I have a question if it possible, Let X a tangantial vector field of a riemannian manifolds M, and f a smooth function define on M. Is it true that X(exp-f)=-exp(-f).X(f) And div( exp(-f).X)=exp(-f)〈gradf, X〉+exp(-f)div(X)? Thank you

Let $\mathcal{K}$ be a category.

**Prop 1.** If $\mathcal{K}$ has a (strong) generator then there is a faithful (and conservative) functor $U: \mathcal{K} \to \text{Set}$ preserving connected limits.

**Def 2.** *A virtual (strong) generator* on $\mathcal{K}$ is a a faithful (and conservative) functor $U: \mathcal{K} \to \text{Set}$ preserving connected limits.

Generators are a useful notion, and even more important, it occurs quite often that a category of interest has a generator. In my research, sometimes I study the properties of those categories having a faithful functor to set, that's how I came up with this definition.

**Q1.** Is there any known example of a category with a virtual generator which has no real generator?

**Q2.** How well behaved is the notion of virtual generator? To answer this question, do you have any testing lemma? For example, I was thinking about the following.

**Prop 3.** Let $L: \mathcal{K} \rightleftarrows \mathcal{C}: R$ be an adjunction where $R$ is faithful (conserative). Then if $\mathcal{K}$ has a (strong) generator, also $\mathcal{C}$ has a (strong) generator.

**Q3.** Is the virtual analogous of this statement true?

I would like to give a better justification of my definition with the following Proposition.

**Prop 4.** Let $\mathcal{K}$ be a complete category. Then the following are equivalent:

- $\mathcal{K}$ has a generator.
- $\mathcal{K}$ has a virtual generator verifying the solution set condition.

It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.

If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$

**Question 1**:
Then do we have
$$\Omega_d^{spin}(BG)_p = ko_d(BG)_p?$$ for $p=2$ and free part, for $d\le 7$? (how about higher $d>7$?)

And $$ \Omega_d^{spin}(BG)_p = \Omega_d^{SO}(BG)? $$ for $p \neq 2$ and $p$ is an odd prime?

Namely, the 2-torsion and free part of $Mspin$ and $KO$ is the same. If there is an odd $p$ torsion, we need to consider localization at odd prime by $MSO$ cohomology. Is this correct?

**Question 2**:
If this is a statement about the spectra, not just about stable homotopy groups, and thus within these spin cobordism and ko theory, do they completely coincide for any dimensions $d$, instead of just $d \leq 7$?

This is probably a simple-minded question, but I haven't been able to prove it or find a counterexample. This old question seems to dance around my question, but I don't think any of the answers address exactly my situation. (Please tell me if I am wrong!)

Suppose $\varphi: X\to Y$ is a surjective morphism of algebraic varieties (reduced, irreducible, separated schemes, finite type over an algebraically closed field), and furthermore assume that $Y$ is smooth. Also, I can assume that $\varphi$ is finitely presented.

If the fiber $X_y$ is smooth and equidimensional of dim $n$ for any $y\in Y$, is the morphism flat? I know that equidimensionality alone does not mean flat, but I wonder if the smoothness assumptions are enough. Obviously, I want to conclude that $X$ is smooth and this is enough.

The equidimensionality assumption rules out the blow-up examples in the above cite problem, and the normalization of a node on a curve is ruled out by smoothness of $Y$. I would be happy with a counterexample though.

Let $G$ be a hyperbolic group with a fixed (finite, symmetric) generating set and suppose that $\varphi : G \to \mathbb{R}$ is a group homomorphism. Write $W_n = \{ g \in G: |g|=n\}$, where $|g|$ denotes the word length of $g \in G$.

M. Coornaert proved that the growth of $W_n$ is purely exponential i.e there exists $\lambda, C>1$ such that $$\frac{1}{C}\lambda^n \le \#W_n \le C \lambda^n$$ for all $n \in \mathbb{Z}_{\ge 0}$ and where $\#W_n$ denotes the cardinality of $W_n$.

Fix $M \in \mathbb{R}$. I'm interested in the growth rate of the quantity $$\#(\varphi^{-1}([-M,M]) \cap W_n)$$ and in particular how this compares to the growth rate of $\#W_n$. It seems plausible to me that the above quanitity takes up an asymptotically vanishing proportion of $\#W_n$ i.e $$ \frac{\#(\varphi^{-1}([-M,M]) \cap W_n)}{\#W_n} \rightarrow 0.$$

However, I am unable to prove this or to provide a counterexample. Is anything known about this problem?

Any thoughts and/or suggestions would be greatly appreciated. Many thanks.

Let $ 1<a<b$ and denote by $P_{p}(\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$ with finite p-th order moment, endowed with the Wasserstein topology.

If $(\mu_{n})$ satisfies a large deviation principle on $P_{p}(\mathbb{R}^{d})$ with good rate function $f$ for all $p\in[a,b)$, prove that the principle can be extended to any index in $[1,b)$ .

I would like to write my idea and tell me if it's ok.

My approach is to prove that the principle also holds for $p^{'} \in [1,a)$. Consider the identity map $ I : P_{p}(\mathbb{R}^{d}) \longrightarrow P_{p^{'}}(\mathbb{R}^{d})$, and since convergence in $P_{p}(\mathbb{R}^{d})$ is stronger this map is continuous. So the contraction principle yields that the principle holds for $p^{'} \in [1,a)$.

Let $S$ denote the set of natural numbers $m$ with the property that for all prime powers $p^k || m$ we have $k \equiv 1 \pmod{2}$.

What is the asymptotic density of $S$?

Note that $S$ contains all prime numbers and more generally, all square-free numbers, so that $\liminf_{X \rightarrow \infty} \frac{\# (S \cap [1,X])}{X} \geq \frac{6}{\pi^2}$.

Is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities?

This is in connection with a quote from someone on the web that I saw a long time ago. At that time I had contacted the author, but they chose not to answer.

The quote:

In some sense, the space of all possible linear PDE's can be viewed as a singular algebraic variety, where Hormander's theory applies only to generic (smooth) points and the most interesting and heavily studied PDE's all lie in a lower-dimensional subvariety and mostly in the singular set of the variety.

Any pointers/refs on any of the points made in the quote would be gratefully received...

At the right column of the page 654 of the paper, R. Palais, The Visualization of Mathematics: Towards a mathematical Exploratorium, Notice AMS it is written "There can be no eversion of a circle (or any odd dimensional spheres)".

In that paper the eversion is defined as a regular homotopy between identity and the antipodal map.

I wonder what is my mistake to think that $f_t(z)=e^{it}z, \quad t\in [0, \pi]$ is a regular homotopy between the identity and the antipodal map on the circle?

Dirichlet Pigeonhole says given prime $p$ and vector $(v_1,v_2,\dots,v_n)\in\mathbb Z^n$ there is an integer $m$ such that the vector $m(v_1,v_2,\dots,v_n)\bmod p$ lies in box $[-p^{(n-1)/n}-1,p^{(n-1)/n}+1]$.

Is it true that if $p$ is a prime and $(v_1,v_2,\dots,v_n)=(a_1,b_1)\otimes(a_2,b_2)\otimes\dots\otimes(a_t,b_t)$ (note $n=2^t$) where each pair $a_i,b_j$ is coprime, each pair $a_i,a_j$ is coprime and each pair $b_i,b_j$ is coprime holds where $p^{1/n}+1<a_i,b_j<2p^{1/n}$ then there is an $m\in\mathbb Z$ such that the vector $$m(v_1,v_2,\dots,v_n)\equiv(r_1,\dots,r_n)\bmod p$$ as vector in $(r_1,\dots,r_n)\in\mathbb Z^n$

is in box $[-p^{(n-1)/n}-1,p^{(n-1)/n}+1]$ (usual pigeonhole)

is independent of $(v_1,v_2,\dots,v_n)$ over $\mathbb R$?

Through exponential sums we might possibly show box with length at most $p^{1-\frac{t}{n^2}}$ suffices to contain an independent $(r_1,\dots,r_n)\in\mathbb Z$. Dirichlet box principle says $(r_1,\dots,r_n)\in\mathbb Z$ can lie in as small as $p^{1-\frac{1}{n}}$ length box.

If we can show the Dirichlet box also contains an independent vector the gap can be closed. Is it possible to close the gap here? Note that the Dirichlet box bound is tight at $t=1$. Is it tight at general $t\in\mathbb N$?

I'm having trouble understanding why the "essential image" is defined the way it is.

The nlab article gives the following definition:

(A concrete realization of) the essential image of a functor $F: A\to B$ between categories or $n$-categories is the smallest replete subcategory of the target $n$-category $B$ containing the image of $F$

This obviously breaks the principle of equivalence as stated later in the article.

When I think about how to define the essential image, the first thing that comes to mind is if we remember the identification of the functor as a new functor from the source to the essential image - then we don't have to break equivalence:

Say: The "essential image" of a functor $F:A\to B$ is the initial object in the category of triples $(D, i:D\hookrightarrow B, F':A \to D)$ such that $F= i\circ F'$ (with morphisms the same as in $Cat/B$ that also require post composing them with $F'$ of the domain gives the $F'$ of the codomain)

It exists as the classical image of $F$ satisfies the universal property.

All the different choices of an initial object form a contractible groupoid whose skeleton yields a subcategory of the classical essential image, as the classical image satisfies the universal property and is contained in the classical essential image (but this is not obvious these are equivalent, I suspect in general they are not, but came short of examples).

**Questions:**

- How large is the difference between this definition and the classical one? To break this down a little:

a. For what categories we have different subcategories of the two constructions (I believe this highly depends on AC to build pathological cases)

b. (Aside) Is this difference between the definitions equivalent to AC?

- Which definition behaves better (this is vague in purpose as there are many interpretations to the question that give different points of view on "which" definition is "the right one").

An argument for this new one is that this definition respects the principle of equivalence.

An argument for the classical definition is simplicity - the classical definition is a subcategory and not an equivalence class of subcategories equipped with extra information)?

Other arguments I can think about can come from higher categorial properties:

For example, gluing essential images (in both definitions) of different functors into a pseudofunctor - when can you do this?

Given a finite Borel measure $\mu$ on $\mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$, define its Fourier coefficients by

$$ \hat\mu(n) = \int e^{2i\pi nx} d\mu(x) \qquad\forall n\in \mathbb{Z}.$$

Clearly, $(\hat\mu(n))_n$ is bounded.

- What sufficient conditions on a bounded sequence $(a_n)_n$ are known that ensure that there is a finite measure with $\hat\mu(n)=a_n$ ?
- same with probability measures instead of finite ones;
- same with necessary conditions.

I would guess that characterizations are out of reach, but maybe I am wrong?

**Added in Edit:** Yemon Choi rightfully asks what kind of sufficient or necessary condition I am after. Any is good for my culture, but I am especially interested in sufficient condition that enable one to construct measure satisfying constraints on Fourier coefficient.
To be honest, one of my goals was to understand why it is not easy to disprove Furstenberg's $\times 2$, $\times 3$ conjecture by simply picking Fourier coefficients $(c_n)_n$ such that $c_{2^p3^qm}=c_m$ (and $c_0=1$) inside the set of Fourier series of probability measures. I think I am starting to get the point.

What are the advantages of a hyperbolic program over a semi definite program? SDPs can be used to represent a wide variety of algebraic constraints. Are there constraints that can be represented in a hyperbolic program but not a semi definite program?

Is there a reference that describes an application or applications of hyperbolic programming?

Has anyone developed a hyperbolic programming solver? If so, is there a reference describing its implementation?

We know that for any continuous function $f:S^1\to S^1$ there are many continuous functions like $g:S^1\to S^1$ with a fixed point which $f$ and $g$ are homotopic(why?). Can we conclude somehow from this to answer the question below?

Call $\mathscr{S}:S^1\to S^1$ a stretching rotational $(SR)$ continuous function when mapping a circle points by continuously expansion, contraction and rotation them to the circle.

Is it true to say every continuous function $\mathscr{C}:S^1\to S^1$ have a fixed point or is $SR$ or a combination of them?

It is well know that if $\lambda, \mu >0$ are the two leading eigenvalues of two $s \times s$ matrices $A$ and $B$ respectively then \begin{align} \lambda - \mu \leq \left(\|A\| + \|B\|\right)^{1- \frac{1}{s}}\|A-B\|^{\frac{1}{s}} ----------(1) \end{align} if $\lambda > \mu$. This is called spectral variation bound.

Now, assume the following: (A1) $A$ is irreducible and non-negative

(A2) $A$ is invertible

(A3) $A$ is positive semidefinite

(A4) $\min_i \sum_{j} a_{ij} \min_i \sum_j b_{ij} \geq \pi_{i=1}^{s} a_{ii}$

Then one can show that $$\lambda \mu - \lambda \alpha(A) \|A-B\| \leq det(A)-------------(2)$$

Now if $A = (a_{ij})_{s \times s}$ with $a_{ij} = p$ if $i=j$ and $a_{ij} =q$ otherwise and $b_{ij}=q$ for all $i,j$ with $p > q$. Then for large $s$, I can show that my bound is better than the spectral variation bound if $1/3 \leq q < 1$. Now, spectral variation bound is a classic result. Therefore I sense that something may be wrong. Not able to figure it out.

Let $(R,\mathfrak m)$ be a Noetherian , local UFD of Krull dimension $3$ .

If every $\mathfrak m$-primary ideal $J$ (i.e. $\sqrt J=\mathfrak m$) with $J^2=\mathfrak mJ$ satisfies $J^2=\mathfrak m^2$, then is it true that $R$ is regular (https://en.wikipedia.org/wiki/Regular_local_ring) i.e. $\mu(\mathfrak m)=\dim_k(\mathfrak m/\mathfrak m^2)$ is $3$ (where $k=R/\mathfrak m$) ?

Certainly $\mu (\mathfrak m)\ne 1$ because a Noetherian local domain with principal maximal ideal is a PID, hence have Krull dimension $1$. Since by Krull's Height Theorem, we have $\mu(\mathfrak m)\le \dim R=3$, so to prove regularity of $R$ (if that is at all true ) we only need to show $\mu(\mathfrak m)\ne 2$ ...

Let $G$ be a connected reductive group split over a field $k$. Let $T$ be a maximal split torus of $G$. Consider $N_G(T)$, the normalizer of $T$ in $G$, we have $N_G(T)/T \cong W$, the Weyl group of $G$. Assume $|k|$ is large enough and consider $N_G(T)(k)$, the $k$-points of $N_G(T)$. Given $w \in W$, does there exist a regular semisimple element (w.r.t. $G$) $n \in N_G(T)(k)$ whose image in $W$ is $w$? It is easy to see this for classical groups by some matrix calculations but I wonder if there is a nice, uniform way to prove this in the positive.

- The Arf invariant is a nonsingular quadratic form over a field of characteristic 2.

The form that I looked at was: $$ S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;i\; q(x)] $$ see Kirby-Taylor, Pin structures on low-dimensional manifolds.

The $M^2$ is an oriented 2 dimensional manifold with spin structures that has $\mathbb{Z}_2$ valued quadratic forms on $H_1(M^2,\mathbb{Z}_2)$, which obeys $$q(x + y) = q(x) + x ∩ y + q(y) \mod 2,$$ here $x ∩ y$ denotes the $\mathbb{Z}_2$ intersection pairing. The bordism invariant is the Arf invariant.

- Stiefel-Whitney class is $\mathbb{Z}_2$-characteristic class associated to real vector bundles. For example, we can take the $w_i(TM)$ for the tangent bundle of the base manifold $M$.

**Question**:

Is there some precise way to pair the Arf invariant $S(q)$ (or any

**variation**of Arf) as a $\mathbb{Z}_2$-cohomology class $\text{arf}$ with Stiefel-Whitney class $w_i(TM)$? How would one write and define them formally as a topological invariant? Like $$ \text{arf} \cup w_{d-2}(TM)? $$ where the $M$ is a $d$-dimensional manifold? How do one cup product this two objects formally?Must this $\text{arf} \cup w_{d-2}(TM)$ be a $\mathbb{Z}_2$-characteristic class?

What will be a nontrivial manifold generator $M$ for such an invariant $\text{arf} \cup w_{d-2}(TM)$?

How to view the Poincare dual PD for the 2 and $d-2$ submanifolds from $\text{arf}$ or $w_{d-2}(TM)$?

This is a variation of the problem from a previous Math.SE post which receives 0 feedback.

Is there any known or interesting relation between critical points (possibly degenerate, or maybe only nondegenerate) of a function on a manifold and generators/relations of high homotopy groups? I only know about the relation to representations of a fundamental group....

I am wondering about the equivalence of two notions of a "nilpotent orbit".

The first notion, which I am familiar with, is as follows: given a lie group $G$ and a lie algebra $\frak{g}$, the orbit of a nilpotent element $n \in \frak{g}$ under the adjoint action of $G$ on $\frak{g}$ is called a $\textit{nilpotent orbit}$.

I encountered a seemingly different notion in Example 4.2 of the following survey on variations of Hodge structures. http://people.math.umass.edu/~cattani/ICTP/cattani_vhs.pdf In this setting, a nilpotent orbit is a certain map from a product of copies of the upper half plane to a certain space of filtrations of a complex vector space.

These two notions don't seem to obviously coincide, but since they share the same name, I would hope that they are related.

Is this the case?