Abstract: In this meeting, 1) We will present potential project ideas 2) Participants will share project ideas if wanted 3) We will start working on team formation
To-do before session: - Continue reading the Clifford hierarchy paper: https://arxiv.org/pdf/2501.07939 - If you have project ideas you'd like to share, come ready with a short pitch (no more than 2-4mins), or send your idea to Daniel, Ari and Kyle ahead of Monday
Abstract: The Wasserstein metric over a metric space X is an optimal-transport based distance on the set of probability measures on X. Metric spaces for which the optimal transport problem is "easiest" to solve are trees, meaning the Wasserstein metric on trees isometrically embeds into L1. Property A is a coarse invariant of metric spaces introduced by Yu as an approach to solving the coarse Baum-Connes conjecture. We prove a new characterization of bounded degree graphs X with Property A as precisely those that are coarsely equivalent to another space Y whose Wasserstein metric admits a biLipschitz embedding into L1. Applications to group actions on Banach spaces will be discussed. Based on ongoing work with Tianyi Zheng and Ignacio Vergara.
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: Let X be a compact Kähler manifold. Calabi asked whether a given Kähler class on X contains a "canonical" Kähler metric, such as an extremal metric. Roughly speaking, the Yau-Tian-Donaldson conjecture states that if the Kähler class is the first Chern class of an ample line bundle, then the existence of an extremal metric should be governed by an algebro-geometric stability condition. I will present joint work with S. Boucksom, where we prove a version of this conjecture.
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.
Abstract: We consider a first passage percolation model in dimension 1 + 1 with potential given by the product of a spatial i.i.d. potential with symmetric bounded distribution and an independent i.i.d. in time sequence of signs. We assume that the density of the spatial potential near the edge of its support behaves as a power, with exponent κ > −1. We investigate the linear growth rate of the actions of optimal point-to-point lazy random walk paths as a function of the path slope and describe the structure of the resulting shape function. It has a corner at 0 and, although its restriction to positive slopes cannot be linear, we prove that it has a flat edge near 0 if κ > 0. For optimal point-to-line paths, we study their actions and locations of favorable edges that the paths tend to reach and stay at. Under an additional assumption on the time it takes for the optimal path to reach the favorable location, we prove that appropriately normalized actions converge to a limiting distribution that can be viewed as a counterpart of the Tracy–Widom law. Since the scaling exponent and the limiting distribution depend only on the parameter κ, our results provide a description of a new universality class.
Abstract: The Cremona dimension of a finite group G is the minimal dimension of a rationally connected variety which admits a faithful action of G. In this talk, based on joint work with Giulio Bresciani and Angelo Vistoli, I will discuss new lower bounds on the Cremona dimension of a finite p-group.
Abstract: "On-scale" stretched exponential estimates for fluctuations of models within the Kardar-Parisi-Zhang universality class have played a particularly important role in mathematical work seeking to make physically motivated heuristic arguments about random growth models rigorous.
This talk will explain a simple coupling argument which recovers a moment generating function identity originally discovered by Rains using integrable probability techniques in the context of last-passage percolation. This coupling approach has proven to generalize significantly since it was first introduced about five years ago. The talk will then discuss how identities of this type can be used to derive the type of stretched exponential bounds mentioned above and when these bounds are sharp. The discussion will highlight some recent and in-progress improvements and extensions of older work.
Based on joint work with Elnur Emrah and Timo Seppäläinen.
Abstract: We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. Our approach relies on two new techniques. The first, called the PDE-ODI principle, converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This framework bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. The second combines a new leading mode condition combined with an "induction over thresholds" argument" to obtain even finer asymptotic estimates (This is joint work with Yi Lai).
Abstract: Given a closed negatively curved manifold, we consider the extent to which dynamical data associated to its closed geodesics (equivalently, periodic orbits of its geodesic flow) determines the underlying metric up to isometry. For instance, the lengths of closed geodesics, marked by their free homotopy classes, are conjectured to characterize the underlying metric up to isometry. In this talk, we consider a dynamically flavored variant of this marked length spectrum rigidity problem. We introduce the marked Poincare determinant, which associates to each free homotopy class of closed curves a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. Our main result is that near hyperbolic metrics in dimension 3, this invariant determines the metric up to homothety. This is joint work with Erchenko, Humbert, Lefeuvre, and Wilkinson.