Atlas-like websites on specific areas of mathematics

Question

In this post, we look for the existing atlas-like websites providing well-presented classifications or database about some specific areas of mathematics. Here are some examples:

GroupNames: https://people.maths.bris.ac.uk/~matyd/GroupNames

Finite groups of order ≤500, group names, extensions, presentations, properties and character tables.

Atlas of Finite Group Representations: http://brauer.maths.qmul.ac.uk/Atlas/v3/

This ATLAS of Group Representations has been prepared by Robert Wilson, Peter Walsh, Jonathan Tripp, Ibrahim Suleiman, Richard Parker, Simon Norton, Simon Nickerson, Steve Linton, John Bray, and Rachel Abbott (in reverse alphabetical order, because I'm fed up with always being last!). It currently contains information (including 5215 representations) on about 716 groups [mainly finite simple groups or almost simple].

Atlas of subgroup lattice of finite almost simple groups: http://homepages.ulb.ac.be/~dleemans/atlaslat/

This atlas contains all subgroup lattices of almost simple groups $G$ such that $S≤G≤Aut(S)$ and $S$ is a simple group of order less than 1 million appearing in the Atlas of Finite Groups by Conway et al. Some simple groups and almost simple groups or order larger than 1 million have also been included, but not in a systematic way.

The L-functions and Modular Forms Database: https://www.lmfdb.org/

Welcome to the LMFDB, the database of L-functions, modular forms, and related objects. These pages are intended to be a modern handbook including tables, formulas, links, and references for L-functions and their underlying objects [like field extensions and polynomial Galois groups].

The Inverse Symbolic Calculator: https://isc.carma.newcastle.edu.au/

The Inverse Symbolic Calculator (ISC) uses a combination of lookup tables and integer relation algorithms in order to associate a closed form representation with a user-defined, truncated decimal expansion (written as a floating point expression). The lookup tables include a substantial data set compiled by S. Plouffe both before and during his period as an employee at CECM.

If you know such a website on any area of mathematics, please put it as an answer (with a short description).

You might be interested in Bridget Tenner's talk Fingerprinting Richard Stanley. — Timothy Chow, Mar 07 '20 at 03:02
Also in the Notices article "Fingerprint Databases for Theorems" by Tenner and Billey. — Russ Woodroofe, Mar 10 '20 at 10:36
@RussWoodroofe's reference: Billey and Tenner - Fingerprint databases for theorems (why reverse the author names?). — LSpice, Apr 21 '20 at 14:59
An Atlas of Even Primes would fill a much-needed gap in the literature. — Gerry Myerson, Apr 30 '20 at 03:53
@LSpice, I think I put Tenner first because I was replying to Timothy Chow's comment on Tenner's talk. (But you're right, I should've used the canonical order.) — Russ Woodroofe, May 19 '20 at 13:29
The ISC is mentioned as a prime example of this. I would be interested in a currently operating version of the ISC. — Gerald Edgar, May 01 '22 at 13:29
@GeraldEdgar http://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html — Sebastien Palcoux, May 02 '22 at 16:17

score 44 · Answer 1 · answered Mar 06 '20 at 19:14

44

A classic: The On-Line Encyclopedia of Integer Sequences (OEIS).

answered Mar 06 '20 at 19:14

Tobias Fritz

5,785

2

I'd be very surprised if anyone here hasn't heard of this – BlueRaja Mar 08 '20 at 03:43
14

@BlueRaja https://xkcd.com/1053/ – user3067860 Mar 09 '20 at 17:45
6

I'm one of the lucky people today <3 – Student Apr 30 '20 at 00:06

score 33 · Answer 2 · edited May 01 '22 at 01:59

33

Atlas of Lie Groups and Representations: info on lie groups, there is a nice article on the history of this atlas
Atlas of Finite Group Representations: info on finite groups actually the succesor of a book (already mentioned by OP but added it anyway to keep it symmetric with the first point)
Complexity zoo:list of complexity classes and what is know about them
Quantum algorithm zoo: list of quantum algorithms
Knot Atlas: list of knots
Encyclopedia of triangles center: list of triangle centers
Bilbao chrystallographic server: list of space groups and info although more physics oriented
Cantor's Attic: A wiki about large cardinal axioms, large countable ordinals, and infinity in general
Googology wiki: Info on googology
Digital Library of Mathematical Functions: a spiritual successor to a book too and there is also a nice article on the history

edited May 01 '22 at 01:59

C7X

1,276

answered Mar 06 '20 at 21:57

Dabed

256

https://math.la.asu.edu/~jj/localfields/ – Dabed Jun 14 '21 at 13:15
https://polytope.miraheze.org/wiki/Main_Page – Dabed Sep 02 '21 at 19:28
https://xlinux.nist.gov/dads/ – Dabed Sep 03 '21 at 05:13
https://www.2dcurves.com/ – Dabed Sep 05 '21 at 03:55
Does this also count as an atlas page? http://log.chez.com/text/logic/links_ordinals.html – C7X May 05 '22 at 14:19
@C7X I wouldn't call it an atlas page but I wouldn't say either that it less useful because it isn't, feel free to add it to the answer if you want, I have left some other pages as comment not because a particular criteria but only because while I want to mention them at the same time I want to avoid appearing in the feed of active answers. – Dabed May 07 '22 at 04:57
https://errorcorrectionzoo.org/ – Dabed May 20 '23 at 02:01

Katja Berčič · Accepted Answer · 2020-03-10T10:24:07.690

25

This catalogue of mathematical datasets could be of some interest to you - at least some of the entries are atlas-like websites. It includes several of the websites mentioned above, and I'm slowly adding more to it.

edited Mar 10 '20 at 10:24

answered Mar 10 '20 at 10:17

Katja Berčič

126
2
7

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Although mere thanks in comments are discouraged, I cannot resist: very, very useful work, thank you very much for your initiative and effort! – მამუკა ჯიბლაძე May 04 '20 at 07:40
1

It seems OP is not only collecting datasets but also working to standarize them, great work! – Dabed Jan 20 '21 at 04:25

score 21 · Answer 4 · answered Mar 06 '20 at 19:08

21

Here's another one.

The Manifold Atlas Project: http://www.map.mpim-bonn.mpg.de/Main_Page

answered Mar 06 '20 at 19:08

Igor Khavkine

20,710

score 20 · Answer 5 · answered Mar 06 '20 at 22:50

20

Forking and dividing is a map of the model theory universe, visually classifying some 63 first-order theories.

answered Mar 06 '20 at 22:50

score 17 · Answer 6 · answered Mar 06 '20 at 19:12

17

PolyDB is a database of polytopes. It can either be accessed through the website or directly from polymake.

answered Mar 06 '20 at 19:12

Tobias Fritz

5,785

score 16 · Answer 7 · edited Jun 15 '20 at 07:27

16

Another one is the Catalogue of Lattices.

This data-base of lattices is a joint project of Gabriele Nebe, RWTH Aachen university (nebe(AT)math.rwth-aachen.de) and Neil Sloane. (njasloane(AT)gmail.com).

Our aim is to give information about all the interesting lattices in "low" dimensions (and to provide them with a "home page"!). The data-base now contains about 160,000 lattices!

edited Jun 15 '20 at 07:27

Community

1

answered Mar 06 '20 at 19:25

pregunton

976

score 16 · Answer 8 · answered Mar 06 '20 at 22:23

16

I am trying to keep up with the area of symmetric functions, and various generalizations: The symmetric functions catalog

answered Mar 06 '20 at 22:23

Per Alexandersson

15,598

score 15 · Answer 9 · answered Mar 06 '20 at 20:01

15

The Information System on Graph Classes and their Inclusions: http://www.graphclasses.org/

This is a database of graph classes with a java application that helps you to research what's known about particular graph classes.

answered Mar 06 '20 at 20:01

Thomas Kalinowski

2,946

score 14 · Answer 10 · edited Mar 29 '23 at 19:19

14

$\pi$-base has examples of topological spaces and their properties.

edited Mar 29 '23 at 19:19

Steven Clontz

963

answered Mar 07 '20 at 15:32

Bart Michels

348

5

Here are two more on general topology: http://hyperspacewiki.org/index.php/Main_Page https://topospaces.subwiki.org/wiki/Main_Page – erz Mar 08 '20 at 06:17

score 13 · Answer 11 · answered Mar 06 '20 at 19:36

13

The graded rings database

http://www.grdb.co.uk

A database of varieties (toric varieties, Fano varieties, Calabi-Yaus).

answered Mar 06 '20 at 19:36

Jonny Evans

6,935

score 11 · Answer 12 · answered Mar 06 '20 at 19:21

11

Here is a database of Vertex Operator Algebras and Modular Categories, though it is still in an early version.

answered Mar 06 '20 at 19:21

pregunton

976

score 11 · Answer 13 · answered Mar 06 '20 at 22:22

11

The Blocks of Finite Groups wiki, which aims to classify the Morita equivalence classes of blocks with a given defect group. This is in part to understand Donovan's Conjecture better.

answered Mar 06 '20 at 22:22

Carl-Fredrik Nyberg Brodda

6,235

score 10 · Answer 14 · answered Mar 06 '20 at 19:16

10

Maybe not exactly an "atlas", but there is a classification of some certain types of PDEs studied in analysis at

The Dispersive PDE Wiki: https://kvm16.pims.math.ca/DispersiveWiki/index.php?title=Main_Page
WikiWaves: https://wikiwaves.org/

answered Mar 06 '20 at 19:16

Igor Khavkine

20,710

rschwieb · Answer 15 · 2023-04-04T15:01:48.793

The Database of Ring Theory (and Module Examples) features

over 100 examples of rings
tracking of over 100 different ring theoretic properties each can have.
Information on important subsets (i.e. zero divisors, Jacobson radical)
notions of dimension (composition length, Krull dimension, etc.)
Links to articles and/or books for each ring and property
Properties about the properties (being Morita invariant, passing to polynomial rings, etc.)
Sundry information about ring theory and theorems, errata in literature, diagrams of implications

I always welcome suggestions for additions: there is too much work for one person to do.

Update

The site now tracks examples of modules over the rings, too. Not all of the features tracked above are fully supported yet but that is the plan.

score 7 · Answer 16 · answered Mar 07 '20 at 15:18

A classic by Stanisław Radiszowski:

= = = = = = Small Ramsey Numbers = = = = = =

https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1/pdf

Last update on 2017-03-03. All known Ramsey numbers or their estimates are provided for several different types of graphs and colorings.

score 7 · Answer 17 · answered Mar 07 '20 at 19:44

7

There's the Reverse Math Zoo.

answered Mar 07 '20 at 19:44

Henry Towsner

7,115

score 7 · Answer 18 · answered Mar 09 '20 at 13:09

7

Pieter Belmans at the University of Bonn maintains a couple of such websites.

There is Fanography, "a tool to visually study the geography of Fano 3-folds."

And le superficie algebriche, "a tool for studying numerical invariants of minimal algebraic surfaces over the complex numbers". Johan Commelin also contributed to this one.

answered Mar 09 '20 at 13:09

Will R

101

https://pbelmans.ncag.info/blog/atlas/ – Dabed Jun 14 '21 at 13:17

Kristof Huszar · Answer 19 · 2020-03-08T01:10:18.483

Regina and SnapPy are free program suits for research in low-dimensional topology (particularly in dimension 3), and are accompanied by large census databases of closed 3-manifolds, knots and links. Some highlights:

Knot tables up to 19 (!) crossings (Regina)
Closed, orientable 3-maifolds with at most 11 tetrahedra (Regina)
Knots and links with up to 14 crossings (SnapPy)
Various censuses of hyperbolic 3-manifolds (Regina, SnapPy)

score 5 · Answer 20 · answered Mar 07 '20 at 18:52

5

The Cunningham Project seeks to factor the numbers $b^n \pm 1$ for $b = 2$, $3$, $5$, $6$, $7$, $10$, $11$, $12$, up to high powers $n$.

answered Mar 07 '20 at 18:52

Oscar Cunningham

3,077

6

(No relation.) – Oscar Cunningham Mar 07 '20 at 18:54

score 5 · Answer 21 · answered Mar 07 '20 at 20:58

5

KnotInfo https://knotinfo.math.indiana.edu/

LinkInfo https://linkinfo.math.indiana.edu/

These are databases of all knots and links up to a certain number of crossings, along with many different computed invariants and properties.

answered Mar 07 '20 at 20:58

Kyle Miller

413

score 5 · Answer 22 · answered Mar 08 '20 at 21:48

The Atlas of Small Regular Polytopes (Michael Hartley) "contains information about all regular polytopes with $n$ flags where $n$ is at most 2000, and not equal to 1024 or 1536".

The website of Dimitri Leemans has

The website of Marston Conder has many catalogs of information related to maps, hypermaps, polytopes, symmetric graphs, and surface actions.

Combinatorial Data (Brendan McKay) has complete collections of various classes of graph, up to some number of vertices, in graph6 format.

Combinatorial Catalogues (Gordon Royle) has class 2 graphs up to 9 vertices, trees up to 16, bipartite up to 14, 3-regular up to 22, and more.

Regular Graphs (M. Meringer) has information and shortcode files for "simple connected $k$-regular graphs on $n$ vertices and girth at least $g$ with given parameters $n,k,g$."

The House of Graphs is a searchable "Database of interesting graphs".

Tobias Fritz · Answer 23 · 2020-03-09T13:49:28.223

5

There is a database of matroids. For example, it includes a list of all $4,886,380,924$ isomorphism classes of matroids of rank 4 on 10 elements!

edited Mar 09 '20 at 13:49

answered Mar 09 '20 at 13:26

Tobias Fritz

5,785

score 4 · Answer 24 · answered Mar 07 '20 at 14:02

If you are interested in historical things,

John Jones' number fields database

and

William Stein's modular forms database (which also includes elliptic curve data, such as the Cremona database) used to be standard such websites. Now this information is subsumed into the

L-functions and modular forms database (LMFDB)

mentioned by the OP.

score 4 · Answer 25 · answered Mar 08 '20 at 00:08

4

A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.)

The site also provides links to similar databases.

answered Mar 08 '20 at 00:08

Andrés E. Caicedo

32,193
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score 3 · Answer 26 · answered Mar 07 '20 at 15:34

3

Another classic, this time about prime numbers, it provides some theoretical results and a lot of data:

= = = = = = https://primes.utm.edu/ = = = = = =

There are many more pages and portals devoted to elementary number theory.

answered Mar 07 '20 at 15:34

Wlod AA

4,686
16
23

score 3 · Answer 27 · answered Mar 07 '20 at 23:37

3

Enumerating Order Types for Small Point Sets with Applications

Database (maintained by Oswin Aichholzer, TU Graz)
Background paper (by O. Aichholzer, F. Auerhammer, H. Krasser (2002))

answered Mar 07 '20 at 23:37

Kristof Huszar

11

score 3 · Answer 28 · answered Mar 08 '20 at 06:41

Many Internet pages provide (or used to) lotto design tables. In particular, the following paper seems solid:

= = = Lotto Design Tables (by P. C. Li and G. H. J. van Rees),

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.5.4298&rep=rep1&type=pdf

score 3 · Answer 29 · answered Mar 11 '20 at 22:25

Table of absolutely maximally entangled states.

Abbreviated as AME(n,D), the table lists quantum states of n parties with D levels each that show maximal entanglement across every bipartition. These states are equivalent to pure $((n,1,n/2+1))_D$ quantum error correcting codes, and are in the case of n even also known as perfect tensors or multi-unitary matrices. They can be seen as a type of self-dual quantum codes.

score 1 · Answer 30 · answered Dec 29 '21 at 20:56

1

http://www.polychora.de/wiki/images/ has renders of cross-sections, vertex figures and their duals, and cellets for many (possibly all) of the 1849 uniform polychora known at the time it was created.

answered Dec 29 '21 at 20:56

Daniel Sebald

2,689

score -3 · Answer 31 · answered Mar 07 '20 at 15:36

-3

You can look at HandWiki https://handwiki.org/ too. It is a dedicated resource for mathematics

answered Mar 07 '20 at 15:36

IraS

101

5

How is that different from Wikipedia? Looking at a few random articles suggests that most or all of the content has been copied from Wikipedia. – Tobias Fritz Mar 09 '20 at 13:22

Atlas-like websites on specific areas of mathematics

31 Answers31

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