Without demeaning your accomplishments, undergraduate mathematics tends to mean you're barely getting into academia and mathematical research. In fact, I am in the middle of my PhD with some published papers and that's still the case. Because of this, what you're learning/researching in undergrad is not as esoteric as it may seem. The topics at REUs are usually chosen because they are easily approachable by undergraduates and widely applicable.
Let's take Lp-regularity. Sure, you can dig deep into pure-math, looking into Lp-regularity of PDEs on manifolds etc., but this is actually a very relevant topic.
You say you're interested in quantitative finance? One of the biggest areas of research and application in quantitative finance is stochastic (partial) differential equations (SPDEs). Many models in the field (including the Black-Scholes model) use different forms of stochastic differential equations. Understanding the regularity results of these equations is crucial to understanding the error estimates for the numerical methods which are used to solve these models. One of the biggest recent results in the field (which resulted in a Fields Medal in 2014) is Holder-regularity of some solutions to SPDEs (and using a Holder-regularity approach, "Regularity Structures", to famously solve the KPZ equation), and people have already shown how this framework can be used to make numerical methods for previously-intractable SPDEs. In fact, in many cases one can only understand "the solution to SPDEs" via Sobolev spaces and Lp-regularity, making it essential to understand the numerical methods and simulation. So simulating financial models with low error requires the mathematics you learned.
But many PDEs require understanding Lp-regularity since many PDEs require talking about Sobolev spaces. In fact, one of the main uses of modern mathematics in industry is Finite Element Methods (FEM, or Finite Element Analysis, FEA). All of these numerical methods are derived for solutions in Sobolev spaces, this is not a fringe topic: this is central to the simulations used in the areospace industry, the petroleum industry, NASA, etc.
Also, some of the main methods in Data Analysis (especially in manifold learning) these days are closely related to optimization on PDEs, usually proving convergence in some weak norm and using facts about the eigenvalues of LaPlacians that you'd be familiar with.
All of this said, I think the best thing to do would be to scramble through Wikipedia and find out how what you learned is related to all of these different fields. Or pick up the Princeton Companion to Applied Mathematics and see how the study of PDEs is showing up in systems biology, genomics, medicine, finance, computer science, etc. You don't need to know all of it, but a good understanding of where you currently stand would give you a good perspective on how you're useful. I am not an industry person so I can't recommend on how exactly to put this in your resume, but with this understood you should be able to cater resumes to the jobs you're applying for.
Even if you don't end up doing something directly related to PDEs, I think most people would like that you have some understanding all the new stuff that's going on. At the very least, you will seem "experienced with math and number stuff".