Explore the concept of re-expansion maps as direct connections in this 55-minute lecture by Sylvie Paycha, presented as part of the Hausdorff Junior Trimester Program on Randomness, PDEs, and Nonlinear Fluctuations. Delve into the generalization of direct connections on vector bundles to groupoids and their interpretation in regularity structures. Examine the concept of "gaugeoid fields" in gauge groupoids and their relation to gauge fields. Investigate jet bundles in polynomial regularity structures through topics such as geometric variety structure, symmetric maps, translations, and geometric polynomials. Compare the presented concepts with 3DS gauges and gain insights from this collaborative work with S. Azzali, Y. Boutaïb, and A. Frabetti.