Mathematical conjectures on which applications depend What are some examples of mathematical conjectures that applied mathematicians assume to be true in applications, despite it being unknown whether or not they are true?
 A: An important specific conjecture is that you cannot factor large integers fast. Many security systems for Internet and other transactions, depend on this.
A: It is an open problem to resolve a question formalized by G. Shephard in 1975:

Q. Can the surface of every
  convex polyhedron be cut along edges and unfolded flat to one non-self-overlapping polygon in the plane? 

It is usually called Dürer's problem because there is a sense
in which it goes back to Dürer. But I am not sure it is fair to call it
a "conjecture," one reason this is not an ideal answer to the posed query.
It is also a bit of a stretch to claim there is a direct application.
But I did find a Ph.D. thesis in mechanical engineering that lamented, 
"there is no theorem or efficient algorithm that can tell if a given 3D shape is unfoldable [without overlap] or not." Despite Dürer's problem being unresolved,
any engineer or architect who wants to unfold a 3D shape—say, to punch it out of aluminum—breaks it into convex pieces (one piece if already convex), and easily edge-unfolds the pieces without overlap. The lack of a resolution to the open problem is no impediment to actually finding edge-unfoldings
in practice. 

               


               

Fig. from Geometric Folding Algorithms, p.299:
Soccer ball unfolding.



J.O'Rourke, "Dürer's problem."
  In Marjorie Senechal, editor, Shaping Space: Exploring Polyhedra
    in Nature, Art, and the Geometrical Imagination, pages 77--86. Springer,
    2013. (Springer link.)

A: The use of RSA for public-key encryption is widely believed to rely on the assumption that factoring is hard. Actually it relies on a stronger assumption than this, namely that the RSA problem is hard. 
The RSA problem is the following: given a semiprime $N = pq$ and an exponent $e$ such that $\gcd(e, \varphi(N)) = 1$, efficiently compute $e^{th}$ roots $\bmod N$. It is widely believed that the only way to do this is to compute $e^{-1} \bmod \varphi(N)$ where $\varphi(N) = N - p - q + 1$, and in turn it is widely believed that the only way to do this is to factor $N$ so as to compute $\varphi(N)$. However, strictly speaking both of these are conjectures which are independent of the conjecture that factoring is hard.
So it may be that factoring is hard but that the RSA problem is easy because there is some clever way to avoid these steps and solve the RSA problem without factoring $N$. Note also that we only need to factor semiprimes; it may also be that factoring is hard but that factoring semiprimes is easy. 
A: The Miller-Rabin primality test works very well in practice as a probabilistic algorithm for finding "practical" (not provable) primes in cryptography, but the algorithm would become an efficient polynomial-time deterministic algorithm if the generalized Riemann hypothesis is true for all Dirichlet $L$-functions (well, maybe just for even characters is enough). In practice I don't think anybody in cryptography cares about GRH being true or not in order to be content using this primality test precisely because of the probabilistic version of it that does not depend on GRH, so this example might not strictly be an answer to the original question, but I think it is a good approximation to an answer.
A: Although we have a proof now, I suspect grocers had intuitively stacked oranges in the most efficient way prior to the proof of Kepler's conjecture.  I don't know for sure if any grocer was an applied mathematician.
https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/
A: Navier Stokes equations are believed to be well-posed.
A: I think the biggest example is $P \neq NP$. Security experts routinely assume this to be true when designing security algorithms. The correct functioning of millions of machines depend on this.
I think its also the biggest example because if someone proves it false, then, depending on the details of the proof, the applications literally stop performing their intended function (or not, depending on the details of the proof).
A: Many cryptographic protocols are based on the problem which seem to be hard. Among them Discrete logarithm problem and its variants (Diffie–Hellman problem, Decisional Diffie–Hellman assumption, Elliptic Curve Discrete Logarithm Problem), Quadratic residuosity problem, lattice problems (Short integer solution problem, Shortest basis problem,...),...
A: Tonelli–Shanks algorithm needs quadratic nonresidue $n$ modulo prime $p$. If the extended Riemann hypothesis is true, then the first quadratic nonresidue $n_p$ is always less than $3(\log p)^2/2$ (Wedeniwski 2001) for $ p>3$. Under this assumption Tonelli–Shanks algorithm becomes polynomial. Probabilistic heuristics (presuming that each non-square integer has a 50-50 chance of being a quadratic residue) suggests that $n_p$ should have size $O( \log p )$. The best unconditional estimate (Burgess) is $n_p\ll p^\theta$ for any $\theta >1/4\sqrt e$. 
See also extended discussion at The least quadratic nonresidue, and the square root barrier.
