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Abstract: The HRT conjecture predicts that finite Gabor systems generated by time-frequency shifts of a nonzero function are linearly independent. I will discuss a recent result by Oussa (Trichotomy for the HRT Conjecture for mixed integer configuration (August 2025)) proving the conjecture for mixed integer configurations - that is, sets of lattice points together with one off-lattice point - for Schwartz class functions.

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Abstract: Since Hironaka's famous resolution of singularities in characteristics zero in 1964, it took about 40 years of intensive work of many mathematicians to simplify the method, describe it using conceptual tools and establish its functoriality. However, one point remained quite mysterious: despite different descriptions of the basic resolution algorithm, it was essentially unique. Was it a necessity or a drawback of the fact that all subsequent methods relied on Hironaka's ideas essentially?

The situation changed in the last decade, when a logarithmic, a weighted and a foliated analogues and generalizations were discovered in works of Abramovich-Temkin-Wlodarzcyk, McQuillan, Quek, Abramovich-Temkin-Wlodarzcyk-Belotto and others. At this stage we can already try to figure out general ideas and principles shared by all these methods and the picture is quite surprising -- it seems that each method is quite determined by its basic setting consisting of the class of geometric objects and basic blowings up one works with. In particular, the classical method is probably the only natural resolution (via principalization) method obtained by blowing up smooth centers in the ambient manifold.

In my talk I'll describe the settings and the methods on a very general level. If time permits, I will add some details about the simplest dream (or weighted) method, which has no memory and improves the singularity invariant by each weighted blowing up. Thus, the algorithm becomes simplest possible and the (modest) price one has to pay consists of extending the setting of varieties (or schemes) and blowings up along smooth centers to the setting of orbifolds and blowings up weighted centers.

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Abstract: Can we find a predictable, polynomial improvement of a scientific machine learning system‘s performance, measured by its test error, as it scales with increases in model(surrogate) size, dataset(collocation) size, and computational resources? In this talk, I’ll talk about why we would expect an optimal scaling law and how to achieve a numerical scaling law. We first present several information optimally results of scaling scientific machine learning and demonstrate the hardness of scaling that hides in optimization. This inspires us to design an optimizer whose hardness is independent of the neural network. Besides, we also show another scaling law via allocating computation at inference time. We build a two-scale Monte Carlo algorithm to debias the ML at inference time that dynamically refines and debiases the SCiML predictions during inference by enforcing the physical laws. We also show how this methodology can be used for diffusion model and language model inference time scaling.

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Abstract: Last-mile delivery is a critical component of logistics networks, accounting for approximately 30%–35% of costs. As delivery volumes have increased, truck route times have become unsustainably long. To address this issue, many logistics companies, including FedEx and UPS, have resorted to using a “driver aide” to assist with deliveries. The aide can assist the driver in two ways. As a “jumper,” the aide works with the driver in preparing and delivering packages, thus reducing the service time at a given stop. As a “helper,” the aide can independently work at a location delivering packages, and the driver can leave to deliver packages at other locations and then return. Given a set of delivery locations, travel times, service times, jumper’s savings, and helper’s service times, the goal is to determine both the delivery route and the most effective way to use the aide (e.g., sometimes as a jumper and sometimes as a helper) to minimize the total routing time. We model this problem as an integer program with an exponential number of variables and an exponential number of constraints and propose a branch-cut-and-price approach for solving it. Our computational experiments are based on simulated instances built on real-world data provided by an industrial partner and a data set released by Amazon. The instances based on the Amazon data set show that this novel operation can lead to, on average, a 35.8% reduction in routing time and 22.0% in cost savings. More importantly, our results characterize the conditions under which this novel operation mode can lead to significant savings in terms of both the routing time and cost. Our computational results show that the driver aide with both jumper and helper modes is most effective when there are denser service regions and when the truck’s speed is higher (≥10 miles per hour). Coupled with an economic analysis, we come up with rules of thumb (that have close to 100% accuracy) to predict whether to use the aide and in which mode. Empirically, we find that the service delivery routes with greater than 50% of the time devoted to delivery (as opposed to driving) are the ones that provide the greatest benefit. These routes are characterized by a high density of delivery locations. https://pubsonline.informs.org/doi/epdf/10.1287/msom.2022.0211

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Abstract: Coding theory has been both a source of intriguing problems in distance geometry and combinatorics and a beneficiary of the methods developed in these areas. Recent developments at the intersection of the three areas have highlighted several new, rapidly developing directions in each of them. The evolution of quantum coding has shed a new light on classical families of error- correcting codes such as Reed-Muller codes and their extensions, and has led to intriguing developments in the construction and applications of expander graphs and their higher-dimensional generalizations. There are interesting connections between finite point configurations, extremal combinatorics and questions arising from combinatorics of root systems, which again can be viewed in the language of codes. The workshop will bring together senior and junior participants who work in these areas, to share their ideas and current research, and advance the understanding of the interplay between distance geometry, combinatorics, and coding theory.

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Abstract: Emerging infectious diseases can drive severe host population declines, yet recovery outcomes vary widely across species and communities. To examine the roles of host defense strategies in shaping these dynamics, we developed a mechanistic eco-evolutionary epidemic model that explicitly incorporates an environmental pathogen reservoir and considered heritable variation of disease tolerance in host. In this single-species system, without costs of tolerance, the coexistence of multiple defense classes is possible only when the pathogen ultimately goes extinct, whereas persistent infection selects for maximal tolerance. Introducing a reproductive cost removes such disease-free coexistence and allows for endemic equilibria with multiple strategies. Motivated by discrepancies between these analytical predictions and real-world observations, we extend the model to multi-species communities with habitat overlap and overlap-dependent competition. This reveals that resistant hosts suffer larger population declines and slower recovery under greater overlap, although communities dominated by resistant species rebound faster and more evenly because tolerant hosts promote pathogen persistence. Competition linked to habitat overlap reduces species-level declines, but increases unevenness in community recovery. Our results, supported by classical reproduction number analyses yet driven by transient dynamics, demonstrate how host defense strategies, habitat structure, and inter-specific interactions jointly determine post-outbreak resilience across biological scales.

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Abstract: In a cryo-electron microscopy experiment, a biomolecular sample is prepared in solution, flash frozen, and imaged. In a processed dataset, each resulting image captures an individual biomolecule in a particular conformation. These images can be averaged to reconstruct the 3D structure of the biomolecule at atomic resolution. With recent advances in microscopes and algorithmic development, the cryo-EM community has focused on the much more ambitious task of "heterogeneity analysis": estimating the probabability distribution of the biomolecule from the images. The "heterogeneity" problem has become extremely popular, with many classical numerical methods to new machine learning approaches. Here, we focus on two aspects of the problem: our work on (1) illuminating a statistical fallacy common in heterogeneity analysis, and (2) a non-parametric maximum likelihood approach which can be used to post-process many existing heterogeneity methods. We highlight theoretical guarantees and numerical simplicity of our approach, and outline the many future opportunities of numerical methods for this inverse problem.

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Abstract: Shimura Varieties are of fundamental importance in number theory, by virtue of the fact that their cohomology provides the only known geometric incarnations of the global Langlands correspondence over number fields. When the Shimura Variety is non-compact, it is often very desirable to work with the intersection cohomology of its minimal compactification, as this can be related to L^2-cohomology of the associated symmetric space, and the latter is computed in terms of the (g,K)-cohomology of the discrete spectrum of the space of automorphic forms via the work of Borel-Cassleman. In this talk, we investigate the relationship of intersection cohomology with a structure coming from the incarnation of the Shimura variety as a p-adic manifold. Namely, we study how the intersection cohomology interacts with Mantovan’s filtration, which generalizes the filtration on the cohomology of the modular curve coming from excision with respect to the ordinary and supersingular locus. We accomplish this by describing the intersection cohomology as a certain Hecke operator applied to a sheaf on Bun_{G}, the moduli stack of G-bundles of the Fargues-Fontaine curve and then explicitly describing the stalks in terms of an intermediate extension from Igusa varieties to their affine partial minimal compactifications.  The aforementioned filtration can then be understood via excision applied to the Harder-Narasimhan stratification of this sheaf on Bun_{G}. This leads to the hope that Borel-Cassleman’s description of the L^{2}-cohomology can be refined to a description of our sheaf in terms of direct sums of certain “sheared” Hecke eigensheaves on Bun_{G}, where the shearing encodes the known relationship between the (g,K)-cohomology and the Arthur SL_{2} of the A-parameters attached to the discrete spectrum via the work of Adams-Johnson and Vogan-Zuckerman. This is joint work in progress with Ana Caraiani and Mingjia Zhang.

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Abstract:  In the aperiodic tilings proposed by Penrose in 1974, the
ratio of tile frequencies is equal to the golden ratio. Several
aperiodic tilings discovered since then, such as Ammann's tilings
(1980s) or Jeandel-Rao's tilings (2015), are also associated with the
golden ratio. This is also the case for the aperiodic monotile called
"hat" discovered in 2023, whose ratio of frequencies of the two
orientations of the monotile is well-defined and is equal to the 4th
power of the golden ratio. In this presentation, we will explain how
one-dimensional symbolic dynamics can be used to understand and obtain
new results in the theory of aperiodic tilings. We also propose a new
family of aperiodic tilings that goes beyond the golden ratio. This is
associated with the metallic means, that is, the positive roots of the
polynomials $x^2-nx-1$ for any positive integer $n$.

The first part will be a general introduction to the topics. In the
second part, we will focus on the metallic mean Wang tiles introduced
recently in https://doi.org/10.1017/fms.2025.10069 and
https://doi.org/10.1017/fms.2025.10098

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Abstract: We introduce a model for 1d wave propagation undergoing strong self-interaction at locations that have been randomly sprinkled throughout the surrounding medium. We then consider the problems of well-posedness and homogenization for this model. This is joint work in part with Rowan Killip and Monica Vișan and in part with Maria Ntekoume.

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Abstract: A periodic function on the real line can be decomposed as a sum of trigonometric waves. But suppose we care about the two-dimensional repeating patterns we see on wallpaper and textiles? With each symmetry type, there is a specific collection of functions that build up its patterns. In this talk, we shall derive these functions.