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Abstract: The primary goal of the conference is to bring together experts from p-adic and complex geometry with a focus on Simpson's correspondence. In particular, the conference will combine expository/instructional talks with research talks. There will be two mini-courses on Ben Heuer's proof of the p-adic Simpson correspondence, a mini-course on the Simpson correspondence in characteristic p, a mini-course on the complex-analytic Simpson correspondence, and a few research talks.

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Abstract: In this talk, I will introduce the concepts of greedy and quasi-greedy approximation in Banach spaces and discuss transfinite refinements of these notions using Schreier families. Konyagin and Temlyakov (1999) characterized greedy bases as precisely those that are both unconditional and democratic. A basis is called quasi-greedy if, for every vector in the space, the sequence of m-term greedy approximants converges in norm to the vector itself. Considerable work has focused on distinguishing greedy bases from quasi-greedy ones. In recent joint work with Hung Chu, Thomas Schlumprecht, and András Zsák, we construct many new examples of quasi-greedy but non-greedy bases. I will describe some of these constructions and outline several natural open problems that arise from them.

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Abstract: Given a data distribution which is concentrated around a one-dimensional structure, can we infer that structure? We consider versions of this problem where the distribution resides in a metric space and the 1d structure is assumed to either be the range of an absolutely continuous curve, or is a connected set of finite 1d Hausdorff measure. In each of these cases, we relate the inference task to solving a variational problem where there is a tradeoff between data fidelity and simplicity of the inferred structure; the variational problems we consider are closely related to the so-called "principal curve" problem of Hastie and Steutzle as well as the "average-distance problem" of Buttazzo, Oudet, and Stepanov. For each of the variational problems under consideration, we establish: existence of minimizers, stability with respect to the data distribution, and consistency of a discretization scheme which is amenable to Lloyd-type numerical methods. Lastly, we consider applications to estimation of stochastic processes from partial observation, as well as the lineage tracing problem from mathematical biology.

This talk includes joint work with Anton Afanassiev, Young-Heon Kim, Forest Kobayashi, and Geoff Schiebinger.

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Abstract: Thurston revolutionized the study of diffeomorphisms of Riemann surfaces with a myriad of tools: hyperbolic geometry, geometric group theory, Teichmüller theory, and much more. One goal of this work is to see how much of his vision persists in higher dimensions. In this talk, I will tell a few stories about 4-manifolds and their mapping class groups, with an eye towards the Nielsen realization problem for certain algebraic surfaces. Some of the results I plan to share are joint work with Seraphina Lee and Tudur Lewis.

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Abstract: The “One Learning Algorithm Hypothesis” summarizes strong experimental evidence that the human brain processes visual, acoustic, and haptic signals, for perception and cognition with a workflow that corresponds to the same abstract architectural model. We present our most recent results on the development and performance evaluation of a universal machine learning architecture inspired by this hypothesis. The abstract architecture proposed is comprised of a multi-resolution processing front, followed by a feature extractor, followed by two “local” learning modules (first an unsupervised one, followed by a supervised one), followed by a deterministic annealing module. There are two global feedback loops, one to the multiresolution processor and one to the feature extractor. Innovative analytical methods and results include: multi-resolution hierarchy, use of Bregman divergences as dissimilarity measures, multi-scale stochastic approximation, multi-scale approximation to Bayes decision surfaces, optimization-information duality. We demonstrate the superior performance and characteristics of the resulting algorithms including: domain agnostic, on-line progressive learning, interpretability, robustness to noise and adversarial attacks, computable performance-complexity tradeoff. We present several applications in signal processing, graph problems, estimation and control.

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Abstract: The use of pre-exposure prophylaxis (PrEP), where approved antivirals are administered to uninfected high-risk individuals, is universally regarded as a promising strategy to prevent susceptible high-risk individuals from acquiring HIV infection from their infected partners. A number of antiviral drugs (and their combinations) have been developed and are being used as prophylaxis against the HIV epidemic here in the U.S. and globally. We will first present a risk-structured mathematical model for assessing the population-level impact of PrEP in an MSM (men who have sex with men) population. An extended model, which considers several high-risk populations, will also be presented and used to assess the potential spillover effect, where the administration of PrEP to individuals in one risk group induces a reduction of disease burden in other risk group(s). The central aim is to determine whether the use of the aforementioned strategies can aid the End the HIV Epidemic initiative, aimed at eliminating the disease in the U.S. by 2030.

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Abstract: The cylindrical singularities are prevalent but complicated in geometric flows. We discuss one of the simplest extrinsic flow, the mean curvature flow, and illustrate how the local dynamics of the singularities influence the singular set itself, and the geometry and topology of the flow. This talk is based on joint works with Zhihan Wang (Cornell) and Jinxin Xue (Tsinghua).

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Abstract: Continuous data assimilation tackles time-dependent problems with unknown initial conditions by integrating observational data into a nudging term. In this work, we focus on the heat
equation with spatially varying conductivity and Neumann boundary conditions, and propose
assimilation schemes that utilize non-interpolant observables, diverging from traditional approaches. These generalized nudging techniques are particularly effective for problems that
lack regularity beyond the minimal framework. We establish that the spatially discretized
nudged solution converges exponentially in time to the true solution, with a convergence
rate determined solely by the nudging strategy—independent of the discretization method.
Moreover, the long-term discrete error achieves optimality, aligning with known estimates for
problems with limited regularity and known initial conditions. We numerically investigate
three nudging strategies:
Nudging via a conforming finite element subspace;
Nudging using piecewise constant functions on the boundary mesh;
Nudging based on the mean value of the solution.
These strategies are tested on three benchmark problems, including one with Dirac delta
forcing and the Kellogg problem characterized by discontinuous conductivity.

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Abstract:  The classical dynamics of billiard flows on rational polygons has been
successfully studied by reduction to straight line flows on
translation surfaces. Nevertheless, the quantum counterpart of this
picture remains much less understood. We will discuss joint work with
Athreya and Forni studying quantizations of straight line flows on
translation surfaces and applications to some classical problems of
Furstenberg in ergodic theory.

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Abstract: Image-on-scalar regression has been a popular approach to modeling the association between brain activities and scalar characteristics in neuroimaging research. The associations could be heterogeneous across individuals in the population, as indicated by recent large-scale neuroimaging studies, e.g., the Adolescent Brain Cognitive Development (ABCD) study. The ABCD data can inform our understanding of heterogeneous associations and how to leverage the heterogeneity and tailor interventions to increase the number of youths who benefit. It is of great interest to identify subgroups of individuals from the population such that: 1) within each subgroup the brain activities have homogeneous associations with the clinical measures; 2) across subgroups the associations are heterogeneous; and 3) the group allocation depends on individual characteristics. We propose a latent subgroup image-on-scalar regression model (LASIR) to analyze large-scale, multi-site neuroimaging data with diverse sociodemographics. LASIR introduces the latent subgroup for each individual and group-specific, spatially varying effects, with an efficient stochastic expectation maximization algorithm for inferences. We demonstrate that LASIR outperforms existing alternatives for subgroup identification of brain activation patterns with functional magnetic resonance imaging data via comprehensive simulations and applications to the ABCD study. We further discuss how insights into subgroup-specific heterogeneity can inform the generalizability of findings in population neuroscience.

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Abstract: Mathematical biology is a vast field which has experienced rapid growth over the past 50 years. But what is mathematical biology? What questions can be (or have been) answered, and what are researchers investigating now? In this talk, we will present a brief survey of the field and then focus on a few recent student-led works in ecology, oncology, and infectious disease modeling.