Abstract: Expected shortfall, measuring the average outcome (e.g., portfolio loss) above a given quantile of its probability distribution, is a common financial risk measure. The same measure can be used to characterize treatment effects in the tail of an outcome distribution, with applications ranging from policy evaluation in economics and public health to biomedical investigations. Expected shortfall regression is a natural approach of modeling covariate-adjusted expected shortfalls. Because the expected shortfall cannot be written as a solution of an expected loss function at the population level, computational as well as statistical challenges around expected shortfall regression have led to stimulating research. We discuss some recent developments in this area, with a focus on a new optimization-based semiparametric approach to estimation of conditional expected shortfall that adapts well to data heterogeneity with minimal model assumptions. The talk is based on joint work with Yuanzhi Li and Shushu Zhang.
Abstract: When Deligne gave his second proof of the Weil conjectures ("Weil II") in terms of etale local systems on algebraic varieties over finite fields, he observed that the Langlands correspondence for GL(n) over a global function field (now a theorem of L. Lafforgue) would imply that under mild conditions, any etale local system on a curve always has a "geometric origin" in the sense that it appears in the relative cohomology of some family of varieties over the curve. Roughly speaking, this means that the Frobenius action on this local system counts points on some family of varieties parametrized by the curve.
Over a higher-dimensional base, such a statement remains unknown, but Deligne conjectured some concrete corollaries which have subsequently all been proven. We state some of these corollaries and how they follow from work of Deligne, Drinfeld, the speaker, et al.
Abstract: In the 2000s, it was discovered by Fargues and Fontaine that the Galois theory of a mixed-characteristic local field can be described by a certain "curve" (i.e., a one-dimensional regular scheme which is "proper" in a suitable sense); this description is central to the proposed geometrization of local Langlands by Fargues-Scholze. We give a historical account of how the work of Fargues-Scholze emerged naturally from prior results, such as the field of norms construction of Fontaine-Wintenberger (which nowadays is viewed in terms of the "tilting correspondence" in the terminology of Scholze) and a result of the speaker which is retroactively equivalent to the classification of vector bundles on the Fargues-Fontaine curve.
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