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Abstract: Dieudonné theory is the study of families of group schemes via linear-algebraic data.  In this talk, I will begin by recalling some motivation for Dieudonné theory, with examples.  Then I will explain some new classification and smoothness results for certain close relatives of p-divisible groups known as n-smooth groups, which affirmatively answer conjectures of Drinfeld. This is joint work with Joshua Mundinger.

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Abstract: Networks of coupled dynamical systems are fundamental models across the sciences, from physics to neuroscience. Despite their success, the governing equations of such systems are often unknown, limiting our ability to predict and control their dynamics. In many applications, only time series data from the network is accessible, and learning the governing equations from data becomes an inverse problem. In this talk, inspired by compressive sensing techniques, I will show how learning network dynamics from data can be formulated as a convex optimization problem. By exploiting structural information encoded in the network dynamics, such as sparsity, statistical properties, and symmetries, we characterize the minimum amount of data required for learning the network dynamics exactly (and robustly). We illustrate these ideas using networks of coupled chaotic maps and oscillators.

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Abstract: Starting from the celebrated results of Eells and Sampson, a rich and flourishing literature has developed around equivariant harmonic maps from the universal cover of Riemann surfaces into negatively curved target spaces. In particular, such maps are known to be rigid, meaning that they are unique up to natural equivalence. This rigidity property fails when the target space has positive curvature, and comparatively little is known in this framework. In this talk, given a closed Riemann surface with strictly negative Euler characteristic, and a unitary representation of its fundamental group on a separable complex Hilbert space H which is weakly equivalent to the regular representation, we aim to discuss a lower bound on the Dirichlet energy of equivariant harmonic maps from the universal cover of the surface into the unit sphere S of H, and to give a complete classification of the cases in which the equality is achieved. As an interesting corollary, we obtain a lower bound on the area of equivariant minimal surfaces in S, and we determine all the representations for which there exists an equivariant, area-minimising minimal surface in S. The subject matter of this talk is a joint work with Antoine Song (Caltech) and Xingzhe Li (Cornell University).

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Abstract: Marron: We present a two-producer, one-consumer ecological stoichiometry model to investigate how differences in producer quality, nutrient enrichment, and space limitation interact to shape community dynamics. The model tracks the biomass and stoichiometry of a preferred, higher-quality producer and a less palatable, lower-quality producer alongside a single consumer, in a nutrient-closed system with dynamic producer nutrient:carbon ratios. Under a Holling Type II functional response, nutrient and space enrichment drive the system through a cascade of bifurcations distinct from those observed in classical two-species stoichiometric models, including an abrupt transition to a state in which both the preferred producer and the consumer are lost. Hysteresis analyses reveal that once this collapse occurs, no subsequent manipulation of space or nutrient availability can recover the coexistence state, underscoring the path-dependence of the system. These findings have implications for real-world systems undergoing climate-driven shifts in vegetation community composition, such as Arctic shrubification, where the encroachment of less palatable woody vegetation may push herbivore communities past critical thresholds from which recovery is not possible through environmental intervention alone.

Nicole: Visualizations of COVID-19 hospitalization data often struggle to synthesize temporal, geographic, and hospitalization information, all of which are critical for understanding pandemic impact and informing mitigation efforts. In this talk, I will demonstrate how Mapper, a tool from Topological Data Analysis, effectively captures these multidimensional relationships by representing data as a simplified simplicial complex. By applying the Mapper algorithm to COVID-19 hospitalization data from 2023-2024, I reveal how this topological approach identifies distinct geographical trends over time that traditional methods may overlook.

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Abstract: Two 20 minute talks. The first is from P Resnik

Title: Large Language Models are Biased Because They are Large Language Models

I argue that harmful biases are an inevitable consequence arising from the design of any large language model as LLMs are currently formulated, all the way down to their mathematical formulation. To the extent that this is true, it suggests that the problem of harmful bias cannot be properly addressed without a serious reconsideration of AI driven by LLMs, going back to the foundational assumptions underlying their design.
here are links to the relevant paper and to a short video. https://www.google.com/url?q=https://direct.mit.edu/coli/article/51/3/885/128621/Large-Language-Models-Are-Biased-Because-They-Are&sa=D&source=calendar&ust=1776713805261960&usg=AOvVaw2to4d8XjzLVNmr_IAi_3Ax

The second is from Maria Molina

Learning Without Labels: New Insights into Climate and Extremes

Abstract:
Climate variability and weather extremes pose profound challenges for prediction, preparedness, and resilience. Traditional approaches often rely on predefined indices or supervised learning methods, which can overlook unexpected patterns or reinforce biases inherent in labeled datasets. This seminar explores how unsupervised learning techniques can uncover hidden patterns in high-dimensional climate data. I will highlight recent innovations that adapt established methods to reveal properties not captured by conventional architectures, offering new perspectives on modes of variability and extreme events. For instance, a knowledge-guided autoencoder can disentangle distinct Pacific climate modes with differing spectral signatures, while a custom hyperparameter search can optimize self-organizing maps to produce smooth, interpretable pathways among weather regimes. Together, these advances help uncover processes and mechanisms that may underlie established climate and weather phenomena. Ultimately, unsupervised learning provides a powerful lens for scientific discovery, with implications for understanding, prediction, and decision-making in a changing climate.

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Abstract: Recall that a subset X of a group is said to be an approximate group if the product set XX:=\{ab:a,b\in X\} can be covered by finitely many translates of X. I will discuss a work-in-progress which attempts to give a structure theory for approximate groups definable in NIP theories. First I will present an example of an approximate group X definable in the universal cover of SL(2,R) such that has no type-definable subgroup of bounded index; this gives a NIP counterexample to a conjecture of Massicot-Wagner, first disproved by Hrushovski, Krupiński, and Pillay. I will then discuss some tentative positive structural results, some of which are of interest even in the case of definable groups. Among other things I will discuss Hrushovski’s “generalized locally compact models” in the NIP setting, and give a new proof of Gismatullin’s result on the existence of G^{000} for NIP groups G.

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Abstract: Modeling three-dimensional magnetohydrostatic equilibria in stellarators is of paramount importance in the design and conduction of fusion experiments. The workhorse methods of the community at this point (VMEC and DESC) simplify the search for equilibria to configurations with nested flux surfaces. It is known, however, that relevant configurations in experiments feature magnetic islands and chaotic regions, violating this assumption. Magnetic relaxation codes provide an opportunity to solve for more general equilibrium solutions.
We introduce a structure-preserving magnetic-relaxation solver that does not require a nested flux surface assumption. It combines several crucial features for the first time.
(i) Through structure-preserving mixed finite-elements, we retain central features of the relaxation problem in the discrete setting, such as divergence-free magnetic fields to machine precision, guaranteed energy-dissipation and helicity preservation.
(ii) The use of B-Spline bases and tools from the isogeometric analysis field allows us to use high-order basis functions in strongly shaped geometries.
(iii) The code is pure Python using the JAX ecosystem, making it very easy to install, run, and extend as well as performant on GPUs. It is also fully differentiable for future inverse design and optimization applications.
The proposed magnetic relaxation solver is tested in several stellarator geometries at low and high plasma beta-values. Future work will address the integration of this code for 3D equilibrium optimization for modeling magnetic islands and chaos in stellarator fusion devices.

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Abstract: Igusa stacks are $p$-adic geometric objects, recently introduced byMingjia Zhang, that roughly parametrize ways to $p$-adically
uniformize (global) Shimura varieties by local Shimura varieties. In
joint work with Patrick Daniels, Pol van Hoften, and Mingjia Zhang, we
construct Igusa stacks for all abelian type Shimura data and use them
to the study of $\ell$-adic cohomology of Shimura varieties. I will
discuss the geometric ingredients that go into the construction as
well as how it naturally fits into Fargues--Scholze's framework of
categorical local Langlands.

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Abstract: Loewner Dynamics is a tool used in many areas of research such as the study of growth processes in the Complex Plane, Schramm-Loewner Evolutions (SLE) and more. I will describe a newly introduced toolbox that studies the Loewner Dynamics via resolvents of matrices. The method works for both deterministic and random matrices. In the random case, one of the first applications of this method, allows us to connect two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory (RMT). This machinery opens new avenues of research that allow the use of techniques from one field to another. In addition, the regular dynamics in the matrix space provides an alternative to the singular driver dynamics of SLE theory. I will describe the first application of our technique, and a short overview of interactions of our method with other areas of active research, which I aim to explore in a sequence of separate recent projects with collaborators from Europe, Asia and North America. More at https://margarintvlad.com/

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Abstract: Vaughan Jones proved in the 1980s that the index of a subfactor of a von Neumann factor has to be in the set $\{ 4\cos^2 \pi/n,\ n=3,4,\ ...\ \}\cup [4,\infty]$. This turned out to be the tip of the iceberg of an incredibly rich structure. It has been realized for some time  that the classification of subfactors runs parallel to the classification of unitary tensor categories and their module categories. We review the current state of classification. This will include recent results about classification of module categories of an important class of fusion categories, closely related to Drinfeld-Jimbo quantum groups and loop groups.