A difficult question in the local Langlands framework is to understand the interplay between the characters of irreducible smooth representations of a reductive group over a local field and the geometry of the dual space of Langlands parameters. An important invariant of the character (viewed as a distribution, i.e, a continuous linear functional on the space of smooth compactly supported functions) is the wavefront set, a measure of its singularities along with their directions. Motivated by the work of Adams, Barbasch, and Vogan for real reductive groups, it is natural to expect that the wavefront set is dual (in a certain sense) to the geometric singular support of the Langlands parameter. Dan Ciubotaru will give an overview of these ideas and describe recent progress in establishing a precise connection for representations of reductive p-adic groups. 

Modern ML methods for derivatives sit at a delicate interface between market prices, implied-volatility (IV) surfaces, and the simulated environments produced by market generators. To date, these models have largely operated in one of two coordinate systems: price space, where markets quote and no-arbitrage constraints are most naturally enforced, and IV space, where surfaces are smoothed, regularized, and evaluated. This talk presents a technique that unifies learning across both coordinates — using gradients from each via a differentiable Jäckel operator and a low-vega gating mechanism — enabling end-to-end batch training without the error-prone, expensive, hand-engineered filtering usually needed to discard incompatible IV values. I will present PIVOT (Price-Implied Volatility Operator Transform), a differentiable Jäckel IV operator that preserves the accuracy of the standard "Let's Be Rational" (LBR) solver in the forward pass while supplying implicit gradients through the Black–Scholes/Black-76 price map. This gives neural volatility-surface models a principled bridge between price-space and IV-space objectives, with explicit handling of the low-vega singular regime. Second, I will  present Fast-Vollib ( https://pypi.org/project/fast-vollib/), a CUDA-accelerated option-pricing library with NumPy, PyTorch, and JAX interfaces, built for high-throughput pricing-label generation in AI/ML batch training.