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.
Abstract: We propose an autoregressive framework for modelling dynamic networks with dependent edges. It encompasses models which accommodate, for example, transitivity, density-dependent and other stylized features often observed in real network data. By assuming the edges of network at each time are independent conditionally on their lagged values, the models, which exhibit a close connection with temporal ERGMs, facilitate both simulation and maximum likelihood estimation in a straightforward manner. Due to the possible large number of parameters in the models, the initial MLEs may suffer from slow convergence rates. An improved estimator for each component parameter is proposed based on an iteration employing a projection which mitigates the impact of the other parameters. Leveraging a martingale difference structure, the asymptotic distribution of the improved estimator is derived without a stationarity assumption. The limiting distribution is not normal in general, and it reduces to normal when the underlying process satisfies some mixing conditions. Illustration with a transitivity model was carried out in both simulation and a real network data set. Joint work with Jinyuan Chang, Qin Fang, Peter MacDonald and Qiwei Yao.
Abstract: Many biological systems exhibit common structural behaviors driven by long-range attraction and short-range repulsion. In the continuum limit of large interacting populations, these dynamics are naturally described by aggregation–diffusion equations governing the evolution of population density. These equations capture the balance between diffusive effects, which model population spreading due to high local density, and aggregation forces arising from attraction–repulsion interactions. They can also be interpreted as gradient flows of free-energy functionals, linking them to principles from statistical physics. This talk reviews recent advances in the mathematical understanding of aggregation–diffusion equations, with a particular focus on their role in modeling cell population dynamics and related applications in mathematical biology.
4176 Campus Drive - William E. Kirwan Hall
College Park, MD 20742-4015
P: 301.405.5047 | F: 301.314.0827