangian), let alone to mathematically formalize.

But I have also heard from mathematicians who are interested in Theory $\mathcal X$ for what I assume are entirely mathematical reasons. I came away with the impression that even though we can't construct it, there are ways to study it to yield interesting results in pure mathematics; but I don't know any examples of such results.

So my question is: **what are some purely mathematical takeaways from the story of Theory $\mathcal X$?** And, if it's known, what
would be some expected mathematical consequences of a construction of Theory $\mathcal X$ in a physics sense?

I get the impression that various dimensional reductions of Theory $\mathcal X$ should include several commonly-studied TQFTs and QFTs, so in a sense studying Theory $\mathcal X$ generalizes the study of those TQFTs. So one possible answer is that there could be theorems about those TQFTs whose proofs were inspired by some conjectured aspect of Theory $\mathcal X$ — but is this accurate?

In the following paper: **Representation type of the blocks of category $\mathcal{O}_S$**
https://www.sciencedirect.com/science/article/pii/S0001870804002853

On p.196, it states that "When $\mu$ is regular we may write $\mathcal{O}^{\text{reg}}_S$ for $\mathcal{O}^{\mu}_S$; by Translation Principle these blocks are all Morita equivalent."

In the book "**Symmetry: Representation Theory and Its Applications**", p.144. The Translation Principle is called "Jantzen-Zuckerman translation principle".

What is meant by Morita equivalent here?

The parabolic Kazhdan-Lustzig-Vogan polynomials is defined on section 3.4 of the paper: **Kostant modules in blocks of category $\mathcal{O}_S$**.
https://arxiv.org/pdf/math/0604336.pdf
${}^SP_{x,w}(q):=\sum_{i\ge 0}q^{\frac{\ell(x,w)-i}{2}}\dim\text{Ext}_{\mathcal{O}^\mathfrak{p}}(N_x,L_w)$ where $N_x,L_w$ is defined in section 2.2-2.3.

Does Morita equivalent implies that ${}^SP_{x,w}(q)$ is independent of the choice of regular antidominant weight $\mu$?

Consider relative curves $X \to S$, defined to be flat, integral, projective schemes of relative dimension 1 over $S$. When are these objects determined by their fibers?

So if $X,Y$ are $S$-schemes with isomorphic fibers $X_s \cong Y_s, \forall s\in S$, when can we conclude that $X \cong Y$ (as $S$-schemes)?

I am most interested in the case where $S$ is a Dedekind scheme, but more general cases are interesting as well. I do not want to stipulate that the $S$-schemes are smooth, but I could tolerate the stipulation to be normal or regular. The quasiprojective case would be interesting too.

In the cases when I can conclude $X \cong Y$, how might I go about constructing a relatively explicit isomorphism, if I have relatively explicit isomorphisms of the fibers?

[EDIT:]

Thanks to all those who added comments. In light of the useful information they provided, I would like to refine the question to the following:

Can a relative curve $X \to \mathrm{Spec}(\mathcal{O}_K)$ over a number ring (sometimes called an arithmetic surface) be determined by its fibers, when the fibers have genus 0? What about higher genus?

*This is a crosspost from this MSE question from a year ago.*

Finite groups are cancellable from direct products, i.e. if $F$ is a finite group and $A\times F \cong B\times F$, then $A \cong B$. A proof can be found in this note by Hirshon. In the same note, it is shown that $\mathbb{Z}$ is not cancellable, but if we only allow $A$ and $B$ to be abelian, it is (see here).

I would like to know if there are any groups that can be cancelled from free products rather than direct products. That is:

Is there a non-trivial group $C$ such that $A*C \cong B*C$ implies $A \cong B$?

It is certainly not true that every group is cancellable in free products. For example, if $A$, $B$, $C$ are the free groups on one, two, and infinitely many generators respectively, then $A*C \cong C \cong B*C$ but $A\not\cong B$. Many non-examples can be constructed this way, but they are all infinitely generated.

As is discussed in the original MSE question, it follows from Grushko's decomposition theorem that if $A$, $B$, and $C$ are finitely generated, then $A*C \cong B*C$ implies $A \cong B$.

Let $M$ be a submanifold of a Riemannian manifold $\widetilde{M}$. Let $A$ be the second fundamental form of $M$.

Suppose that, for all $p \in M$, the linear map $A(v, \cdot)\: \colon T_{p}M \to N_{p}M$ has rank $k$ for every nonzero $v \in T_{p}M$.

Does this condition define some well-known class of submanifolds (if any)?

Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = R\Gamma(C,df:T_C\to f^*T_X)$. Explicitly, this complex may be realized using the Dolbeault resolution of $T_C$ and $f^*T_X$. In this realization, there are three terms in this complex:

$L^0 = \Omega^0(C,T_C)$, $L^1 = \Omega^{0,1}(C,T_C)\oplus\Omega^0(C,f^*T_X)$ and $L^2 = \Omega^{0,1}(C,f^*T_X)$ with the differentials $L^0\to L^1$ and $L^1\to L^2$ given by a sum of pushforward by $df$ and the canonical $\bar\partial$ operator on a holomorphic vector bundle.

By some general philosophy (for example in the deformation theory book by Kontsevich-Soibelman), $L^\bullet$ should carry the structure of a differential graded Lie algebra (DGLA) such that the deformations of $f$ over a local Artin ring $(A,\mathfrak m)$ with residue field $\mathbb C$ can be seen as solutions $\omega\in L^1\otimes\mathfrak m$ to the Maurer-Cartan equation $d\omega + \frac12[\omega,\omega] = 0$ modulo the gauge action of $\exp(L^0\otimes\mathfrak m)$.

Can we realize the DGLA structure in this case explicitly? In particular, what is the explicit expression for the bracket $[\cdot,\cdot]:L^1\otimes L^1\to L^2$? I am able to see that the degree zero bracket $L^0\otimes L^0\to L^0$ should be simply the usual commutator Lie bracket of vector fields.

Fix an integer $p\ge 2$ throughout. Suppose $S_1,S_2$ are Polish spaces and we have Borel measurable $f_n:~S_1^p \rightarrow \mathbb{R}$ and $f:~S_2^p \rightarrow \mathbb{R}$.

For each $n$, let $\xi_n=(\xi_{i,n}, ~i\in \mathbb{Z}_+)$ be a sequence of i.i.d. random elements taking value in $S_1$ and let $\eta=(\eta_i,~i\in \mathbb{Z}_+)$ be a sequence of i.i.d. random elements taking value in $S_2$.

Let $D_p=\{I=(i_1,\ldots,i_p):~ i_1<\ldots<i_p,~ i_j\in \mathbb{Z}_+\}$.
Suppose we have the joint weak convergence $(f_n(\xi_{I,n}),~I\in D_p)\Rightarrow (f(\eta_I), ~I\in D_p)$ as $n\rightarrow\infty$, where, e.g., $\eta_I=(\eta_{i_1},\ldots, \eta_{i_p})$.

Then by the Skorokhod representation with an additional coupling (see e.g., Corollary 6.12 of Kallenberg 2002), there exist random elements $\xi_n^*:=(\xi_{i,n}^*)\overset{d}{=}\xi_n$ and $\eta^*:=(\eta^*_i)\overset{d}{=}\eta$, such that $f_n(\xi_{I,n}^*)\rightarrow f(\eta^*_I)$ a.s. as $n\rightarrow\infty$ for each $I\in D_p$.

**Goal**: can one strengthen the coupling, so that $(\xi_{I,n}^*,\eta_I^*)$ has the same distribution for different $I\in D_p$?

Remark: this is easily achievable if $p=1$, in which case one does not have the dependence due to overlapping $I$'s.

**Weaker goal**: if one knows the moments of $f_n(\xi_{I,n})$ converge to those of $f(\eta_I)$, can one strengthen the coupling so that $\lim_n\|\xi_{I,n}^*-\eta_{I}^*\|_2=0$ uniformly?

The question is related to the question: detecting weak equivalences in a simplicial model category

Suppose that we have a simplicial model category $M$ and let denote $M^{f}$ the full simplicial subcategory of fibrant objects. Suppose that $R$ is a subcategory of $M^{f}$ such that for any object $m\in M^{f}$ there exists an objects $r\in R$ such that $r$ is (zigzag) equivalent to $m$ i.e. $r$ and $m$ are isomorphic in $Ho(M)$ the homotopy category of $M$. Let $w: a\rightarrow b$ be a morphism in $M$ such that for any object $r\in R$ the induced map of simplicial sets $w^{\ast}:Map_{M}(b,r)\rightarrow Map_{M}(a,r)$ is a weak homotopy equivalence of simplicial sets. Can we conclude that $w$ is a weak equivalence in the model category $M$ ?

If $R=M^{f}$ this is true and is proved in Hirschhorn's book.

Because mathematics has been extremely well-developed in XX.-th and continues to do so in XXI.-th century and because there is an enormous number of open problems and conjectures and hypotheses posed from the beginning of XX.-th century till this time of ours, I am thinking that it would be nice to know something about conjectures that survived all the efforts of attack from the time of before the XX.-th century till the present day.

This question is posed with the hope that, in addition to the problems that we all know of that were most probably posed before the beginning of the XX.-th century, there are some problems that are unsettled but known only to some specialists in some branches of mathematics, so it would be nice to collect them here.

We all know about the problem of existence of an infinite number of Mersenne primes, about Goldbach˙s conjecture, about the problem of existence of an odd perfect number, also, all four of Landau´s problems were most probably posed before the beginning of the XX.-th century, also there is a famous conjecture of de Polignac, and problem of Brocard, and so many others, including the Riemann hypothesis.

Let us also take the interval (antiquity, 1900) to be open on the right side because Hilbert posed his problems in 1900 so some of his problems are not includable in the problems that you mention in your answer(s).

Let $R$ be a non-noetherian domain. $S$ be a multiplicatively closed subset of $R$. Let $S^{-1}R$ be a localisation of $R$, where all element of $S$ is invertible. Suppose we have an ideal $I$ of $R$. Let us denote by $I^{\mathrm{e}}$ the extended ideal $IS^{-1}R$ in the ring $S^{-1}R$. Then I would like to ask

Q. Does the following inequality holds? $\colon$ ${\mathrm{ht}}(I) \geq {\mathrm{ht}}(I^{\mathrm{e}})$, where ${\mathrm{ht}}(I)$ is considered in the ring $R$, and ${\mathrm{ht}}(I^{\mathrm{e}})$ in $S^{-1}R$.

Does anyone know the rationale behind the name of "commutative diagrams"? To be precise, what is(are) the reason(s) for calling those diagrams "commutative" and in what sense?

I have previously asked the question here and one can see the discussion that followed.

Let $A$ be a $\sigma$-unital $C^*$-algebra and $A_s:=A\otimes K$ its stabilization (where $K$ is the algebra of compact operators on a separable Hilbert space). Is it true that there exist an approximation of unity $P_n\in A_s$ with $P^*_n=P_n=P_n^2$, in general?