functional programming – What are the implications of Homotopy Type Theory?

I’ve recently come across the topic of homotopy type theory and I’m interested to learn more. I have a very limited background in type theory.

Can anyone tell me, in functional programming terms or through practical examples, how exactly is HoTT going to change the way we view mathematics, and what are the implications of HoTT on proof assistants? Thanks!

sp.spectral theory – References on discrete Sturm-Liouville eigenvectors convergence

Let $ L : u_n mapsto a_n u_{n + 1} + b_n u_n + a_{n – 1} u_{n -1} = nabla ( a_n Delta u_n ) + (b_n + a_n + a_{n – 1}) u_n $ be a discrete Sturm-Liouville operator, with $ nabla u_n := u_{n + 1} – u_n $ and $ Delta u_n := u_n – u_{n -1} $. Let us suppose that $ L $ depends on a parameter $ t $ that tends to $ +infty $ and that with a good scaling $ n = (alpha(t) x + beta(t)) $, one has convergence of $ L $ towards a continuous Sturm-Liouville operator $ mathbb{L} : f mapsto (Af’)’ + B $ where $ A $ and $ B $ are good functions (for instance polynomials, if it can help). I precise that I have Dirichlet boundary conditions $ u_0 = 0 $ and that one can suppose $ L $ to be self-adjoint for a good scalar product, for instance $ ell^2(mathbb{N}) $.

I am looking for results of convergence of eigenvectors of $ L $ towards eigenvectors of $ mathbb{L} $. Are there any references, a theory, etc. ? Some simple examples with explicit computations are welcome, for instance convergence towards some hypergeometric operator $ mathbb{L} $. In fact, are there references on these discrete Sturm-Liouville operators ?

set theory – What is first-order logic with Dedekind-finite sets of variables?

The usual set up of first-order logic is with an infinite reservoir of variables which we can use in formulas. This is one of the annoying reasons why we need to put $aleph_0$ into the cardinal equations, but it also provides us with the expressive freedom that we need.

It is not hard to see that there is no reason to consider any other cardinal, except $aleph_0$, in this case: since formulas are finite, and proofs are finite, in any kind of proof there will only be finitely many variables. So anything larger than $aleph_0$ is kind of irrelevant. But this assumes that all cardinals are comparable, and the Axiom of Choice makes things kinda nice.

Assuming that the Axiom of Choice fails, and badly, what happens when we substitute the reservoir of variables with some Dedekind-finite set? In particular, a set $A$ whose finite subsets (and finite injective sequences) form a Dedekind-finite set, or even an amorphous set?

Can we prove some interesting results (read: not entirely equal to standard first-order logic) in $sf ZF$ (+ whatever set we needed exists), or at least some consistency results? For example, we don’t need choice to prove that every theory in a finite language has a complete theory extending it, or that it has a model. What happens when we switch to this abominable version of first-order logic?

ct.category theory – Are generators defined in Tohoku paper equivalent to that defined in Wikipedia (Which I believe is a more widely used definition)

As I was reading Grothendieck’s Tohoku paper(translated by M.L.Barr and M.Barr), I found that the definition of a generator in the category differs from that defined in wikipedia.
Let $mathbf{C}$ be a category(It may be necessary that $mathbf{C}$ is a locally small category), a family of generators {$U_i$}$_{iin I}$ with $I$ being an index set, according to Tohoku paper, are a collection of objects such that for any object $A$ and any subobject $B neq A$, there is $iin I$ and a morphism $ucolon U_i rightarrow A$ which does not come from $U_i rightarrow B$. While in wikipedia, it is defined in a way such that for any $f,gcolon Arightarrow B$ with $fneq g$, there is an $iin I$ and $ucolon U_irightarrow A$, such that $fcirc u neq gcirc u$.
What I would like to know is that are these 2 definitions equivalent, or is the definition in Wikipedia stronger than that in Tohoku paper?

ct.category theory – Finite sets and relations with boolean matrix

My comment was misleading, and written in haste. Here’s a better version. Recall that the objects of $mathrm{Mat(Bool)}$ are the natural numbers, and a morphism $nto m$ is an $mtimes n$ matrix of booleans, which we can identify with $0,1$. There is an inclusion functor
mathrm{Mat(Bool)} hookrightarrow mathbf{FinRel}

sending $nmapsto {0,ldots,n} =: mathbf{n}$, and the matrix $A$ to the subset of $mathbf{m} times mathbf{n}$ whose indicator function is given by $A$, which is interpreted as a function $mathbf{m} times mathbf{n} to {0,1}$. Matrix multiplication corresponds to composition of relations. Every relation from $mathbf{n}$ to $mathbf{m}$ can be uniquely reconstructed from such a indicator function/matrix. This means the inclusion is fully faithful. The inclusion is essentially surjective since there is a bijection between a finite set and one of the standard finite sets $mathbf{n}$.

nt.number theory – What is the meaning of the $L$-group?

Langlands’ functoriality conjecture predicts that to a suitable homomorphism of $L$-groups
psi : ^LG to ^LH

there should be a transfer of automorphic representations from $G$ to $H$. For the purposes of discussion, let’s take $^LG$ to be the Weil form
^LG = hat{G}(mathbb C) rtimes W_{mathbb Q}

where $W_{mathbb Q}$ is the Weil group of $mathbb Q$. This conjecture, as we know, has revealed many connections between disparate objects in representation theory, geometry, and number theory, and also works to explain various phenomena that we observe. My question is more on a philosophical level: setting aside the reasoning along the lines of “we believe it because it works,” why should functoriality be true?

To narrow the question a little, what is the meaning of the $L$-group? How should we think of the semidirect product? What category does it live in? It blends a complex reductive group with the arithmetic of $mathbb Q$, which is crucial to the entire framework of the Langlands program. As Casselman pointed out here, Langlands’ letter to Weil already established that Langlands understood the centrality of the $L$-group, but this fact seems to have revealed itself through Langlands’ deep experimental knowledge of Eisenstein series. Later work in geometric and $p$-adic Langlands reveal that the geometry of the $L$-group certainly realizes functoriality in certain senses, but I don’t think it quite explains (for me, at least) the question of why.

The picture gets even muddier if we replace $W_mathbb Q$ by the conjectural automorphic Langlands group $L_mathbb Q$ as Langlands’ reciprocity conjecture (perhaps) demands.

ho.history overview – Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?

We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced results in many branches of ordinary mathematics if we only work with sets and set-membership relation in our language, or otherwise only rely on set theory. To put it differently: it seems that in order to get results in many branches of mathematics one might not need to be very familiar with set theory at all, let alone being able to translate everything to the language of sets or to heavily rely on set theory.

I’m wondering if there are cases where an open/a difficult problem in other branches of mathematics (e.g., number theory or real analysis) has been solved mostly/only because of the insight that set theory has offered, directly or indirectly (say, through branches that heavily appeal to set theory, such as model theory). Even a historical incident will be helpful: a problem of the sort that was first solved thanks to set theory, but later on more accessible solutions have been found that don’t deal much with sets.

Thank you very much!

number theory – Radix Economy of Complex Bases

If we extend the allowed bases for a numerical system to the complex numbers, is e still the most economic base? If not, what would it be?

There’s the well-known formula for radix economy: enter image description here

Where b is the base, and N is a given number.

I don’t know if this formula is still valid for complex numbers. Nonetheless, the only local minimum this function seems to have, even extended over the complex numbers, is at b = e + 0i.

complexity theory – Are all reductions from NP-complete problems either NP-complete or are contained in P?

Let’s say we have a problem $A in mathsf{NP}$. Now let’s say we have a reduction $f(mathsf{SAT}): A leq mathsf {SAT}$.

So, assuming that $A$ is not $mathsf{NP}$-complete we have that $f(mathsf{SAT})$ is $mathsf{FNP}$-hard:

  1. $mathsf{exists C: {A in C subseteq FNP}, f(mathsf{SAT}) in mathsf C}$.
  2. $mathsf{NP subseteq C}$.

Since you can use $f(mathsf{SAT})$ to solve $mathsf{NP}$-complete problems, $mathsf C$ can only be equal $mathsf{FNP}$.

Although assuming that $A$ is $mathsf{NP}$-complete the reduction is polynomial-time deterministic reduction. I.e. $f(mathsf{SAT}) in mathsf {FP}$.

But what about cases when $f$ is $mathsf{FNP}$-intermediate? Are they inexistent?

number theory – $n!+1$ is composite for infinitely many odd $n$

It’s the number theory problem from Thailand Mathematical Olympiad. That require one to prove that $n!+1$ is composite for infinitely many odd $n$.

It’s true if $n$ is even from Wilson’s theorem directly that $p text { } Big|Big((p-1)!+1Big)$ but for odd $n$ I have no idea about it. I’ve tried for prime $p$ of the form $4k+3$ and we’ll have that $left(dfrac{p-1}{2}right)!equivpm 1pmod p$ and maybe there is infinitely many cases that $left(dfrac{p-1}{2}right)!equiv -1pmod p$, but I don’t know how to prove it and I don’t think this is a good way to tackle this problem.

I, actually, saw someone’s proof before but I didn’t pay attention at much and didn’t jot down the idea. All in my memory is that he used Bertrand postulate and something in the double factorial like $(n!+1)!+1$. Any help would be really appreciate and thank you in advance.