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Tata Lecture notes on Nori Motives

Mon, 11/27/2017 - 04:36

Does anyone have the Tata Lecture notes on Nori Motives?

This note was taken during a lecture series by Madhav Nori himself at TIFR (Notes taken by N Fakhruddin).

Explicit examples of warped products Gromov converge to a cone

Mon, 11/27/2017 - 02:48

It's well known that a sequence of two dimensional Riemannian manifolds with uniform sectional curvature lower bound can Gromov-Hausdorff converge to a cone.

Let $y=|x|$, by rotating around the y-axis, we get a cone $X$. Now I want to construct functions $y=f_i(x)$ such that their rotations around the y-axis (denoted by $M_i$) are smooth Riemannian manifolds and Gromov-Hausdorff converge to the cone $X$. I also want to compute the Gaussian curvature of $M_i$.

I considered $f_i(x)=|x|^{a_i}$ such that $a_i>1$ decrease to 1, but $f_i''(0)=\infty$, so it's not smooth enough, and the curvature at $x=0$ can't be defined. I also considered mollification of $|x|$ by convolution of $\eta(x)=c\exp(\frac{1}{|x|^2-1})$, but we can't write down the explicit form of the integration.

Can any one give good approximations such that $f_i(x)$ can explicit be written down?

Thanks to Thomas Richard's answer, below we compute the Gaussian curvatures. Let $y=\sqrt{t^2+a^2}$, then the length of the curve $$ l(t)=\int_0^t \sqrt{1+[y'(t)]^2}=\int_0^t (2t^2+a^2)^{\frac12}(t^2+a^2)^{-\frac12}dt $$ The metric can be written as $dl^2+t^2 d\theta^2$, by the curvature formula of the warped products, and note that $t''(l)=-\frac{l''(t)}{(l'(t))^3}$, the Gaussian curvature $$ K=-\frac{t''(l)}{t}=\frac{l''(t)}{(l'(t))^3 t}=\frac{2a^2}{(2t^2+a^2)^2}. $$ At $t=0$, $K=\frac{2}{a^2}\to \infty$ as $a\to 0$.

Pseudo-polynomial potentials for Schrödinger operators

Mon, 11/27/2017 - 01:21

Consider the one dimensional Schrödinger hamiltonian $\mathcal{H}=-\frac{\hbar^2}{2} \frac{d^2}{dx^2} + V(x)$.

Suppose that $V:\mathbb{R} \rightarrow \mathbb{R}^+$ is a continuous and confining potential $\displaystyle \lim_{\lvert x\rvert \to +\infty} V(x)=+\infty$

It is well known that $\mathcal{H}$ has pure discrete spectrum $\lambda_1< \cdots \leq \lambda_n$ with $\lambda_n \to +\infty$ as $n\to + \infty$.

Furthermore, if $V$ is a polynomial of even degree $2s$, then the eigenvalues obey the asymptotic formula

$$\lambda _k \sim {C_{s,\sigma}\, k}^{\frac{2s}{s+1}}$$

I want to know if the above asymptotic formula holds if, instead of having a polynomial potential as above, I consider a "pseudo-polynomial" in the sense that $V$ is another function on a segment $[a,b]\subset \mathbb{R}$ (like $\sin(x)$) and outside $\mathbb{R}\setminus[a,b]$ its an even polynomial such that the result is a continuous piecewise function.

A weak version of the Erdös-Faber-Lovasz conjecture

Mon, 11/27/2017 - 01:02

A hypergraph is a pair $H=(V,E)$ where $V$ is a nonempty set, and $E\subseteq {\cal P}(V)\setminus\{\emptyset\}$ is a collection of non-empty subsets of $V$.

Strong colorings. If $\kappa$ is a cardinal and $H=(V,E)$ is a hypergraph, we call a map $c:V\to \kappa$ a strong coloring if for all $e\in E$ the restriction $c|_e$ to $e$ is injective (that is, for any edge $e$, its members get all different colors).

Weak colorings. If $\kappa$ is a cardinal and $H=(V,E)$ is a hypergraph, we call a map $c:V\to \kappa$ a weak coloring if for all $e\in E$ with $|e|>1$ the restriction $c|_e$ to $e$ is non-constant (that is, for any edge $e$ with more than $1$ element, the elements of $e$ do not all receive the same color).

For any positive integer $n\in\mathbb{N}$ we say $H=(V,E)$ is an $n$-Erdös-hypergraph if $|E|=n$, and $|e| < n$ for all $e\in E$, and $|e_1\cap e_2| \le 1$ for $e_1\neq e_2\in E$.

A version of the Erdös-Faber-Lovasz conjecture states:

Any $n$-Erdös-hypergraph $H=(V,E)$ has a strong coloring $c:V\to \{0,\ldots,n-1\}$.

This conjecture has been open for more than 40 years.

Question. Does any $n$-Erdös-hypergraph $H=(V,E)$ have a weak coloring $c:V\to \{0,\ldots,n-1\}$?

Good introduction to statistics from a algebraic point of view?

Mon, 11/27/2017 - 00:41

There are already lots of questions on this subject like

Is there an introduction to probability theory from a structuralist/categorical perspective?

and related field called ergodic theory which in fact study different things.

However, as a new category theorist with almost no statistics background I don't aim to learn these advanced topics, but to understand very basic notions like random variable and expectation from a algebraic perspective.

For example, we can define a type family

Rand: Type->Type

, and a real random variable can be defined as

randnum : Rand Real

and Expectation as

E : Rand a -> Real

It seems that statistics is the one of the most recalcitrant subject for algebraic approach, but I think it is not the case, we can just treat it as any other abstract object, and define axioms on this abstract random type. The notations and formulas in every introduction book of statistics I have read soon become utterly ugly due to lack of a proper foundation, which is really painful for someone ingrained with abstract algebra and functional programming. However,statistics is extremely useful for machine learning and the modelling of human brain and many others.

Sums of four coprime squares

Sun, 11/26/2017 - 23:25

The Four Squares Theorem says that every natural number is the sum of four squares in $\mathbb Z$. What is known about coprime representations? Here we call a presentation $n=a^2+b^2+c^2+d^2$ coprime if the g.c.d. of the four numbers $a,b,c,d$ is 1. Does every natural number have a coprime presentation? If not, is there a simple criterion characterising the numbers that have coprime presentations? What is known about the number of different coprime presentations of a given $n$?

Number of solutions to $n^ 2 = x^2 + y^2 + 2z^2$

Sun, 11/26/2017 - 15:49

I was intrigued by this result in Journal of Number Theory from 2013. Let $n = 2^{\lambda_2}\prod p^{\lambda_p}$.

Theorem The number of $(x,y,z) \in \mathbb{Z}^3$ such that $n^ 2 = x^2 + y^2 + 2z^2$ is given by $$ b(\lambda_2) \prod_p \left[ \frac{p^{\lambda_p+1}-1}{p-1} - \left( \frac{-2}{p}\right)\frac{p^{\lambda_p}-1}{p-1} \right]$$ the symbol $b$ is a variant of the Legendre symbol are given by $$ b(\lambda_2) = \left\{ \begin{array}{cl} 1 & \lambda_2 = 0 \\ 3 & \lambda_2 \geq 1\end{array} \right.$$ and the necessary values of the classical Legendre symbol are given by $$ \left( \frac{-2}{p}\right) = \left\{ \begin{array}{cl} 1 &p \equiv 1,3 \pmod 8 \\ -1&p \equiv 5,7 \pmod 8 \end{array} \right.$$

I think this problem is slightly difficult because you have to sieve out the squares, but otherwise, shouldn't this result fall out of Hasse principle or a theta function of some kind? I could solve this equation possibly over each prime (as well as $p=2$) and then multiply the result.

$$ p^2 = x^2 + y^2 + 2z^2 $$

There's no guarantee even of one solution here, but for this enumerative result we're simply stating that we could multiply the result from all places.

Was it new to count solutions over all primes? Does the number of solutions $r_{(1,1,2)}(p^2)$ for all primes not sufficient to determine $r_{(1,1,2)}$ in general? I remember reading that the class number was either 1 or 2 (but I don't even know what that means).

There is an approach using half-integer weight forms that solved a few cases in 2014.

And possibly a resolution of all cases where the genus has only 1 or 2 elements in 2017.

A conjecture regarding prime numbers

Sun, 11/26/2017 - 08:32

For $n,m \geq 3$, define $ P_n = \{ p : p$ is a prime such that $ p\leq n$ and $ p \nmid n \}$ .

For example : $P_3= \{ 2 \}$ $P_4= \{ 3 \}$ $P_5= \{ 2, 3 \}$, $P_6= \{ 5 \}$ and so on.

Claim: $P_n \neq P_m$ for $m\neq n$.

While working on prime numbers I formulated this problem and it has eluded me for a while so I decided to post it here. I am not sure if this is an open problem or solved one. I couldn't find anything that looks like it. My attempts haven't come to fruition though I have been trying to prove it for a while. If $m$ and $n$ are different primes then it's clear. If $m \geq 2n$, I think we can find a prime in between so that case is also taken care of. My opinion is that it eventually boils down to proving this statement for integers that share the same prime factors. My coding is kind of rusty so would appreciate anybody checking if there is a counterexample to this claim. Any ideas if this might be true or false? Thanks.

PS: I asked this question on mathstackexchage and somebody recommended I post it here as well. Here is the link to the original post

Asymptotics for a special sum with a divisor function

Sun, 11/26/2017 - 02:40

$\Huge I\ saw\ terrible,\ it\ looked\ at\ me. Do\ you\ belive?$

Coming out as transgender in the mathematical community

Sat, 11/25/2017 - 17:01

I don't know if MO is the right place to ask such a question, but anyway it's my only hope to get an answer, and it's very important for me (not to say 'vital'); so let's try.

I'm at this time a Ph.D. student, and I plan to defend in the spring of 2018. I'm currently looking for a postdoc position for next year. I am, at this time, known as a man in the mathematical community, but I'm actually a trans woman, beginning my gender transition. I have two problems. Firstly, I will have to come out as transgender, in at most a few years, in the mathematical community, and I'm quite fearful about the consequences (for example, for my career). Secondly, I have to ensure before applying for a postdoc that in the country where I apply, I will be able to pursue my transition, I will be accepted as I am at the university, and that there won't be any major threat to my security (because of the policy of the country regarding trans people, for instance). For this reason, having contacts in these countries who are reaserchers in maths and are familiar with transidentity questions would be very helpful for me, as I have no other means to get the info I need.

So my first question is: are there, here, trans mathematicians who would be willing to talk with me, in private, about how they came out (if they had to) in the mathematical community, how it was accepted, what has been the consequences for their career, and more generally what was their experience as trans mathematicians? (I also have other specific questions like, for instance, how to deal with a change of your first name when you already have published under your former name?) Even if you're not trans, if you have information about all of this (if you know a trans mathematician for example), I would be interested.

My second question is: in the countries where I am interested in applying for a postdoc, that is Spain, the Czech Republic, Canada, the US, and Brazil, do you have any contacts, in the academic world, who are familiar with LGBT questions, and who could give me an idea about the situation of trans people in their country, and especially at the university? (In order for me to know if it's safe to apply there or not?)

If some of you are yourselves why I don't ask these questions directly to researchers of the universities where I want to apply: that's simply because it's not safe. Trans people have to face a lot of discrimination and you never know if speaking about your transidentity with someone you don't know is safe or not - that's the reason why I choosed to ask it anonymously here, first.

(You can contact me in private at rdm.v[at]yahoo[dot]com.)

Does there exist an algorithm that decomposes a matrix into a minimal number of elementary matrices for $F_{2}$?

Sat, 11/25/2017 - 10:31

If $i \neq j$, then let $C_{i,j} : F_{2}^{n} \to F_{2}^{n}$ be the elementary linear transformation defined by

$$C_{i,j}(x_{1},\dots,x_{i},\dots,x_{j},\dots,x_{n}) :=(x_{1},\dots,x_{i},\dots,x_{i}\oplus x_{j},\dots,x_{n})$$

which applies the CNOT gate $(x,y)\mapsto(x,x \oplus y)$ to the $i$-th and $j$-th bits.

Does there exist an efficient algorithm that takes a non-singular linear transformation $L:F_{2}^{n} \to F_{2}^{n}$ and outputs a decomposition

$$L = C_{i_{r},j_{r}} \circ \cdots \circ C_{i_{1},j_{1}}$$

such that $r$ is minimized or nearly minimized?

I would like to find such an algorithm in order to optimize circuits composed almost exclusively of CNOT gates. A more general version of this question has been asked here.

Are pseudo-Anosov foliations dense?

Fri, 11/24/2017 - 13:00

A pseudo-Anosov foliation of a compact orientable surface $F$ is a one whose class in the space $\mathcal{PMF}(F)$ of projective measured foliations is preserved by some pseudo-Anosov homeomorphism of $F$. I saw it casually mentioned that pseudo-Anosov foliations are dense in $\mathcal{PMF}(F)$. What is a proper reference for that result?

Calculus problem [on hold]

Fri, 11/24/2017 - 08:54

I was wondering if someone could help me find the velocity of this quadratic function: f(x)=-0.08x2 +0.568x+7.7818 I already solve it using projectile motion, but I was wondering if I could do it with calculus...

Restricted sumsets $(h\land S) \cap (k\land S) \neq \emptyset$ with $h\gg k$

Fri, 11/24/2017 - 08:46

Let $S$ be an infinite set of positive integers and, for each $n \in \mathbf{N}^+$, define $$ n\land S:=\{s_1+\cdots+s_n: s_1,\ldots,s_n \in S \text{ and }s_1<\cdots<s_n\}. $$

Question. Is it true that for each constant $C>0$ there exist integers $h,k\ge 2$ such that $h\ge Ck$ and $$ (h\land S) \cap (k\land S) \neq \emptyset\,\,\,? $$

The answer is affirmative replacing $n\land S$ with the classical sumset $nS$ (and fixing $k=2$ in the above Question). However, the proof cannot be adapted to this case.

Studying the limit of a sequence of spectra knowing their BP-Homology

Fri, 11/24/2017 - 08:32

Let $X_i$ be the spectrum such that $BP_*(X_i) = \Sigma^{i+1} BP_* / (v_0^2, v_1^2 , \dots, v_{i-1}^2)$ and $X = \bigvee_i X_i$.

First question: Can i always say that $X$ and $X_i$ are suspension spectra? Why?

Now let's consider the following fibration:

$ \bigvee_i X_i \to L_n \bigvee X_i \to \Sigma C_n \bigvee X_i $

which gives me the following inverse system of short exact sequences:

$$ 0 \to \bigoplus_{i \leq n} BP_*(X_i) \longrightarrow \ \bigoplus_{i \leq n} BP_*L_n(X_i) \ \longrightarrow \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to 0$$

$$ 0 \to \bigoplus_{i \leq n-1} BP_*(X_i) \to \bigoplus_{i \leq n-1} BP_*L_{n-1}(X_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{n-1}X_i) \to 0$$

with vertical maps $$ \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{i-1}X_i)$$

So here is my second question: How can i study the inverse limit of the right vertical tower $$lim_n \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i)?$$

Thank you

Differentiability of step function

Fri, 11/24/2017 - 08:14

Let $f$ be the fractional part function, i.e. $f(x)=x-\operatorname{floor}(x)$

then I define the integral function $ g(h):=\int_0^1 f(x/h) dx$.

I would like to know: For which values of $h>0$ is the function $g$ differentiable?

The jumps of the function $f$ are somehow giving me a hard time.

In what respect are univalent foundations "better" than set theory?

Fri, 11/24/2017 - 08:03

It was an ambitious project of Vladimir Voevodsky's to provide new foundations for mathematics with univalent foundations (UF) to eventually replace set theory (ST).

Part of what makes ST so appealing is its incredible conciseness: the only undefined symbol it uses is the element membership $\in$. UF with its type theory and parts of higher category theory seems to be a vastly bigger body to build the foundation of mathematics. To draw from a (certainly very imperfect) analogy from programming: ST is like the C programming language (about which Brian Kernighan wrote: "C is not a big language, and it is not well served by a big book"), but UF seem more like the vast language of Java with all its object-oriented ballast.

Questions. Why should mathematicians study UF, and in what respect could UF be superior to ST as a foundation of mathematics?

Hölder continuity of holomorphic motions and J(ulia set)-stability

Fri, 11/24/2017 - 07:29

I am currently working with Shishikura's paper "The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets" ( Two questions arose about details the author doesn't elaborate on.

Firstly, in lemma 3.1 (page 234), it is clear from the proof why the holomorphic motion is Hölder continuous. However, it is not clear to me why the same holds for the inverse maps $i_\lambda^{-1}$.

Secondly, in the proof of theorem 1 (page 236, bottom), I find it difficult to see that $f_\lambda^N(c_\lambda) \in X_\lambda$ does indeed imply that the family $(f_\lambda)$ is not J-stable.

Any help is greatly appreciated. Cheers!

What is currently feasible in invariant theory for binary forms?

Fri, 11/24/2017 - 07:25

When Paul Gordan became a professor in 1875 he could show the binary form in any degree has some finite complete system of (general linear) invariants, but he could not actually give a complete system above degree 6. So far as I can tell (by reading Kung, Sturmfels, Derksen, and Eisenbud and some correspondence with them) advances in theoretical and computer algebra have not really changed that. The complexity of calculation rises so quickly with degree that the limit of degree 6 has not been passed by much if at all. But perhaps my information is incomplete or out of date.

Is it now possible to calculate specific complete systems of invariants in higher degrees? What is the state of that?

Answers to the question Algorithms in Invariant Theory give relevant references but they do not give any clear answer. One leads to an arXive article which describes one computer package this way.

The package calculate the set of irreducible invariants up to degree min(18, βd), but in all known computable cases this set coincides with a minimal generating set, see, for example, Brouwer’s webpage∼aeb/math/invar/invarm.html

That refers to the degree of the invariants, not the degree of the form they are invariant for. I did not find a description there of which cases are computable. And that link no longer works.

boolean algebra

Fri, 11/24/2017 - 07:24

was wondering how would you go around solving this question. P+(pqr)+(p¬q+r)+¬(p+¬q) here is what I did Not sure if it is right though. Pqr+p¬q+r= pr(+q¬q)=(1)pr rules used are commutatively and cancellation then pr+¬(p+¬q)= r(p+q¬p) used negative absorption and involution. My final result is r+q+p