I am reading a book titled "Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group" by J.P. Murre. I am in the chapter 4 titled "Fundamental groups". Here he fixes a base locally noetherian scheme $S$ and defines a category $\mathscr{C}$ taking the objects to be locally noetherian schemes with finite etale maps to $S$. Morphisms with in the objects are finite etale maps between them.

At the beginning of this chapter he is stating the basic properties of this category, like existence of fibre product and disjoint union. I am reading the proof of the existence of the quotient by a finite subgroup $\mathfrak{g}$ of the automorphisms of an object $X$. (I am unable to write proper notation for $\mathfrak{g}$, because I do not know what it is. What is the notation?) He states that the problem is enough to be solved locally and goes on to define a finite subgroup of automorphisms of $\operatorname{Spec}(A)$ corresponding to $\mathfrak{g}$. I am unable to understand what this means. It is not even clear to me why the problem is enough to be solved for affine open subschemes. I have attached the snapshot below.

Let $\omega$ be a primitive complex $n^{th}$ root of unity. I am interested in the following quantity $$ \max_{f(n)\leq \ell \leq g(n)} \quad \min_{0<k\leq n-1} \left| 1+\omega^k+\omega^{2k}+\cdots+\omega^{(\ell-1)k}\right|^2, $$ as $n$ goes to infinity. Note that the optimization simplifies to $$ \max_{f(n)\leq \ell \leq g(n)} \quad \min_{0<k\leq n-1} \left| \frac{\sin (\pi \ell k/n)}{\sin (\pi k/n)}\right|^2, $$ and for concreteness case I don't mind taking $f(n)=\log^2 n, g(n)=\sqrt{n}-$not the iterated log but square of log.

I know the minimum will in general can be very small (at least for arbitrary sums of roots of unity as in question by Terry Tao) but what if we are free to look at multiple lengths for the sum, and take consecutive powers, things must improve, but how much?

**Edit:** One can take $n$ prime, if it helps.

For $X = (x_1^T,\ldots,x_N^T)^T \in \mathbb{R}^{Nm \times 1}$, where $x_i \in \mathbb{R}^{m \times 1}$ for $i \in \{1,\ldots,N\}$, $A \in \mathbb{R}^{r \times Nm}$, and $r \geq Nm$, I want to obtain a closed form solution of

$$\max_{X \in \mathbb{R}^{Nm}}\{\|AX\|: \|x_i\|^2=1\}$$

where $\|\cdot\|$ denotes the Euclidean norm. I know how to find a closed form solution of

$$\max_{X \in \mathbb{R}^{Nm}}\{\|AX\|: \|X\|^2=1\}$$

that is, when $\|X\|^2=1$. However, I do not see how to solve it when we have $\|x_i\|^2=1$.

A subset $B$ of a metric space $(M,d)$ is called a **metric generating set** if and only if $$[\forall b \in B, d(x,b)=d(y,b)] \implies x = y \,. $$ A metric generating set $B$ is called a **metric basis** if it has minimal cardinality.

The **metric dimension** of $(M,d)$ is the cardinality of any metric basis.

**Question:** For which metric spaces is metric dimension well-defined? When can we be sure that any metric basis for a metric space has the same cardinality?

Sufficient criteria will suffice for answers, as will necessary criteria, although of course the holy grail of answers would be a non-trivial necessary and sufficient criterion.

**Note:** This is a follow-up to my previous question. There, the accepted answer pointed out that the notion of metric dimension does not make sense in arbitrary metric spaces.

In a matroid, any basis has the same cardinality, but there are metric spaces with metric generating sets of minimal yet non-equal cardinalities.

Nevertheless, it does seem possible that metric dimension may make sense for certain classes of metric spaces, e.g. Euclidean spaces (Murphy, *A Metric Basis Characterization of Euclidean Space*, 1975) or graphs (Ramirez-Cruz, Oellermann, Rodriguez-Velazquez, *The Simultaneous Metric Dimension of Graph Families*, 2015). It is unclear to me what property common to these two types of metric spaces allows the definition to be well-formed/well-defined for them.

In the case of Euclidean spaces, it seems intuitively clear that this notion should be related to that of affine independence, but coordinate-free definitions of affine independence (solely in terms of the metric) are rare (e.g. section 2.6 here), so I am still working on the algebra to show the connection.

Let $T^*_{\mathbb{C}}(Gr_{n,r})$ denote the cotangent space of the Grassmannian of $r$-planes in $\mathbb{C}^n$. Moreover, let $\Lambda^\bullet$ denote the exterior algebra of $T^*_{\mathbb{C}}(Gr_{n,r})$. Condsidering $Gr_{n,r}$ as the homogeneous space $U_n/(U_r \times U_{n-r})$, we have a unique representation of $U_n/(U_r \times U_{n-r})$ on $\Lambda^{\bullet}$ for which the associated homogeneous vector bundle is the direct sum $\bigoplus_{k \in \mathbb{N}} \Omega^k$.

(i) Just as for any homogeneous space, every de Rham cohomology class of $Gr_{n,r}$ has a $G$-invariant representative. Moreover, every $G$-invariant element must be harmonic, and so, gives by Hodge decomposition a cohomology class. Is it correct to conclude from this that the cohomology group $H^\bullet$ is isomorphic as a vector space to the space of $U(r) \times U(n-r)$-invariant elements in $\Lambda^\bullet$?

(ii) With respect to a standard weight basis of $T^*(Gr_{n,r})$, what do the $U(r) \times U(n-r)$-invariant elements look like, and how does this presentation of Schubert calculus relate to the partition presentation given in this question?

Some model theorists / combinatorial geometers like to think about pregeometries (matroids with a weak finiteness condition) associated to first-order theories. But the usual way of constructing a pregeometry is not invariant under biinterpretability in any obvious sense. Is there a suitable notion of equivalence of pregeometries such that biinterpretable theories have equivalent pregeometries?

More precisely, if $T$ is a theory, then by "the pregeometry associated to $T$" I mean the following. Let $U$ be a monster model of $T$, and consider the algebraic closure operator on (the home sort of) $U$. This forms a pregeometry, the one I have in mind. So the question is

**Question:** Is there a notion of equivalence of pregeometries such that the pregeometry associated to a theory in the above sense is invariant under biinterpretability?

But trying to compare the pregeometries of biinterpretable theories is immediately problematic: biinterpretable theories could consider different sorts to be the "home sort". So it's not clear how to even produce a map of geometries (in the obvious sense) from an interpretation. Even if a map is produced, it seems unlikely to be an isomorphism.

Introductions to this topic seem to provide a candidate notion: a "geometry" is a pregeometry such that the closure of a singleton set is itself. Every pregeometry can be quotiented to yield a canonical geometry. But biinterpretable theories need not have isomorphic geometries -- in fact, the quotient seems even more problematic since biinterpretable theories need not agree on what the singletons are. So this doesn't seem to get us anywhere.

I am looking for the bounds on the distance between probability distributions in terms characteristic functions.

For example, I am aware of the following bound \begin{align} d(P,Q) \le \frac{1}{T} \int_{-T}^T \frac{|\phi_P(t)-\phi_Q(t)|}{t} dt+ \frac{q_{max}}{ \pi T} \end{align} where the $d(P,Q)$ is the Kolmogorov-Smirnov distance and $q_{max}$ the maximum value of the pdf of $Q$.

Specifically, am looking for the bonds that would depend on the following difference

\begin{align} | t \phi_P^{\prime}(t)+\phi_P(t)-(t \phi_Q^{\prime}(t)+\phi_Q(t)) | \end{align}

I want to learn mathematics online. Can anyone give me website to learn mathematics from beginner to graduated level online?

Let $F$ and $H$ be normed spaces and let $E$ be a locally convex space. Let $T:F\to H$ and $S:H\to E$ be linear operators, such that $\|T\|= 1$, $S$ is an injective semi-embedding (i.e. $S\overline{B}_{H}$ is closed in $E$), and $ST$ is weakly compact.

Does it follow that $T$ is weakly compact?

The intuition is that if we consider $F^{**}\to ^{T^{**}} H^{**}\to^{S^{**}} E^{**}$, then $S^{**}T^{**}\overline{B}_{F^{**}}$ is equal to the closure of $ST\overline{B}_{F}$ in $E$, due to weak compactness of $ST$. Since $\|T\|=1$, it follows that $ST\overline{B}_{F}\subset S\overline{B}_{H}$, which is closed in $E$, and so $S^{**}T^{**}\overline{B}_{F^{**}}\subset S\overline{B}_{H}$. Hence, perhaps it is possible to show that $T^{**}\overline{B}_{F^{**}}\subset \overline{B}_{H}$, which is equivalent to weak compactness of $T$.

I would like to know how one solves Sturm-Liouville problems on $\mathbb{R}$ numerically for the eigenvalues that are of the form

$$-f''(x)+\frac{1}{\sinh(x)^2}f(x)=\lambda f(x).$$

So even if there is a closed form solution to this problem, I would like to know how to treat singularities like this one.

Consider an irreducible polynomial $f(x) = x^3+3x+1$ over $\mathbb{Z}[x]$. Now, the algebraic factor base consists of all first degree prime ideals represented by a pair $(p,r)$ where $f(r) \cong 0\ ({\rm modulo}\ p)$. In this case, we consider all primes $\leq 10$ and their corresponding root modulo $p$.

So, ${\cal S}= \{ (3,2),(5,1),(5,2),(7,4) \}$.

- How to identify whether a prime is ramified or not and how to detect a $(p,r)$ in set ${\cal S}$ is a first degree prime ideal or not?
- Let $\alpha$ be a complex root of $f(x)$.

We define Norm of first degree ideal as follows:

$N(a-(b(\alpha))) = b^d * f(a/b)$ where $d$ is the degree of the polynomial. Here, ${\cal S}_1= \{ (0,1),(-4,13),(-1,5),(1,1),(-10,1),(9,4) \} $ consists of pairs smooth over Algebraic factor base.

Now, I am getting $\prod\limits_{(a_i,b_i)\in {\cal S}} N(a_i - b_i \alpha )=(180075)^2$ instead of $\prod\limits_{(a_i,b_i)\in {\cal S}} (a_i - b_i \alpha )$, a square, so how to get down from Norm level to product of $(a_i-b_i (\alpha))$ since the objective is latter product should be a square in $\mathbb{Z}[\alpha]$?

There is an algebraic analogue of the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological $K$-theory of a topological space.

For a field $k$, let $X$ be smooth variety $X$ over $k$.

The following spectral sequence will be referred to in the sequel as the *motivic spectral sequence*:
$$E_2^{i,j} := H^{i-j}(X, \mathbf{Z}(-j)) \Rightarrow K_{-i-j}(X).$$

See:

- the Bloch-Lichtenbaum motivic spectral sequence in [BL], and the generalizations by Levine [L] and Friedlander-Suslin [FS] to smooth varieties over $k$.
- the Voevodsky motivic spectral sequence [V].
- the Grayson motivic spectral sequence [G].

For $k$ and $X$ as in the foregoing, we may form the étale hypercohomology of the Bloch complex $z^{j}(X,\bullet)$ ([B]) on $X_{\rm\acute{e}t}$, denoted $H^{\bullet}_{L}(X, \mathbf{Z}(j))$ and usually called *Lichtenbaum cohomology*.

**Questions:**

- Is an "étale analogue" of the motivic spectral sequence from the foregoing, i.e.: $$E_2^{i,j} := H_L^{i-j}(X, \mathbf{Z}(j))\Rightarrow K_{-i-j}^{\rm\acute{e}t}(X)$$ available?
- If the answer to $(1)$ is "yes", what is the currently known generality?
- If the answer to $(1)$ is "yes", references?

**References.**

[BL] S. Bloch, S. Lichtenbaum, *A spectral sequence for motivic cohomology, K-theory*, 1995.

[L] M. Levine, *Techniques of localization in the theory of algebraic cycles*, 2001.

[FS] E. M. Friedlander, A. Suslin, *The spectral sequence relating algebraic K-theory to motivic cohomology*, 2002.

[V] V. Voevodsky, *A possible new approach to the motivic spectral sequence for algebraic K-theory*, 2002.

[G] A. Suslin, *On the Grayson spectral sequence*, 2003.

[B] S. Bloch, *Algebraic cycles and Higher $K$-theory*, 1986.

Let $(\Omega, \Sigma, \mu)$ be a probability space and ${\mathcal B}:=(B(\omega))_{\omega\in \Omega}$ be parameterized measurable sets, i.e. $B(\omega) \in \Sigma$ for all $\omega \in \Omega$. Now one can show that $$K_{\mathcal B}:\Omega \times \Omega \to [0,1],\ (x,y) \mapsto \int_\Omega 1_{B(\omega)}(x) 1_{B(\omega)}(y) d\mu(\omega)$$ defines a reproducing kernel on a Hilbert space $H(K_{\mathcal B})$ (constructed by Moore's theorem).

So we can observe that a system $\mathcal B$ of measurable sets induces a associated reproducing kernel. Now there arises a natural question:

If $K$ is a reproducing kernel can I always find a system $\mathcal B$ of measurable sets such that $K = K_{\mathcal B}$? Obviously that doesn't work, since it is necessary that $$ K(x,y) \in [0,1] \qquad \text{and} \qquad \vert K(x,y) \vert \leq \min\{K(x,x), K(y,y)\}.$$ So my question is if there are already any results that characterize such reproducing kernels? If so I would appreciate some literature on the topic. The problem might be very difficult in general though.

**Definition:** A subgroup $H$ of a group $G$ is said to be pronormal if every $g\in G$, there exists $x\in \langle H, H^g \rangle$ such that $H^x = H^g$ (note: $H^g:= gHg^{-1}$)

Let $G$ be a group and $H \leq G$. Then $H$ prn $G$ if and only if, in every transitive representation of $G$, $N_G(H)$ permutes the symbols left invariant by $H$.

Proof: Suppose that $G$ is represented transitively by permutations of a set $X$. Let $Y$ be the set of all $x\in X$ which are invariant under $H$. If $y\in Y$ and $a\in N_G(H)$, we have $Hy=y$, and so $Hay = aHy = ay$. Hence $ay \in Y$. Thus $N_G(H)$ leaves $Y$ invariant. Let $G_y$ be the stabiliser of $y$ in $G$. If $y$ and $z$ are in $Y$, we have that $gy=z$ for some $g\in G$, since $G$ permutes $X$ transitively. If $h\in H$, then $hy =y$, since $y\in Y$ is invariant under the action of $H$ by hypothesis. This implies that $h \in G_y$, and so $H \leq G_y$. Similarly, $H \leq G_z$. Thus $H \leq G_y \cap G_z$. Consequently, $gHg^{-1} = gG_yg^{-1} = G_z$. If $H$ prn $G$, then $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ and hence in $G_z$ as $\langle H, H^g \rangle \leq G_z$. Therefore there exists $b\in G_z$ such that $H^b = H^g$. This implies that $b^{-1}g \in N_G(H)$. Now $(b^{-1}g)y = b^{-1}(gy) = b^{-1}z = z$, as $b^{-1}\in G_z$ ($b\in G_z$ and $G_z \leq G$). Thus $N_G(H)$ permutes $Y$ transitively whenever $H$ prn $G$.

Conversely, suppose that in any transitive representation of $G$, $N_G(H)$ permutes the symbols left invariant by $H$. Let $g\in G$ and denote $J = \langle H, H^g \rangle$. Then $HJ = J$ and $H^gJ = J$, or equivalently, $Hg^{-1}J = g^{-1}J$. Therefore in the transitive permutation representation of $G$ on the left cosets of $J$ in $G$, the cosets $J$ and $g^{-1}J$ are invariant under the action of $H$. By hypothesis, $J = n(g^{-1}J)$ for some $n\in N_G(H)$. Then $ng^{-1} \in J$, and $(ng^{-1})gHg^{-1}(gn^{-1}) = nHn^{-1} = H$. This shows that $H^g$ and $H$ are conjugate in $J$, and since this is true for all $g\in G$, we deduce that $H$ prn $G$.

Question: Are there any questions one can derive from this result?

Sorry if this might be off-topic or "sketchy".

I really love reading and studying mathematics in my undergraduate and attained top grades in my year in a top university in Canada. But I didn't display any super talent. For example, I did badly on the Putnam and was never able to reach the Putnam fellow state.

I then went on to graduate school, and really struggled through the PhD because I constantly got stuck on my proof and didn't know what to do. I felt it was partly related to my inability to concentrate on a proof for long time.

In my regular coursework, I never have problem concentrating during class or doing homeworks. But in research, I cannot come into the same office every day and concentrate for many hours being stuck on the same problem. I saw an interview of Len Adleman where he mentions that he can concentrate on the same problem for 10+ hours everyday for months at a time.

I feel that maybe if I self-medicated using mild ADHD drugs, then I can improve my concentration and make more breakthroughs in my math career. I don't drink coffee. Maybe that will help. Does anyone have any advice? Sorry if this is long and very personal.

Let $f(x,y)=(ex+fy)(gx+hy); \ x,y,e,f,g,h \in \mathbb{Z}$ be a reducible integral binary quadratic form. Is there a criterion to determine if a number is represented by this form? In particular, does such a criterion exist for if an integral binary quadratic form has square discriminant?

Let $X$ be a smooth projective variety and $W \subset X$ a smooth, projective subvariety. Let $\pi:\tilde{X} \to X$ be the blow-up of $X$ along $W$. Let $E$ be the exceptional divisor of $\pi$ and $i:E \hookrightarrow \tilde{X}$ the natural closed immersion. Let $\alpha \in H^k(\tilde{X})$ such that $i^*\alpha=0$ in $H^k(E)$. Does this imply that $\pi^*\pi_*\alpha=\alpha$?