## complexity theory – Polynomial time verification of Graph Isomorphism problem

Using guess and check method, for two given graphs with the same number of nodes, a NTM selects a permutation of the node set and then checks if the edges are preserved.

The nondeterministic selection of a permutation of the nodes is done in polynomial time.

How is the choice of a function done in polynomial time? This is not clear to me.

## ct.category theory – Isomorphism of modules over an algebra in a monoidal category \$mathcal{C}\$

What requirements are needed on $$mathcal{C}$$ for us to have left $$A$$-modules in $$mathcal{C}$$ be isomorphic to right $$A$$-modules in $$mathcal{C}^text{op}$$?

If $$mathcal{C}$$ is rigid, then every left $$A$$-module $$(M, p)$$ is isomorphic to a right $$A$$-module $$(M^*, q)$$, where $$q : M^* otimes A rightarrow M^*$$ is given by the image of $$p : A otimes M rightarrow M$$ under the iso:

$$Hom(A otimes M, M) cong Hom(M^*, M^* otimes A^*) cong Hom(M^* otimes A, M^*)$$

But can we get away with $$mathcal{C}$$ simply being monoidal for left $$A$$-modules being isomorphic to right $$A$$-comodules?

## set theory – Definability up to isomorphism versus definability of an isomorphic copy

Question: Is it provable in ZFC that every structure ordinal definable up to isomorphism has an ordinal definable isomorphic copy? If not, what are some counterexamples? All structures are set-sized.

A positive answer would likely generalize to a definable procedure for the following: Given a nonempty set $$S$$ of isomorphic structures, choose a structure, possibly outside of $$S$$, that is isomorphic to an element of $$S$$.

Here is an illustration of some of the subtleties. Assuming GCH, up to isomorphism, there is a unique $$ω_2$$-saturated elementary extension of second order arithmetic (equivalently, $$(ℝ,ℕ,+,⋅,=)$$) of cardinality $$ω_2$$. But defining a specific example of the extension seems problematic — every nonstandard integer in real analysis gives a nonprincipal ultrafilter on $$ℕ$$, and consistently with ZFC, no such ultrafilters are definable. Still, A definable nonstandard model of the reals unconditionally gives a definable countably saturated elementary extension.

We can also consider restricting the domain of the definable isomorphic copy:

• If the copy must be in HOD, then there are counterexamples, such as the second order arithmetic (assuming $$ℝ∉text{HOD}$$).
• If the domain of the copy must consist of sets of ordinals, then the copy still has an ordinal definable (OD) linear ordering, so under appropriate assumptions, third order arithmetic would be a counterexample. I think there is even a symmetric generic extension of $$V$$ with an OD non-linearly-orderable countable set of countable sets of reals.
• Requiring the domain of the copy to consist of sets of sets of ordinals does not affect existence of a definable copy.

## ac.commutative algebra – Ring isomorphism of multivariate polynomials/functions

It’s well-known that over an infinite integral domain $$R$$, the ring of univariate polynomials $$Rleft(X_{1}right)$$ is isomorphic to a ring of one-argument “polynomial functions” (see, for example, Mac Lane and Birkhoff’s Algebra).

It seems to me that this result should extend by induction to $$Rleft(X_{1},X_{2},ldots,X_{n}right)$$ with the corresponding function ring of maps from $$R^{n}$$ to $$R$$, but I have been unable to find the more general result written down.

Is it true? If so, can someone provide a citation?

## coq – How do program types such as natural numbers figure into the Curry-Howard Isomorphism?

In Coq, the `nat`, the type of natural numbers, has type `Set`. By the Curry-Howard Isomorphism, all propositions of type `Prop` are types of corresponding proof terms. How do `nat`s or other instances of `Set` figure into this isomorphism?

In other words, is there a correspondent for `Set`s in the Curry-Howard isomorphism, as there is for `Prop`s and proof terms, or are they outside the things that have correspondents in the Curry-Howard Isomorphism?

Sorry for the imprecise wording, I’m struggling to express my question clearly, probably because of a poor understanding of the Curry-Howard Isomorphism, happy to be corrected on any misunderstandings I have expressed above.

## Isomorphism between groups and subgroups.

Say I know that $$G_1,G_2$$ are isomorphic groups, both with normal subgroups $$N_1,N_2$$ respectively. Can I take an isomorphism $$psi$$ between $$G_1,G_2$$ s.t $$psi(N_1)=N_2$$? Meaning that $$psi$$ also defines a isomorphism between $$N_1$$ and $$N_2$$?

The thm I’m trying to prove is if $$G_1 cong G2, N1cong N2,N1unlhd G1, N2 unlhd G2$$ then $$G1/N1 cong G2/N2$$, and proving the above statement I would be able to use the first isomorphism theorem and that’s it. Is it even true?

Any hint would be helpful on how to approach this problem.

## graph isomorphism: Simple argument why it is not NP-complete

I need to provide one simple evidence that graph isomorphism is not NP-complete.

I saw a number of papers on google scholar and answers on StackExchange. However, I have very limited knowledge of graph isomorphism, and I would like to just provide one simple evidence which I both understand and can explain clearly.

Below is what I can think of. Is this a valid argument?

Currently, we can solve GI in Quasipolynomial Time. If GI is NP-complete, then we should be able to solve other NP-complete problems in Quasipolynomial Time as well. However, right now, we are unable to solve any NP-complete problem in Quasipolynomial Time. Therefore, GI cannot be in np-complete.

## Every distance squared between two points in $$mathbb{Z}^n$$ is clearly all $$in mathbb{N}$$. Is the converse true (upto isomorphism)?

The title was rather informal, so clarifying the question:

Is every countably infinite set $$S subset mathbb{R}^n$$ such that

$$forall v_1, v_2 in S, v_1 neq v_2 rightarrow d(v_1, v_2)^2 in mathbb{N}$$

subset of $$L + delta$$, where $$L$$ is some lattice in $$mathbb{R}^n$$ and $$delta in mathbb{R}^n$$
?

I suspect it would be true, here is some approach I’ve took.

1. Fix a point v. translate $$S$$, so that $$v = 0$$. Also transform so that $$dim S = n$$. Define $$S^*$$ as maximal superset of $$S$$ that also satisfies such property. It would be enough to show that $$S^*$$ is a lattice.
2. Fix some set of $$k$$ points, and denote them as $$T$$. Denote set of points $$p$$ in $$mathbb{R}^n$$ such that $$d^2(p, t) in mathbb{N}$$ for every $$t in T$$ as $$P_T$$. By definition, $$S^* subset P_T$$.
3. (choosing nice $$k$$ points). Since $$S^*$$ is countable, set of every size $$k$$ subset of $$S^*$$ is also countable. Therefore we select $$k$$ points $$T$$ that has maximal rank with respect to $$S$$. Formally this would mean selecting $$k$$ points that $$dim (Span(T) cap Span(S^*))$$ is maximum. Here dim is used in lattice sense.
4. It would be enough to prove that for $$k = n$$, such $$T$$ satisfies $$P_T = S^*$$.

There are close problems, but non exact search results…
https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ulam_problem

Seems to be lots of researches with “integral distances” but my google search skills limits me with finding non about distance squared being integral.

https://link.springer.com/article/10.1007/s00454-003-0014-7

## ag.algebraic geometry – The isomorphism classes of lifts of a \$operatorname{PGL}_r\$-bundle to \$operatorname{GL}_r\$

I want to show the following Lemma,

The set of algebraic isomorphism classes of lifts to $$operatorname{GL}_r$$ of an algebraic principal $$operatorname{PGL}_r$$ -bundle $$P$$ on a smooth projective curve $$X$$ over $$mathbb{C}$$ is an $$H^1(X,mathcal{O}_{X}^*)$$ torsor.

Let $$rho:operatorname{GL}_rrightarrow operatorname{PGL}_r$$. By difinition, the lift of $$P$$ is a $$(P^{operatorname{GL}_r},zeta)$$, a pair consisting of a $$operatorname{GL}_r$$-bundle $$P^{operatorname{GL}_r}$$ and an isomorphism $$zeta:rho_{*}P^{operatorname{GL}_r}cong P$$
$$H^1(X,mathcal{O}_X^*)cong check{H}{^1}(X,mathcal{O}_X^{*})$$.Then, I want a the action of $$check{H}{^1}(X,mathcal{O}_X^{*})$$ on the set of lifts.
Let $$(P^{operatorname{GL}_r},zeta)$$ be a lift and $$ain check{H}{^1}(X,mathcal{O}_X^{*})$$. Let $${g_{ij}}$$ be $$operatorname{GL}_r$$-cocycle representing the isomorphism class of $$P^{operatorname{GL}_r}$$, and $${a_{ij}}$$ be a $$mathcal{O}_X$$-cocycle representing $$a$$.

Then, How can I check that $${g_{ij}a_{ij}}$$ defines a principal $$operatorname{GL}_r$$-bundle $$P^{‘}$$, and how can I construct an isomorphism $$zeta^{‘}:rho_{*}P^{‘}cong P$$ ?
In addition, is this action simply transitively ?

Thanks in advence.

## gr.group theory – When is the Natural Map of Tate Cohomolgy is an Isomorphism?

First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate Cohomology Groups which in someway relate Group homology and Group cohomology in one sequence.

My Question is the following:

Is there any result towards the characterization of the finite Groups $$G$$ and $$G$$-Modules $$A$$ such that the natural map $$N:H_0(G,A) rightarrow H^{0}(G,A)$$ appearing in Tate Cohomology is an Isomorphism of Groups?

(My reference is https://en.wikipedia.org/wiki/Tate_cohomology_group)

It seems interesting to me because when $$N$$ is an isomorphism it represents some sort of notion of Duality between group homology and group cohomology at the zeroth level.

I asked this question to the speaker but did not get satisfactory answer . So I am asking here for an answer / Partial answer to my question.

I apologise in advance if my question sounds Stupid.