All Questions

7k views

How seriously should a graduate student take teaching evaluations?

Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...
986 views

State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations $$a^5 + b^5 = c^5 + d^5$$ and $$a^6 + b^6 = c^6 + d^6$$ have no known nontrivial solutions, but $$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$ and ...
2k views

Can you efficiently solve a system of quadratic multivariate polynomials?

Given a system of 2nd-degree polynomials, $P=\{p_1,\dots,p_m\}$ where $p_i: \mathbb{R}^n \rightarrow \mathbb{R}$, can you efficiently find a common zero of all of these polynomials? In other words, ...
1k views

Can one recover a metric from geodesics?

Assume there are two Riemannian metrics on a manifold ( open or closed) with the same set of all geodesics. Are they proportional by a constant? If not in general, what are the affirmative results in ...
27k views

Reading list for basic differential geometry?

I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...
4k views

Does the curvature determine the metric?

Hello, I ask myself, whether the curvature determines the metric. Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that ...
2k views

When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals)

Sorry if this is a silly question. I was wondering, under what axioms of set theory is it true that if $\alpha$,$\beta$ are cardinals, and $2^\alpha=2^\beta$, then $\alpha=\beta$? Do people use these ...
3k views

What is meant by smooth orbifold?

There seems to be some confusion over what the tangent space to a singular point of an orbifold is. On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth ...
378 views

A formula on Kronecker coefficients

Accidentally, I proved the following formula for the Kronecker coefficients using some obscure method. $$g_{lm^n,mn^l}^{m^{nl}}=1,\ \forall l,m,n\in\mathbb{N},$$ where $n^m$ is the rectangle ...
3k views

298 views

Proof of Lomnicki and Ulam on Infinite Product Probability Spaces

Given an arbitrary, nonempty family $(\Omega_i,\Sigma_i,\mu_i)_{i\in I}$ of probability spaces, there exists a probability measure $\mu$ on $\otimes_i\Sigma_i$ such that for every finite set ...
1k views

Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
599 views

Why does the internal singular simplicial space realize to the same thing as the discrete singular simplicial set?

There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations ...
114 views

PA proves that functions are total

Is there a total recursive function $f:N \to N$ such that for no $\Sigma_1$ formula $\phi(x,y)$ which defines it (i.e., defines its graph), is it true that PA proves that "$\phi$ defines a total ...
530 views

Groups of Rational Points on Gaussian Circles

Let a gaussian circle $C_R$ be any circle defined by the equation: $$x^2+y^2 = R, (x,y) \in \mathbb{R}^2$$, where $R$ is the norm of a gaussian integer ($R=a^2+b^2, (a,b) \in \mathbb{Z}^2$). IF $R$ ...
4k views

Conjugate prior of the Dirichlet distribution?

What is the conjugate prior distribution of the Dirichlet distribution?
74 views

Prove that k ≤ log2N [on hold]

I have the following problem and I don't know where to star: Let n ≥ 2 and let n = p1p2...pk be its prime factorization, where the primes are not necessarily distinct. Prove that k ≤ log2N (hint: ...
138 views

A self-square group is a group with extra structure, which encodes the fact that the group is isomorphic to its own direct square. To be exact, the group $G$ has a special element $1$, a unary ...
62 views

Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as \det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ ...
1k views

What axioms are stronger than the Axiom of choice?

What other axioms in set theory are stronger than AC ? I mean what are those axioms that will imply AC ?
68 views

Characterization of global sections (which are not products) of a sheaf which is locally a product

In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...
2k views

How do you calculate the group scheme of E[p] for a an elliptic curve E in characteristic p?

I know that the answer is $\mu_p \times \mathbb{Z}/p\mathbb{Z}$ if $E$ is ordinary, and $\alpha_p$ if $E$ is supersingular, where $\mu_p$ and $\alpha_p$ are the kernels of Frobenius on $\mathbb{G}_m$ ...