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MathOverflow : Recent Questions and Answers (Page 3)

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Fundamental ring of a circle

Starting with fundamental group, say of a circle, let's reflect back to path groupoid a little. The path concatenation operation is partial, but this can be remedied by focusing on...

Asked on 01/01/2022 by Tegiri Nenashi

0 answer

Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds

In the lecture Notions of Scalar Curvature - IAS around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully":Suppose $(X,g_X)$ and ...

Asked on 01/01/2022 by Lawrence Mouillé

1 answer

Should cohomology of $mathbb{C} P^infty$ be a polynomial ring or a power series ring?

Some people define total cohomology of a space $X$ to be $bigoplus_{i geq 0} H^i(X)$, which would make $H^*(mathbb{C} P^infty)$ a polynomial ring in one generator of...

Asked on 01/01/2022 by PowerToThePeople

1 answer

Non-degenerate simplexes in a Kan complex

I have the following question on simplicial sets:a non-constant Kan complex has a non-degenerate simplex in every sufficiently large simplicial degree?It's Exercise 8.2.3 (p. 262) of Charles Weibel's book An...

Asked on 12/29/2021 by Lao-tzu

2 answer

How to find a rational $mathbb{F}_{!q}$-curve on a quite classical Calabi–Yau threefold?

Take a finite field $mathbb{F}_{!q}$ such that $q equiv 1 pmod 3$, i.e., $omega mathrel{:=} sqrt[3]{1} in mathbb{F}_{!q}$, $omega neq 1$. Also, for $i in...

Asked on 12/29/2021 by Dimitri Koshelev

1 answer

Eigenvectors of random unitary matrices

Any unitary matrix $U$ can be diagonalized by another unitary matrix $V$,$$U=VDV^dagger,$$where $D={rm diag}(z_1,z_2,...,z_N)$ is diagonal. If $U$ is taken at random uniformly...

Asked on 12/29/2021

1 answer

Exceptional divisor of the Hilbert-Chow morphism of the punctual Hilbert scheme

Let $X$ be a smooth and projective variety of dimension $d>1$. Let $X^{[2]}$ denote the Hilbert scheme of length two subschemes of $X$. Let $X^{(2)}:=Xtimes X/mathbb{Z}_2$, where $mathbb{Z}_2$ acts by...

Asked on 12/27/2021

1 answer

Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach

Let $Prightarrow M$ be a principal fibre bundle with structure group $G$, $F$ a manifold and $alpha: Gtimes Frightarrow F$ a smooth left action. There is...

Asked on 12/27/2021 by Bence Racskó

1 answer

How to solve numerically a system of 3 interdependent non-linear ordinary differential equations?

As per title, I need to solve this: $$begin{cases}frac{d^2V}{dx^2} = -frac{q}{epsilon}left[p - n + frac{N_0}{1+c_pp+c_nn}right] \\frac{d}{dx}left[mu_nnfrac{dV}{dx} + D_nfrac{dn}{dx}right] - left[frac{n-n_0}{tau_n} - G_{op}right] = 0 \\frac{d}{dx}left[mu_ppfrac{dV}{dx}...

Asked on 12/27/2021

1 answer

Non existence of stable vector bundles on $mathbb{P}^4$ with $c_1=0$ and $c_2=1$

The Horrocks–Mumford bundle is the only known rank 2 vector bundle on $mathbb{P}^4$ which is not split. My question is: How to prove that there is no ...

Asked on 12/27/2021 by MCjr

1 answer

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