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Explore classical problems in orthogonal polynomials and harmonic analysis through the lens of the "unlikely intersection" paradigm in this mathematical lecture. Learn how these problems can be reformulated as questions about counting integers in definable sets, where algebraic components are governed by Ax-Schanuel type theorems. Discover the unique analytical challenges that arise in the irregular-singular context, including the use of steepest descent and asymptotic methods for studying complex "uniformizing maps." Examine how the relevant mathematical sets exist within the o-minimal structure R_{G,exp} of multisummable functions rather than the classical R_{an,exp} setting. Understand how functional transcendence remains controlled by differential Galois theory, but now applied to irregular-singular systems, introducing concepts such as exponential tori and Stokes phenomena into the description of weakly special subvarieties. Gain insights from collaborative research bridging advanced topics in mathematical analysis, algebraic geometry, and differential equations.
