Could you recommend economic papers or books that exhibit exceptionally creative applications of mathematics? In other words, if you were to demonstrate to a mathematician the joy and potential of applying their skills to economic problems, what sources would you suggest?
Turnpike theory uses fairly sophisticated mathematics to obtain results (which may appear surprising to many) about optimal economic growth over the long term. The theory appears to have its origin in J V Neumann's 1945 paper A Model of General Economic Equilibrium. As an introduction to the topic I suggest pages 1-7 of Geshkovski B & Zuazua E (2022) Turnpike in Optimal Control of PDEs, Resnets and Beyond. The paper also includes a detailed list of references.
If you're interested in a book, then this is a good one: Microeconomic Theory by Mascollel, Whinston and Green
But if you just want to show mathematicians the joy and potential of applying their skills to economic problems, you can ask them to solve optimization problems such as utility maximization, profit maximization, finding Pareto efficient allocations, and finding competitive equilibrium in different environments. You can find many examples on economics Stack Exchange. If the mathematician is unfamiliar with these concepts, you may need to define the concepts.
In my opinion, General Economic Equilibrium is one the most important and beautiful theoretical constructions and achievements of applied mathematics of the past century and beyond.
Beginning with the pioneering work by Léon Walras, his successor in the chair at the University of Losanne Vilfredo Pareto, through Hicks, Samuelson, Arrow, Hahn, Debreu and many others, until present days. Hicks, Samuelson, Arrow, Debreu, are all Nobel laureates in economics.
The mathematical importance of General Economic Equilibrium was acknowledged by Stephen Smale, a famous mathematician, Field medalist in 1966, who was interested in economics.
He included a question of General Equilibrium, the problem of price formation, in his list of unresolved mathematical problems for the XXI century:
His 8th problem was: Extend the mathematical model of general equilibrium theory to include price adjustments.
Standard references of modern General Ecomomic Equilibrium are:
Debreu, G., Theory of Value, 1959, Yale University Press.
Arrow, K. J., Hahn, F.H, General Competitive Analysis, 1971 first edition, North Holland.
Debreu was influenced by Bourbaki, and introduced Bourbakism into Economic Theory, with axiomatic method, topology, convexity, instead of analytical tools. This approach led also to controversies, see for instance:
On the opposite side of the abstract and theoretical field of General Equilibrium, if we speak of more applied mathematicians, there is nowadays the vast field of Computational Economics, see for example the journal of the Society for Computational Economics:
Lucas and Rational Expectations. So elegant that it rebuilt macroeconomics, laying the foundations for "modern macro". https://julia.quantecon.org/multi_agent_models/rational_expectations.html
Robert E Lucas, Jr. and Edward C Prescott. Investment under uncertainty. Econometrica: Journal of the Econometric Society, pages 659–681, 1971. http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/adjustment%20costs/LucasPrescott%20Econometrica%2071.pdf
Search-theoretical models of money are a narrow field of application of the above, probably looking for mathematical minds http://homepage.ntu.edu.tw/~yitingli/file/Money/introduction_2020_new_r1.pdf
An algebraic topology formulation: not an "application within economics" but rather a linking from discrete math proof in economic theory (Arrow's impossibility theorem; social choice in MWG) to another area of math. So it's "pure" math! https://www.sciencedirect.com/science/article/abs/pii/S0096300305002936
meta: mathematical elegance (like Lucas) has done more damage to the reputation of the economics profession amongst social sciences and society, than helped move the discipline forward
(belongs in a comment, but i'm not allowed)