# All Questions

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### Cohen's model yet again

It has been discussed already whether a countable OD set necessarily contains an OD element. See e.g. A question about ordinal definable real numbers . A negative answer was obtained in Archive for ...
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### The current status of the conjecture on algebraic matroids

Can anyone point out some articles for the conjecture: the dual of an algebraic matroid is algebraic? Thank you!
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### On balanced bipartite graphs

Of the $2^{n^2}$ balanced bipartite graphs on $2n$ vertices how many of them have $i$ perfect matchings where $i\in\Bbb N\cup\{0\}$ and $0\leq i\leq n!$ holds?
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### A priori estimates for elliptic operators

Suppose $L : L^{m,p}(M)\rightarrow L^p(M)$ is some elliptic operator of order $m$, and $(M,g)$ is a compact Riemannian manifold. Then it is known that there exists a constant $C$ such that we have the ...
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### Applications of arithmetic topology to number theory

There is a well-known analogy between 3-manifolds and number fields, with prime ideals corresponding to knots. Are there any results in number theory that have been proven using topology through this ...
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### Does one need an external, peer-reviewed grant to become tenured faculty in this field? [on hold]

As a secondary question, how important is it to be awarded grants to remain employed as a PhD mathematician at an academic research institution? Is it common or uncommon for PhD mathematicians to have ...
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### Count Functional digraph

Given a set of nodes, how can I count the number of different functional digraph containing a specific number of connected components? With a restriction that no node can have an edge point to itself. ...
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### What is combinatorially possible, for a singular minimal surface in $\mathbf{R}^3$?

A cubic planar graph gives a cell decomposition of a two-sphere, whose dual complex is a triangulation. If I understand Plateau's laws here, a soap film gives a cell decomposition whose dual complex ...
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### weighted restricted integer compositions and extended binomial coefficients [on hold]

proof of d_{S,f}(n,k) = \binom{k}{n}{(f(s)){s\in S}}
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### A specific spanning property of a family of vectors

Let $v_2, v_2, v_3, \dots$ be a family of vectors in $\mathbb R^n$ that span $\mathbb R^n$. Given this family, define the family of vectors \begin{align*} \begin{pmatrix} v_1 \\ \alpha_1 v_1 \end{...
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### Is the assignment of a root system to a complex semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
### Is it possible for a quantum group algebra $U_{q}\left(\mathfrak{g}\right)$ to have a diagonal universal $R$-matrix?
I am writing a research paper and have shown that in the special case when a quantum group algebra $U_{q}\left(\mathfrak{g}\right)$ with the quantum group parameter $q$ not a root of unity has a ...