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3
votes
0answers
33 views

Signatures of latin squares: what about the extremal cases?

For a latin square (LS) of order $n$, we will define a cut (or maybe general transversal, I don't know whether there is an entrenched name for this) as a collection of $n$ cells such that no two share ...
8
votes
1answer
190 views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...
17
votes
1answer
284 views

Lens spaces and generalized Petersen graphs

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a ...
11
votes
1answer
555 views

In what sense is the classification of all finite groups “impossible”?

I think there is a general belief that the classification of all finite groups is "impossible". I would like to know if this claim can be made more precise in any way. For instance, if there is a ...
4
votes
1answer
131 views

Castelnuovo's rationality criterion on singular surfaces?

Let $S$ be a projective surface over an algebraically closed field. Suppose that $q(S)=h^1(\mathcal O_S)=0$ and $P_2(S)=h^0(\mathcal O_S(2K_S))=0$. If $S$ is smooth, Castelnuovo's rationality ...
7
votes
2answers
224 views

Uniform-in-p classification* of p-groups of order p^n for each fixed n?

To what extent is there/can there be a description that is uniform in p (for p sufficiently large) of the p-groups of order $p^n$, for each fixed n? Note 1: I used the word "description" rather ...
2
votes
2answers
200 views

How much information does the multiplicative semigroup of an algebra contain?

How much do we know about an given algebra when we only know its semigroup strucure under the product law? How far can two algebras be distinguished by knowing only their semigroup strucure? The ...
2
votes
1answer
207 views

Are all symmetric idempotent Latin squares known?

Are all symmetric idempotent Latin squares known? There is such a square of order $n$ if and only if $n$ is odd. However, is there a classification of all of them? (The motivation for the question ...
5
votes
1answer
265 views

Taking direct sums in $K$-theory in Kirchberg-Phillips classification

A theorem by Kirchberg and Phillips states that two unital separable nuclear simple purely infinite $C^*$-algebras (so called Kirchberg algebras) satisfying the Universal Coefficient Theorem are ...
7
votes
2answers
438 views

basics of classification of trilinear forms (when is it non-discrete)

Consider tri-linear forms, $\{A_{ijk}\}$ where $i=1,..,n_1$, $j=1,..,n_2$, $k=1,..n_3$, over a field of zero characteristic, up to the equivalence $A\to (U_1,U_2,U_3)(A)$, by three matrices. What is ...
2
votes
1answer
337 views

The PDE $u_t=u_{xx}-u_{yy}$: The simplest linear second-order PDE that isn't elliptic, parabolic, or hyperbolic.

I know that there have been several questions on here and stackexchange about linear PDE's which don't fall into the standard classification, but I had a more focused question which I haven't seen ...
1
vote
2answers
248 views

Finding Decision Boundary from empirical distribution

Based on measuring a certain characteristic, we want to classify measurements as coming from either of two populations. The true population distributions are unknown (and we don't want to take any ...
22
votes
1answer
1k views

Rock-paper-scissors…

A directed graph whose underlying undirected graph is complete is called a tournament. Let us call a (finite) directed graph balanced if every vertex has as many incoming as outgoing edges. The ...
2
votes
0answers
275 views

bielliptic surfaces

Definition: A surface is called $S$ bielliptic if $S \cong (E \times F) /G$, where $E,F$ are elliptic curves and $G$ is a finite group of translations of $E$ acting on $F$ such that $F /G \cong ...
4
votes
1answer
427 views

Del pezzo surfaces in positive characteristic

For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...
10
votes
0answers
210 views

“Locally Euclidean” varieties

Differentiable manifolds can be described in terms of local charts to open subsets of $\mathbb{R}^n$ and transition functions that are diffeomorphisms. Trying to put $\mathbb{A}^n$ (over an ...
15
votes
6answers
2k views

Is there a classification of open subsets of euclidean space up to homeomorphism?

I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among ...
6
votes
7answers
558 views

Classifications of finite simple objects

I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "classification" in the ...