Abstract: Counting problems are among the oldest in algebraic geometry, and many techniques and tools, such as intersection theory, Schubert calculus, moduli theory etc. were developed to find answers to these questions. In the past few decades, the introduction of new ideas from physics, representation theory, derived categories, derived algebraic geometry and logarithmic geometry among others have revolutionized the field of enumerative geometry. The aim of this workshop is to bring together experts to discuss new and exciting developments in enumerative geometry, Gromov-Witten theory, Donaldson-Thomas theory, and related areas.
Abstract: Digital twins/models (DTs) are designed to be replicas of systems and processes. At the core of a digital twin (DT) are physical/mathematical models that capture the behavior of the real system across temporal and spatial scales. One of the key roles of DTs is enabling “what if” scenario testing of hypothetical simulations to understand the implications at any point throughout the life cycle of the process, to monitor the process, to calibrate parameters to match the actual process and to quantify the uncertainties. In this talk, we will present various real-time Scientific Deep Learning (SciDL) approaches for forward, inverse/calibration, and UQ problems. Both theoretical and numerical results for various problems including transport, heat, Burgers, (transonic and hypersonic) Euler, and Navier-Stokes equations will be presented.
Abstract: The risk and intensity of mosquito-borne disease outbreaks are tightly linked to the frequency at which mosquitoes feed on blood, also known as the biting rate. Standard mosquito-borne disease transmission models assume that mosquitoes bite only once per reproductive cycle – an assumption commonly violated in nature. For example, host defensive behaviors or climate factors can increase the occurrence of multiple biting while simultaneously impacting the mosquito gonotrophic cycle duration (GCD), the quantity customarily used to determine biting rates.
We present a framework for incorporating complex mosquito biting behaviors into transmission models, to account for the heterogeneity in and linkages between the biting rate and the multiple biting number. We derive general formulas for the basic offspring number, N0, and basic reproduction number, R0, and introduce specific models arising from empirical, phenomenological, and mechanistic perspectives. Using the gonotrophic cycle duration as a standard quantity to compare these models, we show how assumptions about the biting process strongly affect the relationship between the GCD and R0. This work highlights the importance of behavioral dynamics on mosquito-borne disease transmission while providing a tool for evaluating how individual-level interventions against biting scale up to affect population-level disease risk.
Abstract: Digital Twins (DTs) are adaptive, real-time virtual replicas of physical systems that integrate physics-based models, sensor data, and intelligent decision-making. At their core, DTs can be rigorously framed within PDE–constrained optimization (PDECO).
This talk develops a unified PDECO framework for state estimation and control, employing adjoint-based methods in both deterministic and stochastic settings. To meet the challenges of infinite-dimensional, large-scale optimization, we introduce novel trust-region and augmented Lagrangian algorithms formulated in function spaces.
Beyond these advances, we discuss connections between PDECO and modern machine learning, including how score-based generative models can be interpreted as backward-in-time PDEs. This perspective illustrates how physics-informed modeling and data-driven synthesis can complement one another.
Applications span structural and biomedical systems—from bridges and dams to aneurysm modeling, optimal insulation, electromagnetic cloaking, light bending, and neuromorphic computing—illustrating a pathway toward predictive, adaptive, and trustworthy Digital Twins.
Abstract: We will discuss the long-time dynamics of the derivative nonlinear Schr\"odinger equation. For small, localized initial data, where no solitons arise, we prove dispersive estimates globally in time. Under the same assumptions, we further prove modified scattering and asymptotic completeness. To the best of our knowledge, this is the first result to achieve an asymptotic completeness theory in a quasilinear setting. Our approach combines the method of testing by wave packets of Ifrim and Tataru, a bootstrap argument, and the Klainerman–Sobolev vector field method.
Abstract: Evaluating and validating the performance of prediction models is a crucial task in statistics, machine learning, and their diverse applications, including precision medicine. However, developing robust prediction performance measures, particularly for time-to- event data, poses unique challenges. In this talk, I will highlight how conventional performance metrics for time-to-event data, such as the C Index, Brier Score, and time- dependent AUC, may yield unexpected results when comparing prediction models/algorithms. I will then introduce a novel time-dependent pseudo R-squared measure and demonstrate its utility as a prediction performance measure for both uncensored and right-censored time-to-event data. Additionally, I will discuss its extension to time-dependent prediction performance measures and to competing risks scenarios. Its effectiveness will be showcased through simulations and real-world examples.
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