Termine

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In these two talks I will present my current project on endowing a torsor structure on L-packet of the covering tori. In the first half I will talk about how to construct gerbes on the groupoid of isocrystals, and a formulation of the Langlands correspondence of sharp covering tori on the level of gerbes. In the second half I will talk about how the groupoid of isocrystals of the sharp tori can help to interpolate the L-packets of different extended pure inner forms of the covering tori, which gives rise to the desired torsor structure.

Perfectoid algebras and spaces are fundamental objects in modern p-adic geometry. However, many of the foundational results about them are only true "up to almost mathematics". By contrast, passing to Banach ultraproducts can eliminate this ?almost?. Our aim is to identify the structural properties of these Banach ultraproducts that make this possible. Joint work in progress with Franziska Jahnke.

Minimal surfaces in a random environment arise naturally in the homogenization of models in fracture mechanics. They also are one of the natural higher-dimensional generalizations of planar first-passage percolation, the problem of finding the shortest path in a random environment. Under suitable assumptions on the probability distribution - assuming in particular statistical isotropy of the random medium -, we establish an algebraic rate of convergence for the energy of the minimal surface dividing a box $(-R,R)^d$ in the limit $R\rightarrow \infty$. We also prove a superconcentration principle for energy fluctuations in up to four ambient spatial dimensions: Energy fluctuations of the minimal surface are suppressed, as compared to the the energy fluctuations of a fixed surface like $(-R,R)^{d-1} \times \{0\}$. We discuss possible implications of the superconcentration principle concerning the accuracy of variational models for fracture in the setting of heterogeneous media. joint works with Antonio Agresti, Nicolas Clozeau, Assylbek Olzhabayev, Christian Wagner