What are some extensively studied mathematical objects NOT yet proven to exist?

Question

I am currently making a presentation for some undergraduate students entering a mathematics B.S. program. I'd like to add a slide about the fun of math, and part of that (to myself, at least) is that a mathematician is bounded only by his or her imagination.

So, to that end, I'd like to add a slide introducing some extensively studied mathematical objects with the caveat that an existence proof doesn't exist. Additionally, I myself am pretty curious about this. I've searched the internet and MSE, but couldn't find a good list of such objects.

Don't worry about the complexity of the topic - we aren't going into depth on these!

What are some objects that have had their properties deeply studied, but an existence proof does not yet exist?

EDIT: To clarify, I do not mean "existence" as existing in the physical world. I don't mean things like "do imaginary numbers exist in the real world?" or "do dual numbers have an analogue in the real world?" I mean something more akin to the solution of a differential equation; properties of solutions to differential equations can be described, and these solutions can be proven to obey certain rules, but these proofs are entirely separate from an actual proof of the existence of a solution to such a differential equation. (Sub-note: I am using a differential equation to illustrate my question. I am not only looking for differential equations that aren't proven to have solutions - other mathematical objects, like topological manifolds, algebraic sets, and the like are all welcome!)

Answer 1

This happens a lot when any open problem says that something doesn't exist: to try to make progress on that problem, we prove a lot of things about the thing that doesn't exist, with the hope of getting a contradiction.

An example already mentioned in the comments is odd perfect numbers. Wikipedia has a long list of properties that they'd have to have. For example, one of their citations, Odd perfect numbers are greater than $10^{1500}$ by Ochem and Rao, proves the thing that it promises in the title, and also these numbers must have at least $101$ prime factors (with multiplicity), and also they must have a prime power divisor bigger than $10^{62}$.

For another example, there's Hajós's conjecture that every graph with chromatic number $k$ contains a subdivision of $K_k$. Actually, this conjecture is now known to be false for $k \ge 7$; it's been proven for $k \le 4$. So cases $k=5$ and $k=6$ are open, and more people are working on $k=5$. As a result...

...a Hajós graph is a graph which has no $K_5$ subdivision, but is not $4$-colorable, and has as few vertices as possible subject to these constraints. We don't know if any such graphs exist, and it's quite likely they don't! But you can find many papers proving properties of Hajós graphs.For example, from the four-color theorem, it follows that they're not planar; from Kuratowski's theorem, it follows that they contain a subdivision of $K_{3,3}$; other work has shown that they must be $4$-connected, but cannot be $5$-connected; and so forth.

We can get more examples if "proving that something must be very large" is good enough for us to say that mathematicians are proving things about this object that doesn't exist. Then, for any kind of conjecture about the integers, you get people verifying numerically that it holds for all small cases, and so all counterexamples must be very big.

Answer 2

Perhaps what you mean is something like the Navier-Stokes equations. These equations and properties of their solutions have been extensively studied but it is a famous open problem as to whether smooth global solutions exist for any given smooth initial data.

Answer 3

This used to be a good example but it isn't anymore: there was a period during the search for the classification of finite simple groups where some of the sporadic groups were conjectured to exist but had not been constructed yet, most notably the Monster, which was predicted to exist by Fischer in 1973 and Griess in 1976, and constructed by Griess in 1982. So there was a period where people were publishing results describing properties of these conjectural groups before knowing that they existed.

An example which is still a good example: there's an object in mathematical physics called the 6-dimensional (2, 0) superconformal field theory, or "Theory X," which is predicted to exist because of string-theoretic arguments but has not been constructed at even a physical level of rigor. Nonetheless people have written a lot about its conjectural properties and relationships to other field theories and other mathematical topics such as the geometric Langlands correspondence, which are abundant. The Wikipedia article and nLab page have links to references and you can also check out the videos and lecture notes from this 2014 workshop on the subject, which I had the privilege of attending.

Answer 4

Here is a famous old example which is not an example any more: of all plane closed curves with given length, find the one which encloses the largest area.

If I have got the history correct, Steiner showed that for any such curve which is not a circle, we can find a curve of the same length which encloses a greater area. This does not quite solve the problem, as it is conceivable that there is no maximum area: for any curve at all, we can find a curve with the same length enclosing a greater area. So at this stage, it is possible that the answer to the problem is: "there is no such curve".

Later (not very much later, I think) it was shown that the circle is in fact the solution to the problem.

Answer 5

A good example in combinatorics is the missing Moore graph, a hypothetical regular graph with degree 57, diameter 2 and order 3250. This is the only case for which the existence of an odd-girth Moore graph is not settled (there are only two more non-trivial examples, the Petersen graph with degree 3 and the Hoffman-Singleton graph with degree 7, both of which have diameter 2). Its structure has been researched extensively in a series of papers (for example, a lot is known about its symmetry group). You can find more references in 'A survey on the missing Moore graph' by Dalfo.

More generally, the existence of several smaller strongly regular graphs is an open question and their structure has been studied intensively. An example is Conway's 99-graph problem, which asks for a strongly regular graph with parameters (99,14,1,2).

Answer 6

My favorite is the empty set. It is not proven to exist; that is why an axiom is required to bring it to life.