Questions tagged [classification]
Classification of various mathematical structures. For classification in the sense of statistics / machine learning, use [tag:statistical-classification].
73 questions
4 votes
0 answers
98 views
What are open classification problems that are conjectured to lead to a Dynkin type classification?
Question: What are open problems in mathematics where the solution is conjectured to lead to a Dynkin type classification but this classification is not yet proven or incomplete? Here an example: For ...
- 28.1k
1 vote
1 answer
202 views
Classifying manifolds that are tangentially homotopy equivalent to a product manifold
Let $M,N$ be two smooth manifolds which is parallelizable. Let $S_t(M),S_t(N)$ be the set of "diffeomorphism classes of manifolds tangentially homotopy equivalent to $M,N$". Suppose, $S_t(M)=...
4 votes
1 answer
224 views
Simple Lie algebras of dimension 3 over non algebraically closed fields
Let $K$ be a not algebraically closed field (not necessarily of characteristic 0). Clearly $\mathfrak{sl}_2(K)$ and $\mathfrak{so}_3(K)$ are simple Lie algebras of dimension 3 over $K$ (and they may ...
- 335
3 votes
0 answers
138 views
Finding certain elements in the smooth structure set of a manifold
Let $M$ be a manifold with fundamental group $Z_2$. $S^{diff}(M)$ and $S^{diff}_t(M)$ respectively denotes smooth structure set of $M$ and Tangential smooth structure set of $M$. then, we have 2 maps ...
3 votes
0 answers
162 views
Some details about the classifications of surfaces according Ricci flow
I'm a prospective graduate student organizing some results about Ricci soliton. P. Petersen and W. Wylie in "On the classification of gradient Ricci solitons" gave a remark on R. S. Hamilton'...
- 71
14 votes
1 answer
1k views
Higher Obstruction Theory?
Obstruction theory is an intuitive yet powerful method in homotopy theory. It helps study the space of maps (more generally, sections of bundles) inductively on skeletons. For example, it provides a ...
- 5,748
9 votes
0 answers
262 views
Reduction of the $0$-handle data in Lurie's classification of TFT
A vital part of Jacob Lurie's classification of fully extended topological field theories [1], very roughly, says that any representation of the n-Cobordism category $Z: {\rm Cob}_{{n}} \to C$ depends ...
- 5,748
1 vote
0 answers
182 views
Classify all open affine subschemes of a projective variety
Currently I am pondering a question from algebraic geometry that can be stated in very simple terms: Let $X \subseteq P^n_k$ be a projective variety, subscheme of projective $n$-space over an ...
- 748
4 votes
0 answers
202 views
Can one formalize the prevalence of the Big Five systems of reverse math?
Simpson's systems of second order arithmetic turn out to be five in number; to simplify notation let's denote them A, B, C, D, E. What seems to be an empirical observation is that most theorems in ...
- 18k
1 vote
1 answer
165 views
Bayes classifiers with cost of misclassification
A minimum ECM classifier disciminate the features $\underline{x}$ to belong to class $t$ ($\delta(\underline{x}) = t$) if $\forall j \ne t$: $$\sum_{k\ne t} c(t|k) f_k(\underline{x})p_k \le \sum_{k\ne ...
- 31
0 votes
1 answer
177 views
Classification of all connected simple real Lie groups?
Is there an explicit(!) classification of all(!) connected real simple Lie groups up to isomorphism? Not just simply connected or adjoint, but all of them?
- 319
1 vote
0 answers
125 views
Almost simple groups and their involutions without CFSG
Suppose $A$ is a finite almost simple group (meaning, by definition, that there exists a finite simple group $P$ such that $P \leq A \leq \mathrm{Aut}(P)$). Suppose furthermore that $A$ acts $2$-...
- 4,791
4 votes
0 answers
154 views
Finding inverses of certain elements in the set of normal invariants of a smooth manifold
Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . ...
- 141
4 votes
1 answer
161 views
CFSG-free proof for classifying simple $K_3$-group
Let $G$ be a finite nonabelian simple group. We call $G$ a $K_3$-group if $|G|=p^aq^br^c$ where $p,q,r$ are distinct primes and $a,b,c$ are positive integers. My question is: Is there a CFSG-free ...
- 643
3 votes
0 answers
192 views
Understanding segments in Bernstein-Zelevinsky Classification
All reps shall be admissible in what follows. Let $k$ be a non-arch. field, $n = a\cdot b$ natural numbers and $P = M \cdot N \subset \mathrm{GL}_n(k)$ the standard parabolic subgroup with $$ M = \...
- 778
4 votes
1 answer
483 views
Perceptron / logistic regression accuracy on the n-bit parity problem
$\DeclareMathOperator{\sgn}{sign}$The perceptron (similarly, logistic regression) of the form $y=\sgn(w^T \cdot x+b)$ is famously known for its inability to solve the XOR problem, meaning it can get ...
- 141
17 votes
2 answers
1k views
Counting degrees of freedom in Lie algebra structure constants (aka why are there any nontrivial Lie algebras of dim >5?)
This is a question about the true number of constraints imposed by the Jacobi identity on the structure constants of a Lie algebra. For an $n$-dimensional Lie algebra, there are $\frac{n^2(n-1)}{2}$ ...
- 285
6 votes
1 answer
333 views
Classification results
A typical classification result for a class $C$ of objects looks like that: Theorem. Each object of $C$ is isomorphic to one object of the following list: [insert list here]. Examples are the ...
- 91
13 votes
2 answers
616 views
Wild classification problems and Borel reducibility
My question is whether the archetype of 'wild' problems in algebra, namely classifying pairs of square matrices up to similarity, is 'non-smooth' in the sense of Borel reducibility. This was ...
- 24.5k
2 votes
0 answers
273 views
Centralizers of automorphisms in finite simple groups (reference request)
I would like to have a precise version of the following statement and, if possible, a reference to such a statement in some standard book. Claim 1: Let $G$ be a finite simple non abelian group with ...
- 177
5 votes
2 answers
240 views
Twisted root subgroups in twisted Chevalley groups (reference request)
I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups. Let me first recall the classical set-up. According to Steinberg'...
- 177
3 votes
1 answer
204 views
Is there a classification of the first geodesic nets?
A geodesic net is an embedding of a multigraph $(V,E)$ into a Riemannian manifold $(M,g)$, so that the vertices are mapped to points of $M$ and the edges to geodesics connecting them. Additionally, ...
- 5,163
7 votes
0 answers
306 views
Classification of octonionic reflection groups
I know that there exist classification theorems for real, complex, and quaternionic, reflection groups. There are presentations for the real reflection groups, as well as further presentations for the ...
- 113
13 votes
1 answer
506 views
Properties of finite dimensional, real division algebras that yield only $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$
It is a classical result by Kervaire and Milnor that every finite-dimensional, real division algebra has dimension 1, 2, 4 or 8, with the most prominent examples being $\mathbb{R}$, $\mathbb{C}$, $\...
5 votes
0 answers
206 views
Finite simple groups of automorphisms of finite simple Lie algebras
I begin by briefly recalling some basic facts in order to pose my question in context. According to the classification, the finite simple groups are cyclic of prime order, are alternating on $n \geq 5$...
- 177
2 votes
0 answers
145 views
Deduce Sheffer's classification of orthogonal polynomials of A-type 0
Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's ...
6 votes
1 answer
294 views
Is there a known classification of regular multiplicity-free permutation groups?
The question is in the title, but let me clarify the terminology. I consider a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on a finite set $\Omega$. $\Sigma$ is regular if it acts ...
- 14.5k
3 votes
0 answers
124 views
Are there any zeta functions with concurrent derivative shifts in multiple variables?
Expressions for rational zeta series have been obtained by considering the Taylor series of zeta functions. For instance, one has \begin{align}\zeta(s,x+y) &= \sum_{k=0}^{\infty} \frac{y^{k}}{k!} \...
- 5,025
13 votes
0 answers
816 views
What are the known convex polyhedra with congruent faces?
Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
- 3,352
2 votes
0 answers
137 views
Can all (inverse) trigonometric functions with periodic iterates be characterized?
I wonder whether all (composites of) trigonometric and inverse trigonometric functions with periodic functional iterations can be found. In order to specify what I mean by that, let's introduce some ...
- 5,025
1 vote
2 answers
748 views
Difference between semilinear and fully nonlinear
I'm confused why the Hamilton Jacobi Bellman equation: $$\frac{\partial u}{\partial t}(t,x)+\Delta u(t,x) -\lambda||\nabla u(t,x) ||^{2}=0$$ is considered fully nonlinear, but not semilinear. By ...
- 11
6 votes
1 answer
765 views
On classifying groups of order $p^5$
Can someone suggest me some source where the author has classified all non-isomorphic groups of order $p^5$ ? I need complete classification (not upto isoclinism), and also in finitely presented form ....
- 391
3 votes
1 answer
495 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, ...
- 453
6 votes
1 answer
511 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? ...
- 5,552
3 votes
0 answers
81 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 ...
- 14.5k
3 votes
1 answer
5k 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 ...
- 83
2 votes
1 answer
95 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 ...
- 2,606
9 votes
2 answers
618 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 ...
- 3,808
5 votes
1 answer
577 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
0 answers
136 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 ...
- 623
5 votes
1 answer
214 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. ...
- 395
17 votes
1 answer
874 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 ...
- 5,552
4 votes
0 answers
269 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 ...
- 3,139
2 votes
0 answers
112 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 $\mathbb{R}^3$ ...
- 1,636
28 votes
1 answer
2k 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 ...
- 1,302
2 votes
0 answers
186 views
Classification of finite subgroup of $PGSp_4(\mathbb{C})$
Is there a classification of the finite subgroups of $PGSp_4(\mathbb{C})$?
- 1,066
2 votes
1 answer
868 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 ...
- 815
19 votes
4 answers
2k 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
2 answers
833 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 $\...
- 143
5 votes
0 answers
142 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 ...
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