# All Questions

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### Institutional response to “Esquisse d'un programme”

It is well-known that Grothendieck's "esquisse d'un programme" was submitted in 1984 as part as the author's application for a permanent position of "Directeur de Recherche" at the C.N.R.S. (the main ...
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### Definability in HOD

Let $\mathfrak{M}$ be a countable transitive model of set theory, and consider HOD (the hereditarily ordinal definable elements of $\mathfrak{M}$). Let $x$ be an object $x \in HOD$. So $x$ is ...
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### Expected Value for a Connected Graph

Consider a connected graph of N nodes. Assign randomly to each node a distinct number from 1 to N. For each node consider the maximum adjacent value or itself if all adjacent values are smaller. ...
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### Rate of convergence of the average of an equidistributed sequence

Let $f : \mathbb R\to\mathbb C$ be an $1$-periodic and sufficiently smooth function, which has zero average, and let $\alpha$ irrational. We know the following: a. ...
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### Which Counterexample for the Open Mapping Theorem [on hold]

I would like to ask a counterexample for the open mapping theorem: Find a discontinuous linear mapping $T:X \to Y$ such that $T(X)=Y$ and $X,\;Y$ are Banach but $T$ is not open. Could you help me ...
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### Proving that Brownian motion has no points of increase

I am reading Burdzy's paper on the points of increase of Brownian motion: Burdzy's Paper He is proving that, almost surely, a Brownian motion, has no points of increase. What he actually proves is ...
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### Is a particular set of polynomials dense in a set of functions?

Let us consider the set $\mathcal{F}$ of strictly increasing continuous functions from $[0;1]$ on $[0,1]$ that cancel in $0$ and are equal to $1$ in $1$. So, if $f\in \mathcal{F}$ one has $f(0)=0$ and ...
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### Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper? Since weak ...
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### A Question About Pure States, Support Projections and Central Covers

I am trying to study the paper Consistency of a Counterexample to Naimark’s Problem by Charles Akemann and Nik Weaver, and there is a claim in Lemma 1 of the paper that I am stuck at, which is as ...
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### Does push forward commute with taking dual?

Let $f:X \to \mathbb{P}^1$ be a flat projective morphism between projective Noetherian schemes. Let $\mathcal{L}$ be line bundle on $X$ such that $f_*\mathcal{L}$ is locally free ...
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### Vanishing theorems that work in positive characteristic

Let $X$ be a smooth projective variety over a field of characteristic $p>0$ of dimension at least $2$. I am looking for some examples when $H^2(\mathcal{O}_X)$ vanishes. Is there any standard way ...
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### Irreducible polynomials with a root modulo almost all primes

Let $f \in \mathbb{Z}[x]$ be a non-zero polynomial which is irreducible over $\mathbb{Q}$. Suppose that $f$ has a root in $\mathbb{F}_p$ for almost all primes $p$. Must $f$ be linear? ...