Is there an infinite amount of rules in math? (is math infinite?)

Suppose $f$ is any function supported on $[w/v,xv]$ which is continuous, bounded and piecewise monotonic. Then, why do we have

$$\sum_{n \equiv \alpha \mod q} f(dn) (dn)^{-s}=\frac{F(s)}{dq} +O(|s| \frac{v}{w})$$

with $$F(s)= \int f(\xi) \xi^{-s} d\xi$$

It is stated as equation 18.42 in Kowalski's book.

There was some activity a while ago, like 10 years ago, string theoreists try to relate

- the
**fluid dynamics**, for example, governed by**Navier-Stokes**equation,

to

- the
**Einstein gravity**, and its relations to holography (such as AdS/CFT),

Naively, it looks that the development there is just trying to relate one problem to the other problem. For example, explicit construction that for every solution of the incompressible Navier-Stokes equation in $p+1$ dimensions, there is a uniquely associated "dual" solution of the vacuum Einstein equations in $p+2$ dimensions. See for example, studied in

and References therein. There is also a Breakthrough Prize in Fundamental Physics awarded to the developement of this direction.

My question is that have these efforts from "string theory and Einstein gravity" also affected the mathemtical study of Navier-Stokes equation and fluid dynamics? What are new progresses? What are new and solid lessons have we learned in mathemtics of Navier-Stokes equation and fluid dynamics then after this development? (i.e. the impacts on the community of mathemticians.)

Thanks for the References/answers!

Let $X$ be a quasi-projective variety over $\mathbb{C}$, and let $\mathcal{L}$ be a rank one $\mathbb{C}$ local system on $X$. Does $H^*(X,\mathcal{L})$ come with some mixed hodge structure in general?

Let $G$ be a plane graph, and $G^*$ its dual. Among the $k$ partitions of the nodes of $G$, I'll call the *connected* k-partitions those such that each block of nodes of the partition induces a connected subgraph of $G$.

It is well known that the connected 2-partitions of the vertices $G$ are dual to the simple cycles of $G^*$. (The duality is by sending the partition to the cut edges of that partition. This is also called the bond / simple cycle duality.)

I believe I have a proof that the connected $k$-partitions of the vertices of $G$ are dual to the set of subgraphs $K$ of $G^*$ with

- $H_1(K)$ is of rank $k - 1$
- Each connected component $K$ is $2$ edge connected.

Again, the duality is by sending a partition to the cut edges in that partition. In the other direction, it comes by taking the connected components of $G$ after removing the edges crossing the subgraph $K$.

I use this result in the course of some other proof, and I'm looking for a reference for it (as this seems like the kind of thing that is either wrong, or well known).

A given integer $N$ of $n$ bits can have almost exponentially (in $n$) number of factors of $n/c$ bits where $c\geq2$ is fixed. However take a prime $p\nmid N$ of $n/(2c)$ bits and fix $r$ of at most $n/(2c)$ bits

How many $d\in\mathbb Z$ can there be with $d|N$ and $r\equiv d\bmod p$?

How many coprime pairs of $d,d'\in\mathbb Z$ can there be with $dd'|N$ and $r\equiv dd'\bmod p$?

Is there a formulation of distributions in 1. and 2.? Is it almost always $O_c(1)$ in 1. and 2. or even almost always at most $1$ at $c\rightarrow2$ in 2. at least for square free integers?

our professor told us a week ago about the type theory. I started to think about it a little bit. So if we predict, that this theory (or one of them) could be used to proof theorems on a computer, then there seems to me a little problem. How do these theories solve this: Let us assume there is a proof, which says of itself, that it can't be proved by a computer. Now if we let the computer proof our unsolvable problem, what will it do? If the computer could prove, that he cannot prove the proof, then this is obviously a paradox and in any other case it can't solve it, so the idea holds for this concept. So my question is, how these theories deal with such a concept? Thanks for your advice :)

There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(\mathbb{Z}^d)$ such as $$ -\Delta+\lambda V $$ where $\Delta$ is the discrete lattice Laplacian and the potential $V$ is random say given by a vector $(V_{\mathbf{x}})_{\mathbf{x}\in\mathbb{Z}^d}$ of iid standard normal variables. The constant $\lambda$ is the strength of disorder.

Did anyone study similar random operators $$ (-\Delta)^{\alpha}+\lambda V $$ with a fractional Laplacian?

I am in particular interested in references from the physics literature which provide some heuristics, e.g., a scaling theory à la Abrahams et al.

Given the vectors x1 = [2 1 2], x2 = [1 -1 –2], x3 = [1 1 1]. Test if these are linearly independent (li). If they are li, write x3 as a linear sum of x1 and x2. Is such a linear sum unique? Justify your answer.

A *quandle* is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold.

Q1. a $\star$ a = a

Q2. (a $\star$ b) $\bar\star$ b = (a $\bar\star$ b) $\star$ b = a

Q3. (a $\star$ b) $\star$ c = (a $\star$ c) $\star$ (b $\star$ c)

When we drop out the first axiom we obtain a *rack*, by definition. Quandles generalize basic properties of the conjugation in a group (where $a \star b = b^{-1}ab$ and $a\ \bar\star \ b = bab^{−1}$), but they are also useful in knot theory.

Nevertheless, I'm interested mainly in the homotopy theory of these objects. In fact, following this article by Eisermann, we can define arrows

$a \xrightarrow{b}c$ for each triple $a,b,c \in Q$ with $a \star b = c$.

$a' \xleftarrow{b'}c'$ for each triple $a',b',c' \in Q$ with $a'\ \bar\star \ b' = c'$.

Then we have a notion of homotopy, built in the following way (see the article for details).

First define a *combinatorial path* between two elements $q,q'\in Q$ to be a sequence of arrows going in both ways, such that the first arrow is given by the action of an element of the quandle on $q$ and the last is given by the action of another one on $q'$.

**Definition 1** Let $P(Q)$ be the category having as objects the elements $q\in Q$ and as morphisms from $q$ to $q'$ the set of combinatorial paths from $q$ to $q'$. Composition is given by juxtaposition:
$$(a_0 \to \cdots \to a_m) \circ (a_m \to \cdots \to a_n) = (a_0 \to \cdots \to a_m \to \cdots \to a_n).$$

Then we can construct an homotopy as in the following definition.

**Definition 2** Two combinatorial paths are *homotopic* if they can be transformed one into the other by a sequence of the following local moves and their inverses:

(H1) $a\xrightarrow{a}a$ is replaced by $a$, or $a\xleftarrow{a}a$ is replaced by $a$.

(H2) $a\xrightarrow{b}a \star b\xleftarrow{b}a$ is replaced by $a$, or $a\xleftarrow{b}a \ \bar\star \ b \xrightarrow{b}a$ is replaced by $a$.

(H3) $a\xrightarrow{b}a \star b\xrightarrow{c}(a \star b) \star c$ is replaced by $a\xrightarrow{c} a \star c \xrightarrow{b\star c} (a \star c) \star (b \star c) $

It seems to me that this data can be extended to a simplicial set, whose 0-simplices are elements of $Q$, 1-simplices are arrows between them and higher simplices witness these homotopical information.

My question is

Does $P(Q)$ embed in a simplicial set which keeps track of these information? Is it possible that this simplicial set is actually an $\infty$-category having $P(Q)$ as homotopy category? Is there an analogous construction for racks?

For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition.

The Giambelli identity states that if $a_1 > \dots > a_r$ and $b_1 > \dots > b_r$ are natural numbers with $b_1 \leq n-1$ then the determinant in the representation ring of the matrix $M$ with entries $M_{ij} = V_{ (a_i |b_j)} $ is an irreducible representation of $GL_n$ with highest weight vector $$( a_1 + 1,\dots, a_r +r, r^{b_r}, (r-1)^{b_{r-1} -b_r-1}, \dots, 1^{ b_1 - b_{2}-1} , 0 ^{n-1-b_1}).$$

Technically, the Giambelli identity is an identity in the ring of symmetric functions in infinitely many variables. We deduce this using the fact that the representation ring of $GL_n$ is the quotient ring of symmetric functions in $n$ variables, where each Schur function is sent to an irreducible representation or zero depending on the number of parts.

Let $a_1 > \dots a_r, b_1> \dots > b_{2r}, c_1> \dots c_r$ be natural numbers with $b_1 \leq n-1$.

Let $M$ be the matrix whose entries are $M_{ij} = V_{(a_i|b_j)}$ if $1 \leq i \leq r$ and $M_{ij} = V_{( c_{2r-i} | n-1-b_j )}^\vee$ if $ r+1 \leq i \leq 2r$.

Is $\det M$ an irreducible representation? Is it the one with highest weight vector $$(a_1+1,\dots, a+r+r, r^{b_{2r} }, (r-1)^{b_{2r-1}-b_{2r} -1},\dots, (-r+1)^{ b_1-b_2 -1}, (-r)^{n-1-b_{1}}, -c_r-r, \dots, -c_1-1)?$$

If not, does some similar formula hold for the class of this irreducible representation in the representation ring?

The motivation is that this would help generalize my work in arXiv:1810.01303 on the CFKRS conjecture for integral moments, from the moments conjectures to the ratios conjecture.

It doesn't seem possible to deduce this from any nice identity purely in the ring of symmetric functions, so the same approach might not work here.

We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1^2,x_1,x_2,x_2^2,x_3,x_4,x_3x_4\}$$ or from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\mathbb Z^4$ with $|x_1|<X_1$, $|x_2|<X_2$, $|x_3|<X_3$ and $|x_4|<X_4$. We see that $x_1,x_2$ and $x_3,x_4$ variables do not mix.

It seems the set of $x_1,x_2$ variables in first set satisfy the generalized lower triangle bound on page $16$ http://www.cits.rub.de/imperia/md/content/may/paper/jochemszmay.pdf and the overall set of variables $x_1,x_2,x_3,x_4$ also satisfy the generalized lower triangle bound in both sets.

In here $\lambda_1=\lambda_2=\lambda_3=\lambda_4=2$ and $D=1$.

Assume we have the additional condition that for a given $(x_1,x_2)\in\mathbb Z^2$ with $|x_1|<X_1$ and $|x_2|<X_2$ we have that there is an unique $(x_3,x_4)\in\mathbb Z^2$ with $f(x_1,x_2,x_3,x_4)=0$, $|x_3|<X_3$ and $|x_4|<X_4$ vice versa for a given $(x_3,x_4)\in\mathbb Z^2$ with $|x_3|<X_3$ and $|x_4|<X_4$ we have that there is an unique $(x_1,x_2)\in\mathbb Z^2$ with $f(x_1,x_2,x_3,x_4)=0$, $|x_1|<X_1$ and $|x_2|<X_2$. $W$ is highest absolute value of coefficient of $f(x_1X_1,x_2X_2,x_3X_3,x_4X_4)$.

In this situation can the bound of $X_1^{\lambda_1}X_2^{\lambda_2}X_3^{\lambda_3}X_4^{\lambda_4}\leq W^\frac1D$ be improved to $$\max(X_1^{\lambda_1}X_2^{\lambda_2},X_3^{\lambda_3}X_4^{\lambda_4})\leq W^\frac1D$$ or may be at least $$\max(X_1^{\lambda_1}X_2^{\lambda_2},X_3^{\lambda_3/2}X_4^{\lambda_4/2})\leq W^\frac2{3D}$$ ($\lambda_3/2$ and $\lambda_4/2$ is based on guess that variables are disjoint and have separate control and $x_3,x_4$ do not satisfy generalized triangle bound with $\lambda_3=\lambda_4=1$ and assuming $x_1,x_2$ variables were not present in given polynomial will give $W^{\frac2{3D}}$ bound)?

If not what is the best we can do at least for the case $X_1=X_2=X_3=X_4$?

Consider the cluster algebras $A_n$ and $D_n$. Choose any cluster $x$, is there an explicit formula that express all other cluster variables in terms of $x$?

Suppose that $X\sim \text{Bin}(n,\theta)$. Note that $X$ is the sum of $n$ $iid$ Bernoulli($\theta$) random variables. By the local limit theorem (Theorem 7 here) for the sum of discrete random variables, $$ P(X=t)=\frac{1}{\sqrt{2\pi n\theta(1-\theta)}}\exp\left(-\frac{(t-n\theta)^2}{2n\theta(1-\theta)} \right)+o(n^{-1/2}) $$ for all $n\geq 1$ and uniformly in the integers $t$.

Suppose $t=n\theta+\sqrt{2\theta(1-\theta)n\log m+O(1)}$. I am interested in the relationship between $n$ and $m$, where we can assume, $m>>n>>1$. For example, in my application, $m\approx 20000$ and $n\approx 200$ seems to work well. Intuitively, $m$ should grow much faster than $n$.

I'm interested in finding a theoretical relationship between $n$ and $m$ such that quantity $mP(X=t)=O(1)$. Now if the $o(n^{-1/2})$ remainder were not there, then I can reason the following,

\begin{align*} mP(X=t)&=O(mn^{-1/2}\exp(-O(\log m)))\\ &=O\left(\exp\left(\log m-\log n^{1/2}-O(\log m)\right)\right)\\ &= O\left(\exp\left(\log m^{r}-\log n^{1/2}\right)\right)\text{ for some constant $r>0$}\\ &= O\left(\frac{m^{r}}{n^{1/2}}\right) \end{align*}

This suggests that $m\leq n^\gamma$, where $\gamma = \frac{1}{2r}>0$ for $r>0$ can be a reasonable relationship.

How do I handle that remainder $o(n^{-1/2})$?

Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the ** Clifford algebra** $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ideal generated by $v\otimes v + b(v,v) \mathbb{1}$ for all $v \in V$. $Cl(V,b)$ is a $\mathbb{Z}_2$-graded unital real associative algebra.

Every such $(V,b)$ is isomorphic to $\mathbb{R}^{r+s+t}$ with bilinear form defined by polarisation from the quadratic form $$ q(x) = \sum_{i=1}^{r + s + t} \varepsilon_i x_i^2, $$ where $$ \varepsilon_i = \begin{cases} 0 & i = 1,\dots,r\\ 1 & i = r+1,\dots,r+s\\ -1 & i = r+s+1, \dots, r+s+t .\end{cases} $$ Let $Cl(r,s,t)$ denote the Clifford algebra of $\mathbb{R}^{r+s+t}$ and the above bilinear form.

As $\mathbb{Z}_2$-graded unital real associative algebras, $$ Cl(r,s,t) \cong \Lambda \mathbb{R}^r \hat\otimes Cl(s,t), $$ where $\hat\otimes$ is the $\mathbb{Z}_2$-graded tensor product and where $Cl(s,t):= Cl(0,s,t)$ are the standard Clifford algebras associated to non-degenerate bilinear forms.

The representations of $Cl(s,t)$ are well-known: there are either one or two simple modules (up to isomorphism) depending on $s,t$ and every finite-dimensional module is a direct sum of simples.

I am interested in the representations of $Cl(r,s,t)$ for $r=1$, but more generally for $r>0$.

For $(s,t) = (1,0)$ and $(0,1)$, it is easy to work this out "by hand". The resulting category of representations is no longer semisimple, but it is not hard to show that any finite-dimensional module is a direct sum of indecomposable (but not simple) modules.

But before attempting to study the case of general $(s,t)$, I wonder whether there is some technology out there which can be brought to bear on this problem.

More concretely, I have a couple of

**Questions**

Would a knowledge of the indecomposable modules of $\Lambda \mathbb{R}$ and $Cl(s,t)$ be sufficient to determine the indecomposable modules of their $\mathbb{Z}_2$-graded tensor product? If so, how?

Is there a classification of indecomposable finite-dimensional modules of the exterior algebra $\Lambda \mathbb{R}^r$ for $r>1$? If so, where?

The following theorem is commonly attributed to Hadamard.

Assume $\Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $\Sigma$ is embedded and bounds a convex set.

Many authors refer to Hadamard's *Sur certaines propriétés des trajectoires en Dynamique* (1897)
(for example, J.J.Stoker in his *Über die Gestalt der positiv...* (1936)).

Likely the statement is there, but the paper is long, it is in French and often the statements are not clearly marked; I was searching for it for several days. I asked a friend and she said that it was there 20 years ago, but she could not find it; she also said that it was not easy to extract it from what is written ( = one has to think). [For sure the word *immersion* is not there.]

I hope someone here knows this paper and can help me.

Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(x_i) \cdot n =0$, $i=1,2,3$, where $n$ is a vector normal to the plane $P$?

Suppose $G$ is a group acting freely on a topological space $X$. Take $g$ is an element of $G$. Let $g^*_\mathbb Z$ and $g^*_{\mathbb Z _2}$ are the corresponding induce map on cohomology with $\mathbb Z$ -coefficient and $\mathbb Z _2$-coefficient respectively, i.e., $g^*_\mathbb Z:H^*(X;\mathbb Z ) \to H^*(X;\mathbb Z ) $ and $g^*_{\mathbb Z _2}: H^*(X;\mathbb Z _2 ) \to H^*(X;\mathbb Z _2) $. Suppose $g^*_\mathbb Z$ is the trivial map. My question is: If $g^*_\mathbb Z$ is the trivial map then does it follow that $g^*_{\mathbb Z _2}$ is also trivial? If not under which condition $g^*_\mathbb Z$ is trivial will imply $g^*_{\mathbb Z _2}$ is also trivial?

Let $G<\mathrm{GL_n}$ be a simple linear algebraic group defined over a finite field $K$. Let $\mathfrak{g}$ be its Lie algebra. Assume $\mathfrak{g}$ is simple.

Is it necessarily the case that there is no subspace $\mathfrak{v}\subset \mathfrak{g}$ with $0<\dim(\mathfrak{v})<\dim(\mathfrak{g})$ such that $\mathfrak{v}$ is invariant under $\mathrm{Ad}_g$ for every $g\in G(K)$?

Note: it is clear that there is no $\mathfrak{v}\subset \mathfrak{g}$ with $0<\dim(\mathfrak{v})<\dim(\mathfrak{g})$ such that $\mathfrak{v}$ is invariant under $\mathrm{Ad}$ for every $g\in G(\overline{K})$. It is also clear that the answer to the question above is "yes" when the number of elements of $K$ is larger than a constant depending only on $n$: since $\mathfrak{v}$ is not an ideal, there is a $v\in\mathfrak{v}$ such that all $g\in G(\overline{K})$ such that $\mathrm{Ad}_g(v)\in \mathfrak{v}$ lie in a proper subvariety of $G$.

Note 2: A friend has just proposed over the breakfast table that there are linear algebraic groups with no non-trivial rational points over $K$. That would obviously imply an answer of "no" to my question. I am not convinced that such a thing is really possible, at least not when we are talking about the group $G(K)$, $G$ simple (as opposed to more exotic groups of Lie type).

I am currently reading the paper *Deformation Spaces Associated to Hyperbolic Manifolds* by Johnson and Millson, Section 7, and the highlighted bit below has been giving me difficulty:

Specifically, how do we define this ideal $\frak{a}$ that allows us to get this group $\Gamma=\Gamma(\frak{a})$? I have tried following up the reference to *Geometric construction of cohomology for arithmetic groups* by Millson and Raghunathan, but unfortunately despite my best efforts I have really struggled to follow the latter paper and extract anything useful, since I am not only new to hyperbolic geometry, but also completely unfamiliar with number theory.

If someone could concretely describe how to define such an ideal and the group $\Gamma$, I would really appreciate it!