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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:

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

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].

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.

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 (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).

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    $\begingroup$ You might be interested in Bridget Tenner's talk Fingerprinting Richard Stanley. $\endgroup$
    – Timothy Chow
    Commented Mar 7, 2020 at 3:02
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    $\begingroup$ Also in the Notices article "Fingerprint Databases for Theorems" by Tenner and Billey. $\endgroup$
    – Russ Woodroofe
    Commented Mar 10, 2020 at 10:36
  • $\begingroup$ @RussWoodroofe's reference: Billey and Tenner - Fingerprint databases for theorems (why reverse the author names?). $\endgroup$
    – LSpice
    Commented Apr 21, 2020 at 14:59
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    $\begingroup$ An Atlas of Even Primes would fill a much-needed gap in the literature. $\endgroup$
    – Gerry Myerson
    Commented Apr 30, 2020 at 3:53
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    $\begingroup$ @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.) $\endgroup$
    – Russ Woodroofe
    Commented May 19, 2020 at 13:29

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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.

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    $\begingroup$ Although mere thanks in comments are discouraged, I cannot resist: very, very useful work, thank you very much for your initiative and effort! $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 4, 2020 at 7:40
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    $\begingroup$ It seems OP is not only collecting datasets but also working to standarize them, great work! $\endgroup$
    – Dabed
    Commented Jan 20, 2021 at 4:25
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A classic: The On-Line Encyclopedia of Integer Sequences (OEIS).

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    $\begingroup$ I'd be very surprised if anyone here hasn't heard of this $\endgroup$
    – BlueRaja
    Commented Mar 8, 2020 at 3:43
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    $\begingroup$ @BlueRaja xkcd.com/1053 $\endgroup$
    – user3067860
    Commented Mar 9, 2020 at 17:45
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    $\begingroup$ I'm one of the lucky people today <3 $\endgroup$
    – Student
    Commented Apr 30, 2020 at 0:06
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Here's another one.

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

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

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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!

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

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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.

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

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The graded rings database

http://www.grdb.co.uk

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

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

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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.

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

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The Database of Ring Theory (and Module Examples) features

  1. over 100 examples of rings
  2. tracking of over 100 different ring theoretic properties each can have.
  3. Information on important subsets (i.e. zero divisors, Jacobson radical)
  4. notions of dimension (composition length, Krull dimension, etc.)
  5. Links to articles and/or books for each ring and property
  6. Properties about the properties (being Morita invariant, passing to polynomial rings, etc.)
  7. 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.

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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.

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There's the Reverse Math Zoo.

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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.

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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)

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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 $n$.

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    $\begingroup$ (No relation.)​ $\endgroup$
    – Oscar Cunningham
    Commented Mar 7, 2020 at 18:54
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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.

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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".

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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!

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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.

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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.

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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.

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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))

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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

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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.

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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.

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