Questions tagged [classification]
The classification tag has no usage guidance.
43
questions
2
votes
1answer
166 views
Is there a precise relationship between the goals of moduli theory and the minimal model program?
I want to get into some of the big classification problems in algebraic geometry, but have a very broad question. Ultimately we would like to classify all varieties over some field up to isomorphism, ...
6
votes
1answer
143 views
Comonoids in the category of monoids
Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids?
...
3
votes
0answers
45 views
Matroids which are transitive on minimal basis exchanges
I am looking for matroids in which all minimal basis exchanges look the same, that is, the matroid is transitive on these. Let me explain what I mean by that.
Consider a finite matroid $M$. Define a ...
1
vote
1answer
412 views
Minimum number of support vectors? [closed]
I'm learning SVM and its written everywhere that the minimal number of support vectors is 2? But I couldn't find any formal proofs of that. Why cant there be less than 2 support vectors? Can somebody ...
2
votes
1answer
2k views
Why the VC dimension of triangles in 2D space is not greater than 7?
I understand that there are sets of 7 points on a circle that can be fully
shattered using triangles.But, it is not clear to me why it cannot shatter 8 points.
Is there any intuitive way of arriving ...
2
votes
1answer
65 views
Name for “partially complete” invariants in classification problems?
For any equivalence $\sim$ on some collection of objects $C$ consider the problem of trying to determine if two arbitrary objects $x$ and $y$ in $C$ are equivalent i.e. if $x\sim y$ now by definition ...
7
votes
2answers
286 views
The “Johnson polychora”
Firstly, a definition:
A convex polyhedron, whose faces are regular polygons (2D polytopes).
This includes the 92 Johnson solids, 13 Archimedean solids, 5 Platonic solids and two infinite ...
5
votes
1answer
395 views
Classification of the quotients of the ring Z/4 [X]
Is it possible to classify all cyclic $\mathbb{Z}/4$-algebras, i.e. the regular quotients of $\mathbb{Z}/4 [X]$? A typical example is $\mathbb{Z}/4 [X] / \langle X^n , 2 X^k \rangle$. For my purposes ...
1
vote
0answers
100 views
Generalize characterization of upper semicontinous functions
Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$
It is easy to show that the following two statements are equivalent:
$(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
5
votes
1answer
113 views
Classification of pointed Hopf algebras up to gauge equivalence
The classification of finite-dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero and whose group of group-like elements is abelian is very much completed. ...
12
votes
1answer
391 views
Classification of non-Hausdorff topological vector spaces
It is well-known that up to topological isomorphism there is exactly one Hausdorff topological vector space (say, over $\mathbb{C}$) of a given dimension $n$, namely $\mathbb{C}^n$ with the euclidean ...
3
votes
0answers
172 views
Smallest number $n$ for which we don't know the classification of all groups of order $n$
I noticed that in groupprops and wikipedia there are often given tables of classifications of groups of small order. This motivated me to ask, what is the current state of research in classifying all ...
2
votes
0answers
75 views
Regular homotopy of punctured surfaces
A theorem of James and Thomas (Note on the classification of cross-sections, Topology 4) asserts that the space of immersions, up to regular homotopy, from a compact surface $S$ into $\mathbb{R}^3$ ...
25
votes
1answer
1k views
Has anyone catalogued the “first generation” proof of the classification of finite simple groups?
It has been estimated that the original proof of the CFSG spans around 15,000 journal pages written by hundreds of authors over most of the 20th century. The GLS project attempted to simplify this ...
1
vote
0answers
141 views
Classification of finite subgroup of $PGSp_4(\mathbb{C})$
Is there a classification of the finite subgroups of $PGSp_4(\mathbb{C})$?
2
votes
1answer
351 views
Classification of cubic surfaces in $\mathbb{P}^3$
We know every cubic surface in $\mathbb{P}^3$ is obtained by blowing up $\mathbb{P}^2$ at 6 points in general position. Hence they are all birational to $\mathbb{P}^2$.
My question is: Do we have ...
18
votes
4answers
926 views
Representation theorem for modular lattices?
Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices?
For example, I ...
2
votes
2answers
476 views
Are the closed and unbounded subsets of $\mathbb{R}$ known up to homeomorphism?
I am currently working on a problem for which this knowledge could greatly reduce the number of cases, but I have yet to find anything after searching online. Are the closed unbounded subsets of $\...
5
votes
0answers
111 views
Heegaard diagrams of prime 3-manifolds
Are there some known results which give a classification of closed prime 3-manifolds up to their Heegaard diagrams? (That is, providing a collection of Heegaard diagrams which exhausts all prime ...
4
votes
1answer
264 views
Classification of finite abelian hypergroups and table algebras
Update: Originally, I formulated this question for finite abelian hypergroups, but in a discussion with Geoff Robinson below I realized that the abelian hypergroups defined below are equivalent to ...
5
votes
1answer
718 views
Is there a structure theorem or group law for finite groups generated by two elements?
Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the ...
5
votes
0answers
263 views
A class 3 group of order 243
Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a (...
4
votes
0answers
163 views
A relation on the set of isomorphism classes of finitely generated groups
Let $G$ be the set of finitely generated groups up to isomorphism hence its elements will be noted $[B]$ where $B$ is some finitely generated group.
On this set we put a relation $\mathcal{ND}$ ("...
40
votes
2answers
4k views
How do you *state* the Classification of finite simple groups?
From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite ...
3
votes
3answers
549 views
First Explicit Irreducible Representations
Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is some book with exhaustive tables describing explicit irreducible ...
3
votes
1answer
201 views
What is the growth of the rank of a power of a finite simple group?
Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
7
votes
0answers
171 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 ...
11
votes
2answers
720 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
413 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 ...
19
votes
2answers
2k 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 ...
5
votes
1answer
319 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 ...
8
votes
2answers
385 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 than ...
2
votes
2answers
269 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 ...
4
votes
1answer
345 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 ...
7
votes
1answer
413 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 ...
10
votes
2answers
1k 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 ...
1
vote
2answers
654 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 ...
23
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
529 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 \...
5
votes
1answer
653 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
256 views
“Locally Cartesian” 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 ...
20
votes
6answers
4k 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 ...
7
votes
7answers
812 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 ...
