Abstract: Counting mod p points on Shimura varieties has been for a few decades the main avenue for establishing non-abelian reciprocity laws. This began with the work of Deligne and Langlands on the modular curve, continued with that of Kottwitz on PEL type Shimura varieties, and has culminated in recent work of Kisin and Kisin-Shin-Zhu (KSZ) on varieties of abelian type. All of these results depend ultimately on a serious use of isogenies between abelian varieties with additional structure. In this talk, I’ll explain how to prove variants of the Langlands-Rapoport conjectures formulated by KSZ using structural properties of integral models that avoid any discussion of abelian varieties or even p-divisible groups, and works also for many exceptional Shimura data for large enough p. This is based on joint work with Alex Youcis and also with Si Ying Lee.
Abstract: Mammalian sleep consists of repeated ultradian cycles between non-REM (NREM) and rapid-eye-movement (REM) sleep. Although the timing of these cycles is not fully understood, a key contributor is thought to be REM pressure, a drive for REM sleep that accumulates between REM episodes. Building on prior work in mice, we introduced a REM propensity measure that quantifies the probability of entering REM sleep as a function of accumulated NREM sleep. In mice, REM propensity at REM onset was positively associated with both REM bout duration and the likelihood of short, sequential REM cycles. Here, we extend this framework to human and rat sleep. We show that ultradian cycles in all three species can be classified as either short sequential or longer single REM cycles, and that REM propensity exhibits a conserved dependence on time spent in NREM sleep, rising to a peak and then declining. Across species, higher REM propensity at REM onset predicts longer REM bouts, suggesting a shared role of NREM accumulation in shaping REM duration. Finally, analysis of human sleep reveals systematic variation in the occurrence of sequential and single REM cycles across the sleep episode.
Abstract: Optimal transport has been an essential tool for reconstructing dynamics from complex data. With the increasingly available multifaceted data, a system can often be characterized across multiple spaces. Therefore, it is crucial to maintain coherence in the dynamics across these diverse spaces. To address this challenge, we introduce Synchronized Optimal Transport (SyncOT), a novel approach to jointly model dynamics that represent the same system through multiple spaces. With given correspondence between the spaces, SyncOT minimizes the aggregated cost of the dynamics induced across all considered spaces. The problem is discretized into a finite-dimensional convex problem using a staggered grid. Primal-dual algorithm-based approaches are then developed to solve the discretized problem. Various numerical experiments demonstrate the capabilities and properties of SyncOT and validate the effectiveness of the proposed algorithms.
Abstract: The Workshop will draw together sessions on the following topics: (i) examples from Survey Sampling, where Variance Estimation for Design-based inference from surveys uses resampled or reweighted data replicates, and in current applications reweighting may incorporate machine-learning or network methodologies; (ii) UQ in mechanistic dynamical-system models arising in mathematical epidemiology, incorporating interacting disease-transmission and human behavioral effects, where uncertainty enters through noisy data, stochastic dynamics, and through parameterization and calibration of the model; (iii) bootstrap and other resampling methods in artificial-intelligence and machine-learning use cases, including ensemble learning, resampling for robust learning, and resampling in the context of generative AI; (iv) Uncertainty quantification in Bayesian and variational Bayes methods, including applications to deep neural network models; and (v) nascent uncertainty quantification methods for inference from Networks, using asymptotic statistical properties of estimators and other methods to account for the complex dependency structure of network data.